The Complete Classification of Solutions to the Riemann Problem of the Defocusing Complex Modified KdV Equation

Abstract

The complete classification of solutions to the defocusing complex modified KdV equation with step-like initial condition is studied by the finite-gap integration approach and Whitham modulation theory. All kinds of combination solutions consisting of genus-0 regions, genus-1 regions, or genus-2 regions are found by classifying the Riemann invariants. The behaviors of wave breaking in Riemann problem of the defocusing complex modified KdV equation are much richer and more complicated than those in the nonlinear Schrödinger equation. It is demonstrated that a large oscillating region can be composed of four basic genus-1 dispersive shock waves, a case of solution may be consisted of up to six regions, and the plateau, vacuum, rarefaction wave, and dispersive shock wave can coexist in the same solution region. Moreover, the genus-2 region, produced from the collision of two dispersive shock waves, is described detailedly by the genus-2 Whitham equations. The direct numerical simulations on the defocusing complex modified KdV equation show remarkable agreement with the results from Whitham modulation theory.

1 Introduction

The Whitham theory was first formulated by G.B. Whitham in his seminal publication (Whitham 1965) in which he gave the Whitham modulation equations based on the averaged conservation laws to describe some physical phenomena such as undular bore in water and formed the basis of impressive development of dispersive hydrodynamics. The first application of Whitham theory to Korteweg–de Vries (KdV) equation was achieved by Gurevich and Pitaevskii (1974) who studied the self-similar solutions for dispersive shock wave (DSW), called collisionless shock, whose evolution can be described by the diagonal Whitham equation. One of its edge appears to be a soliton wave, the harmonic wave for its opposite. The simplest expanding oscillating structure described by a Jacobian elliptic function was also obtained in Gurevich and Pitaevskii (1974) with a step-like initial jump known as Riemann problem. The analytical description of DSW that transformed the Whitham equation to Euler–Poisson–Darboux equation for the nonlinear Schrödinger equation (NLS) has been presented in Tian and Ye (1999).

The Riemann problem of the evolution waves has been discussed in various important physical fields. In photon fluid, all the possible wave patterns propagating in the normal fiber has been discussed with account of steepening effects (Ivanov and Kamchatnov 2017). Ivanov et al. (2017) gives the classification of possible flows in two-component Bose–Einstein condensate and the solutions of Riemann problem for Gardner equation (related to modified KdV equation) are completely classified in Kamchatnov et al. (2012) which appears some new structures and more complicated cases compared to the KdV case. Indeed, this can also be found in the case of defocusing complex modified KdV (cmKdV) equation with special step-like initial data (Kodama et al. 2008; Kong et al. 2019). However, the studies on general step-like initial problem of the defocusing cmKdV equation are even more complicated.

Except the pseudo-phase method introduced by Whitham himself, there are several way to average the original equation to get the Whitham equations. For example, Luke (1966) used a perturbation procedure to investigate the nonlinear wave problem, which could recover the Whitham equations of slow variations. Flaschka et al. (1980) extended the finite-gap integration theory to study the multiphase averaging of integrable system of KdV type. Dubrovin and Novikov (1989) proposed a procedure for averaging the local Poisson brackets to derive the Whitham equations. Lax and Levermore (1983) opened another way to describe the DSW rigorously by utilizing the method of inverse scattering transform and Whitham modulation theory. Moreover, the combinations of Whitham modulation theory with numerical techniques have been studied by Grava and Klein (2007) and Ablowitz et al. (2016).

This paper focuses on the complete classification of Riemann problem for the defocusing cmKdV equation with small dispersion

$$\begin{aligned} q_t-6|q|^2q_x+\varepsilon ^2q_{xxx}=0, \end{aligned}$$

(1)

where $q=q(x,t)$ represents the complex wave envelope and $\varepsilon \ll 1$ is a small modulation scale. This equation is analyzed by means of Whitham modulation theory, in which the corresponding Whitham equations are neither strictly hyperbolic nor genuinely elliptic systems (Kodama et al. 2008) compared with the defocusing and focusing NLS equations. Self-similar solutions in such kind of systems have be investigated and discussed in KdV hierarchy (Pierce and Tian 2007), mKdV (Kamchatnov et al. 2012), Landau–Lipshitz equation (Ivanov et al. 2017), Camassa-Holm equation (Abenda and Grava 2005), etc. However, it is found in this work that the solutions in the defocusing cmKdV equation are much richer such as an oscillating shock wave region may be composed of four basic shock wave structures and a case of solution can be consisted of up to six regions, etc. In addition, the whole solutions we have classified are even more than 50 categories, which has never been found before.

The Madelung transformation

$$\begin{aligned} q(x,t)=\sqrt{\rho } e^{i \phi / \varepsilon },\quad \phi _x=v, \end{aligned}$$

(2)

where $\rho$ and v, analogs of density and velocity of the hydrodynamics, are all real functions, maps the defocusing cmKdV equation (1) to the dispersive hydrodynamics-like system

$$\begin{aligned} \left\{ \begin{array}{lr} \rho _t-(3\rho ^2+3v^2\rho )_x=-\varepsilon ^2(4\rho ^{3/4}(\rho ^{1/4})_{xx})_x,\\ v_t-(6\rho v+v^3)_x=-\varepsilon ^2(3v\rho _{xx}/2\rho -3v(\rho _x)^2/4\rho ^2+3v_x\rho _x /2\rho +v_{xx})_x, \end{array} \right. \end{aligned}$$

(3)

which suffice to give the solutions as $\varepsilon \rightarrow 0$ until it develops a shock formed at once when multi-value region appears. After the moment when multi-value region appears, this limit is converted into the Whitham equations in the diagonal Riemann form Grava and Klein (2007), Ablowitz et al. (2016), Pierce and Tian (2007), Abenda and Grava (2005), Hoefer (2014), Ivanov and Kamchatnov (2017), Ablowitz et al. (2020), Bridges and Ratliff (2021) and Congy et al. (2019)

$$\begin{aligned} \frac{\partial \lambda _i}{\partial t}+v_{i}(\lambda _1, \lambda _2,\ldots , \lambda _{2N+2})\frac{\partial \lambda _i}{\partial x}=0,~~i=1,2,\ldots ,2N+2, \end{aligned}$$

(4)

where $v_{i}$ are called the Whitham velocities, $\lambda _i$ are Riemann invariants and N represents the number of phases in the oscillations. The boundaries connecting $N=0$ and $N=1$ regions including in Whitham equations (4) are exactly the same with the diagonal Riemann form of dispersionless limit of hydrodynamics-like system (3), which will be explained below. Here we concentrate only on the case of $0 \le N\leqslant 2$, while the case $N>2$ will be discussed in the future work.

This paper is constructed as follows. In Sect. 2, the zero-phase, one-phase and two-phase periodic solutions and the corresponding Whitham equations are derived by employing the finite-gap integration approach. In Sect. 3, five types of basic rarefaction wave structures and ten types of basic dispersive shock wave structures are proposed by considering the self-similar solutions of the Whitham equations. The complete classification of solutions to the Riemann problem of the defocusing cmKdV equation (1) is investigated analytically and numerically in Sect. 4. We conclude this work in Sect. 5.

2 Finite-Gap Periodic Solutions and Whitham Equations

In this section, the finite-gap periodic solutions and Whitham equations for the defocusing cmKdV equation (1) are derived by the Flaschka–Forest–McLaughlin (FFM) approach (Flaschka et al. 1980; Kamchatnov 1994, 1997) to describe its evolutions of initial discontinuities in Riemann problem. For our purpose, this section only focuses on the zero-phase, one-phase, and two-phase solutions in view of the form of the step-like initial data considered in this work.

It is known that the defocusing cmKdV equation (1) is the second flow in the defocusing NLS hierarchy, which has Lax pair of the form

$$\begin{aligned} \varepsilon \begin{pmatrix} \psi _1\\ \psi _2 \end{pmatrix}_x =\begin{pmatrix} F&{}G \\ H &{}-F \end{pmatrix} \begin{pmatrix} \psi _1\\ \psi _2 \end{pmatrix}, \quad \varepsilon \begin{pmatrix} \psi _1\\ \psi _2 \end{pmatrix}_t =\begin{pmatrix} A&{}B\\ C&{}-A \end{pmatrix} \begin{pmatrix} \psi _1\\ \psi _2 \end{pmatrix}, \end{aligned}$$

(5)

where the entries of the matrices above are

$$\begin{aligned}&F=-i\lambda ,\quad G=q,\quad H=q^*,\quad A=-4i\lambda ^3-2i\lambda |q|^2-\varepsilon q q_x^*+\varepsilon q^* q_x,\nonumber \\&B=4\lambda ^2 q+2i\lambda \varepsilon q_x+2q|q|^2-\varepsilon ^2q_{xx}, \quad C=4\lambda ^2q^*-2i\lambda \varepsilon q_x^*+2q^*|q|^2-\varepsilon ^2q^*_{xx}. \end{aligned}$$

(6)

The linear systems (5) and (6) have two independent basic solutions $(\psi _1,\psi _2)$ and $(\phi _1,\phi _2)$, which can be used to define the “squared” eigenfunctions as follows

$$\begin{aligned} f=-\frac{i}{2}(\psi _1\phi _2+\psi _2\phi _1),\quad g=\psi _1\phi _1, \quad h=-\psi _2\phi _2, \end{aligned}$$

(7)

which dates back to the work of Its and Kotlyarov Kotlyarov (1976), Its and Kotlyarov (1976). Obviously, the “squared” eigenfunctions f, g and h satisfy the following linear systems

$$\begin{aligned} \varepsilon f_x=-iHg+iGh,\quad \varepsilon g_x=2iGf+2Fg, \quad \varepsilon h_x=-2iHf-2Fh, \end{aligned}$$

(8)

and

$$\begin{aligned} \varepsilon f_t=-iCg+iBh,\quad \varepsilon g_t=2iBf+2Ag, \quad \varepsilon h_t=-2iCf-2Ah. \end{aligned}$$

(9)

In fact, the linear systems (8) and (9) compose a three-order Lax pair of the defocusing cmKdV equation (1). Further, it is convenient to prove that the quantity $f^2-gh=(-1/4)(\psi _1\phi _2-\psi _2\phi _1)^2$ is independent of x and t, and is only dependent on the spectral parameter $\lambda$, which can be denoted by

$$\begin{aligned} f^2-gh=P(\lambda ), \end{aligned}$$

(10)

where $P(\lambda )$ is polynomial of parameter $\lambda$.

One merit of the linear systems (8) and (9) is that they can be used to derive the conservation laws of the defocusing cmKdV equation (1). Indeed, the second equations in both equation (8) and (9) can be rewritten as

$$\begin{aligned} \varepsilon (\mathrm{log}(g))_x=\varepsilon \frac{g_x}{g} =2if\frac{G}{g}+2F,\quad \varepsilon (\mathrm{log}(g))_t =\varepsilon \frac{g_t}{g}=2if\frac{B}{g}+2A. \end{aligned}$$

(11)

The compatibility condition of these two equations indicates that

$$\begin{aligned} \frac{\partial }{\partial t}\left( 2if\frac{G}{g}+2F\right) =\frac{\partial }{\partial x}\left( 2if\frac{B}{g}+2A\right) , \end{aligned}$$

(12)

which can be simplified to

$$\begin{aligned} \frac{\partial }{\partial t}\left( \frac{G}{g}\right) -\frac{\partial }{\partial x}\left( \frac{B}{g}\right) =0. \end{aligned}$$

(13)

This is just the conservation law of the defocusing cmKdV equation (1) in term of the “squared” eigenfunction g.

For the AKNS system like the defocusing cmKdV equation (1), it is convenient to expand f, g and h to finite-order polynomials in $\lambda$

$$\begin{aligned} f(x,t,\lambda )=\sum _{j=0}^{N+1}f_j(x,t)\lambda ^{j}, \quad g(x,t,\lambda )=\sum _{j=0}^{N+1}g_j(x,t)\lambda ^{j}, \quad h(x,t,\lambda )=\sum _{j=0}^{N+1}h_j(x,t)\lambda ^{j}. \end{aligned}$$

(14)

The second and third equations in Eq. (8) show that both g and h must be of order N in $\lambda$ (i.e., $g_{N+1}=h_{N+1}=0$), thus the first equation in Eq. (8) indicates that the coefficient $f_{N+1}$ of $\lambda ^{N+1}$ is a constant. Without any loss of generality, one can set $f_{N+1}=1$. In order to derive the N-phase solution, assume

$$\begin{aligned} g(x,t,\lambda )=q \prod _{j=1}^{N}(\lambda -\mu _j(x,t)), \end{aligned}$$

(15)

where $\mu _j=\mu _j(x,t)$ is called auxiliary spectrum, N is the genus of the hyperelliptic curve

$$\begin{aligned} w^2=P(\lambda ). \end{aligned}$$

(16)

Plugging Eq. (15) into Eqs. (8) and (9) and letting $\lambda =\mu _k(x,t)$ $(k=1,2,\ldots , N)$, yield Dubrovin-type equation for $\mu _k(x,t)$ as

$$\begin{aligned}&\epsilon \frac{\partial \mu _k}{\partial x} =-\frac{2i\tilde{G}\sqrt{P(\mu _k)}}{\prod _{j\ne k}(\mu _k-\mu _j)},\nonumber \\&\epsilon \frac{\partial \mu _k}{\partial t} =-\frac{2i\tilde{B}\sqrt{P(\mu _k)}}{\prod _{j\ne k} (\mu _k-\mu _j)},\quad j=1,2,\ldots , N, \end{aligned}$$

(17)

where $\tilde{G}(\mu _k)=G(\mu _k)/q=1, \tilde{B}(\mu _k) =B(\mu _k)/q=4\mu _k^2+2i\varepsilon \mu _k (\mathrm{ln}q)_x +2|q|^2-\varepsilon ^2q_{xx}/q.$

Substituting Eq. (14) with Eq. (15) into the second equation in Eqs. (8) and (9), respectively, yields

$$\begin{aligned} \epsilon q_x&=2iq\left( f_N+\sum _{j=1}^N \mu _j\right) ,\nonumber \\ \epsilon q_t&=2i[4qf_{N-2}+2i\varepsilon q_{x} f_{N-1} +f_N(2q|q|^2-\varepsilon ^2q_{xx})]\nonumber \\&\quad +2\left[ q(\varepsilon q^* q_x-\varepsilon q q_x^*) +4iq \sum _{i<j<k}\mu _i\mu _j\mu _k+2i|q|^2\sum _{j=1}^N \mu _j\right] . \end{aligned}$$

(18)

An algebro-geometric representation of q(x, t) in Eq. (18) can be developed by integrating the Dubrovin-type equation (17) for $\mu _k$ $(k=1,2,\ldots , N)$ with the aid of the Abel transform, which leads to the expressions for $\mu _k$ and q(x, t) in terms of Riemann theta functions depending on phase variables

$$\begin{aligned} \theta _j=\kappa _jx+\omega _jt+\theta _{0j},\quad j=1,2,\ldots , N, \end{aligned}$$

(19)

where $\kappa _j$ and $\omega _j$ are determined by integrating over certain cycles on the Riemann surface of the hyperelliptic curve (16), and $\theta _{0j}$ are constants.

In the framework of Whitham theory, it is vital to derive the Whitham equations of Riemann invariants, which are the zero points $\lambda _i$ $(i=1,2,\ldots , 2N+2)$ of the polynomial $P(\lambda )$ in Eq. (10). The FFM approach (Flaschka et al. 1980) for studying Whitham equations is based on the finite-gap integration theory, which is an important extension of the inverse scattering transform to the problems of periodic boundary conditions (Belokolos et al. 1994). The construction of multiphase averaging in FFM way is further extended by Kamchatnov (1994, 1997) without use of algebro-geometric tools like FFM approach. We now outline the basic procedure for deriving the Whitham equations of the defocusing cmKdV equation (1).

Firstly, normalizing the equation $f^2-gh=P(\lambda )$ according to the transformation $f\rightarrow f/\sqrt{P(\lambda )},$ $g\rightarrow g/\sqrt{P(\lambda )}$ and $h\rightarrow h/\sqrt{P(\lambda )}$ yields

$$\begin{aligned} \left( f/\sqrt{P(\lambda )}\right) ^2-\left( g/\sqrt{P(\lambda )}\right) \left( h/\sqrt{P(\lambda )}\right) =1. \end{aligned}$$

Under the same transformation, the conservation law (13) becomes

$$\begin{aligned} \frac{\partial }{\partial t}\left( \frac{G\sqrt{P(\lambda )}}{g}\right) -\frac{\partial }{\partial x}\left( \frac{B\sqrt{P(\lambda )}}{g}\right) =0. \end{aligned}$$

(20)

Secondly, assume the function $Q=Q(q(x,t))$ to be either a flux or a density in the conservation law and define the average of function Q in the form of

$$\begin{aligned} \langle Q\rangle =\lim _{L\rightarrow \infty }\frac{1}{2L}\int _{-L}^{L}Q(q(x,t))\hbox {d}x. \end{aligned}$$

(21)

In order to describe the modulated waves, two scales should be introduced: a fast scale (x, t) and a slow scale $(X=\epsilon x, T=\epsilon t)$ with $\epsilon$ small. As done in FFM approach (Flaschka et al. 1980), the phase parameters $\kappa _j$ and $\omega _j$ depend only on the slow variables (X, T), but not on the fast variables (x, t). Moreover, the Riemann invariants $\lambda _i$ $(i=1,2,\ldots , 2N+2)$ also depend only on the slow variables (X, T). However, during the averaging procedure, the slow variables (X, T) are frozen. The integral (21) over the spatial variable x can be replaced by the integral over the N-torus as parameterized by the phases $\theta _j$ $(j=1,2\ldots , N)$ variables provided the spatial wave numbers $\kappa _j$ are incommensurate (see Eq. (19)). Thus the integral (21) is written as

$$\begin{aligned} \langle Q\rangle =\frac{1}{(2\pi )^{m}}\int _{0}^{2\pi }\cdots \int _{0}^{2\pi } Q(q(\theta _1,\theta _2,\ldots ,\theta _N))\hbox {d}\theta _1\hbox {d}\theta _2\ldots \hbox {d}\theta _N, \end{aligned}$$

(22)

which is further transformed to the integration over the variables $\mu _j$ $(j=1,2,\ldots , N)$ in Dubrovin-type equation (17)

$$\begin{aligned} \langle Q\rangle =\frac{1}{(2\pi )^{m}}\int _{C_1}\cdots \int _{C_{N}} Q\frac{\partial (\theta )}{\partial (\mu )}\hbox {d}\mu _1\hbox {d}\mu _2,\ldots , \hbox {d}\mu _N, \end{aligned}$$

(23)

where $C_j$ $(j=1,2,\ldots , N)$ are the cycles defined by Abel maps and $\frac{\partial \theta }{\partial \mu }$ is the Jacobian (Flaschka et al. 1980) defined below

$$\begin{aligned} \frac{1}{(2\pi )^{m}}\frac{\partial (\theta )}{\partial (\mu )} =\frac{1}{V}\frac{\prod _{n>k}(\mu _n-\mu _k)}{\sqrt{\prod _{j=1}^{N} P(\mu _j)}}, \end{aligned}$$

(24)

where V is a constant.

Thirdly, imposing the definition of the integral (21) on the conservation law (20) and considering the Riemann invariants $\lambda _i$ $(i=1,2,\ldots , 2N+2)$ as functions of the slow variables X and T give rise to

$$\begin{aligned} \left\langle \frac{G}{g}\right\rangle \frac{\partial \lambda _i}{\partial T} -\left\langle \frac{B}{g}\right\rangle \frac{\partial \lambda _i}{\partial X}=0, \quad i=1,2,\ldots , 2N+2. \end{aligned}$$

(25)

Finally, reminding the average defined in (23) and canceling the small quantity $\epsilon$ in Eq. (25) the desired Whitham equations for Riemann invariants $\lambda _i$ $(i=1,2,\ldots , 2N+2)$ are obtained as follows (Kamchatnov 1994)

$$\begin{aligned} \frac{\partial \lambda _i}{\partial t}+v_i (\lambda _1,\lambda _2,\ldots , \lambda _{2N+2}) \frac{\partial \lambda _i}{\partial x}=0,\quad i=1,2,\ldots , 2N+2, \end{aligned}$$

(26)

where the characteristic velocities $v_i$ are given by

$$\begin{aligned}&v_i(\lambda _1,\lambda _2,\ldots , \lambda _{2N+2}) =-\frac{I_2(\lambda _1,\lambda _2,\ldots , \lambda _{2N+2})}{I_1(\lambda _1,\lambda _2,\ldots , \lambda _{2N+2})}, \nonumber \\&I_1(\lambda _1,\lambda _2,\ldots , \lambda _{2N+2}) =\int _{C_1}\cdots \int _{C_{N}} \frac{G}{g} \frac{\partial (\theta )}{\partial (\mu )}\hbox {d}\mu _1\hbox {d}\mu _2\cdots \hbox {d}\mu _N,\nonumber \\&I_2(\lambda _1,\lambda _2,\ldots , \lambda _{2N+2})=\int _{C_1} \cdots \int _{C_{N}} \frac{B}{g} \frac{\partial (\theta )}{\partial (\mu )}\hbox {d}\mu _1\hbox {d}\mu _2\cdots \hbox {d}\mu _N. \end{aligned}$$

(27)

In the following three subsections, we will take $N=0, N=1$ and $N=2$ to investigate the zero-phase, one-phase, and two-phase solutions of the defocusing cmKdV equation (1), respectively.

2.1 Zero-Phase Solution and Whitham Equations

For $N=0$, take f to be degree one polynomial in spectral parameter $\lambda$ and g, h to be functions independent of $\lambda$, i.e.,

$$\begin{aligned} f(x,t,\lambda )=\lambda +f_0(x,t),\quad g(x,t,\lambda )=g_0(x,t), \quad h(x,t,\lambda )=h_0(x,t). \end{aligned}$$

(28)

Substituting them into Eqs. (8)–(9) and collecting the coefficients of $\lambda$ yield

$$\begin{aligned} f_0(x,t)=\mathrm{constant},\quad g_0(x,t)=q(x,t), \quad h_0(x,t)=q^{*}(x,t), \end{aligned}$$

(29)

and

$$\begin{aligned} \varepsilon q_x=2iqf_0,\quad \varepsilon q_t=(12if_0\rho _0 +8if_0^3)q, \end{aligned}$$

(30)

with $\rho _0=|q|^2$, which has exact solution of the form

$$\begin{aligned} q=\sqrt{\rho _0}e^{\frac{2if_0}{\epsilon }[x+(6\rho _0+4f_0^2)t]}. \end{aligned}$$

(31)

This is the zero-phase solution of the defocusing cmKdV equation (1), in which the function

$$\begin{aligned} \theta _0=\frac{2f_0}{\epsilon }\left[ x+\left( 6\rho _0+4f_0^2\right) t\right] \end{aligned}$$

(32)

is a fast variable. The density $\rho _0$ and phase velocity $-(6\rho _0+4f_0^2)$ are slowly varying, and we have $\phi =2f_0x$ and $v=2f_0$ in the Madelung transformation (2).

In viewing the form of functions f, g and h, we have

$$\begin{aligned} f^2-gh=\lambda ^2+2f_0\lambda +f_0^2-\rho =\lambda ^2-s_1\lambda +s_2, \end{aligned}$$

(33)

where $s_1=-2f_0$ and $s_2=f_0^2-\rho .$ Moreover, assume the term $f^2-gh$ has two roots $\lambda _1$ and $\lambda _2$, i.e., $f^2-gh=P(\lambda )=(\lambda -\lambda _1)(\lambda -\lambda _2)$, then we have

$$\begin{aligned} \lambda _1+\lambda _2=s_1,\quad \lambda _1\lambda _2=s_2. \end{aligned}$$

(34)

The equality $f_0=v/2$ gives $s_1=-v$ and $s_2=v^2/4-\rho$, so the equation (34) can be rewritten as

$$\begin{aligned} \lambda _1+\lambda _2=-v,\quad \lambda _1\lambda _2=v^2/4-\rho , \end{aligned}$$

(35)

which can be solved for $\lambda _1$ and $\lambda _2$ as

$$\begin{aligned} \lambda _1=-\frac{v}{2}-\sqrt{\rho },\quad \lambda _2 =-\frac{v}{2}+\sqrt{\rho }. \end{aligned}$$

(36)

In order to derive the Whitham equation for the slow variables $\lambda _1$ and $\lambda _2$, we transform the conservation law (13) into

$$\begin{aligned} \frac{\partial }{\partial t}\left( \sqrt{P(\lambda )}\frac{G}{g}\right) -\frac{\partial }{\partial x}\left( \sqrt{P(\lambda )}\frac{B}{g}\right) =0. \end{aligned}$$

(37)

In the sense of zero-phase solution (31), the modified conservation law (37) is simplified to

$$\begin{aligned} \frac{\partial }{\partial t}(\sqrt{P(\lambda )}) -\frac{\partial }{\partial x}(\sqrt{P(\lambda )} (4\lambda ^2-2\lambda v+2\rho +v^2))=0. \end{aligned}$$

(38)

Expanding the partial derivatives in the above equation and taking limits $\lambda \rightarrow \lambda _1$ and $\lambda \rightarrow \lambda _2$, respectively, yield the Whitham equations for the slow variables $\lambda _1$ and $\lambda _2$ as follows:

$$\begin{aligned}&\frac{\partial \lambda _1}{\partial t}-\left( \frac{15\lambda _1^2}{2} +3\lambda _1\lambda _2+\frac{3\lambda _2^2}{2}\right) \frac{\partial \lambda _1}{\partial x}=0, \nonumber \\&\frac{\partial \lambda _2}{\partial t}-\left( \frac{15\lambda _2^2}{2} +3\lambda _1\lambda _2+\frac{3\lambda _1^2}{2}\right) \frac{\partial \lambda _2}{\partial x}=0. \end{aligned}$$

(39)

2.2 One-Phase Periodic Solution and Whitham Equations

It suffices to suppose that $P(\lambda )$ is a polynomial of degree four in $\lambda$ for the one-phase periodic solution, that is

$$\begin{aligned} f^2-gh&=P(\lambda )=\prod _{i=1}^4(\lambda -\lambda _i)\nonumber \\&=\lambda ^4-s_1\lambda ^3+s_2\lambda ^2-s_3\lambda +s_4, \end{aligned}$$

(40)

where $s_i~(i=1,2,3,4)$ are called elementary symmetric polynomials related to the four roots of the polynomial $P(\lambda )$. Recalling the Eqs. (8)–(9) for f, g and h, one has

$$\begin{aligned} f=\lambda ^2-f_1\lambda +f_2,\quad g=q(\lambda -\mu ), \quad h=q^*(\lambda -\mu ^*), \end{aligned}$$

(41)

where

$$\begin{aligned} f_{1x}= & {} f_{1t}=0,\quad \varepsilon f_{2x}=i|q|^2(\mu -\mu ^*),\nonumber \\ \varepsilon f_{2t}= & {} 2i|q|^4(\mu -\mu ^*)-i\varepsilon ^2 qq_{xx}^*\mu +i\varepsilon ^2 q^*q_{xx}\mu ^*, \end{aligned}$$

(42)

and

$$\begin{aligned}&\varepsilon q_x=2iq(\mu -f_1),\quad \varepsilon ^2 q_{xx} =-2i(2iqf_2+q_xf_1\varepsilon ), \end{aligned}$$

(43)

$$\begin{aligned}&\varepsilon q_t=2i\varepsilon ^2 q_{xx}f_1+2\varepsilon (|q|^2q_x-q^*_xq^2-2q_xf_2)+4i|q|^2q(\mu -f_1), \end{aligned}$$

(44)

$$\begin{aligned}&\varepsilon (q\mu )_x=-2iqf_2,\quad \varepsilon (q\mu )_t=2i\varepsilon ^2 q_{xx}f_2+2\varepsilon \mu (q_x|q|^2-q_x^*q^2)-4if_2|q|^2q, \end{aligned}$$

(45)

as well as the complex conjugate of all coefficients of Eq. (41). Substituting Eq. (41) into Eq. (40) and comparing the coefficients of $\lambda ^k$, the condition (41) gives the conservation laws

$$\begin{aligned} s_1=2f_1, s_2=-|q|^2+f_1^2+2f_2,s_3=2f_1f_2-|q|^2 (\mu +\mu ^*),s_4=f_2^2-|q|^2\mu \mu ^*, \end{aligned}$$

(46)

which indicates that $f_1=s_1/2,f_2=(|q|^2-f_1^2+s_2)/2$. Thus Eq. (42) for $f_2$ can be reduced to

$$\begin{aligned} \varepsilon \rho _x=2i\rho (\mu -\mu ^*),\quad \varepsilon \rho _t =2i\rho (\mu -\mu ^*) (\frac{3}{2}s_1^2-2s_2), \end{aligned}$$

(47)

where $\rho =|q|^2$ and the evolution of $\mu$ can be expressed by Eq. (45) for $\mu$ as

$$\begin{aligned} \varepsilon \mu _x=-2if(\mu )= -2i\sqrt{P(\mu )}, \quad \varepsilon \mu _t=-2if(\mu )\left( \frac{3}{2}s_1^2-2s_2\right) , \end{aligned}$$

(48)

in which the second equality of the first equation (48) can be achieved by substituting $\lambda =\mu$ into Eq. (40). The relations given by Eqs. (47)–(48) indicate that $\mu$ and $\rho$ depend on the phase

$$\begin{aligned} \xi =x-Vt,\quad \mathrm{with}~~V=2s_2-\frac{3}{2}s_1^2. \end{aligned}$$

(49)

Notice that the defocusing cmKdV equation (1) is the second flow in the defocusing NLS hierarchy, thus following the procedure of Appendix B.1 in the book of Kamchatnov (2000), the one-phase periodic solution $\rho =|q|^2$ can be expressed in term of the elliptic function

$$\begin{aligned} \rho =\rho _3+ (\rho _2-\rho _3)\mathrm{sn}^2 \left( \frac{\sqrt{\rho _1-\rho _3}}{\varepsilon }(\xi -\xi _0),m\right) , \end{aligned}$$

(50)

where $\mathrm{sn}$ is the Jacobi elliptic function, the modulus $m=(\rho _2-\rho _3)/(\rho _1-\rho _3)$, the parameter $\xi _0$ is the phase shift which is actually equal to zero in this work, and the parameters $\rho _1, \rho _2, \rho _3$ are

$$\begin{aligned}&\rho _1=\frac{1}{4}(\lambda _1+\lambda _2-\lambda _3-\lambda _4)^2,\nonumber \\&\rho _2=\frac{1}{4}(\lambda _1-\lambda _2+\lambda _3-\lambda _4)^2,\nonumber \\&\rho _3=\frac{1}{4}(\lambda _1-\lambda _2-\lambda _3+\lambda _4)^2, \end{aligned}$$

(51)

from which it is easy to see that $\rho _1>\rho _2>\rho _3$ provided that $\lambda _1>\lambda _2>\lambda _3>\lambda _4$.

Reminding the derivation of the Whitham equations in Eqs. (26) and (27), the Whitham equations corresponding to the one-phase periodic solution (50) are obtained as

$$\begin{aligned} \frac{\partial \lambda _i}{\partial t} -\frac{I_2(\lambda _1,\lambda _2,\lambda _3, \lambda _{4})}{I_1(\lambda _1,\lambda _2,\lambda _3, \lambda _{4})} \frac{\partial \lambda _i}{\partial x}=0, \quad i=1,2,3,4, \end{aligned}$$

(52)

where

$$\begin{aligned} I_1=\frac{2}{\varepsilon }\frac{\partial L}{\partial \lambda _i}, \quad I_2=\left( \frac{3}{2}s_1^2-2s_2\right) I_1+\frac{2}{\varepsilon } (s_1+2\lambda _i)L, \end{aligned}$$

(53)

where the L represents the wavelength of the one-phase periodic solution (50), i.e.,

$$\begin{aligned} L=\frac{2\varepsilon K(m)}{\sqrt{(\lambda _1-\lambda _3) (\lambda _2-\lambda _4)}}, \end{aligned}$$

(54)

where K(m) is the complete elliptic integral of the first kind and the modulus m of the elliptic function is

$$\begin{aligned} m=\frac{\rho _2-\rho _3}{\rho _1-\rho _3} =\frac{(\lambda _1-\lambda _2) (\lambda _3-\lambda _4)}{(\lambda _1-\lambda _3)(\lambda _2-\lambda _4)}. \end{aligned}$$

(55)

Finally, substituting the wavelength L formulated by Eq. (54) into Eqs. (52)–(53) the Whitham equations for the one-phase periodic solution (50) can be rewritten explicitly

$$\begin{aligned} \frac{\partial \lambda _i}{\partial t} +v_i(\lambda _1,\lambda _2,\lambda _3, \lambda _{4}) \frac{\partial \lambda _i}{\partial x}=0,\quad i=1,2,3,4, \end{aligned}$$

(56)

where the characteristic velocities $v_i=v_i(\lambda _1,\lambda _2,\lambda _3, \lambda _{4})$ $(i=1,2,3,4)$ are

$$\begin{aligned}&v_1=-\left( \sum _{i<j}\lambda _i\lambda _j+\frac{3}{2} \sum _{i=1}^4\lambda _i^2\right) -\left( \sum _{i=1}^4\lambda _i+2\lambda _1\right) \frac{2(\lambda _1-\lambda _2)(\lambda _1-\lambda _4)K(m)}{(\lambda _1-\lambda _4)K(m)+(\lambda _4-\lambda _2)E(m)},\nonumber \\&v_2=-\left( \sum _{i<j}\lambda _i\lambda _j+\frac{3}{2} \sum _{i=1}^4\lambda _i^2\right) +\left( \sum _{i=1}^4\lambda _i+2\lambda _2\right) \frac{2(\lambda _1-\lambda _2)(\lambda _2-\lambda _3)K(m)}{(\lambda _2-\lambda _3)K(m)+(\lambda _3-\lambda _1)E(m)},\nonumber \\&v_3=-\left( \sum _{i<j}\lambda _i\lambda _j+\frac{3}{2} \sum _{i=1}^4\lambda _i^2\right) -\left( \sum _{i=1}^4\lambda _i+2\lambda _3\right) \frac{2(\lambda _2-\lambda _3)(\lambda _3-\lambda _4)K(m)}{(\lambda _2-\lambda _3)K(m)+(\lambda _4-\lambda _2)E(m)},\nonumber \\&v_4=-\left( \sum _{i<j}\lambda _i\lambda _j+\frac{3}{2} \sum _{i=1}^4\lambda _i^2\right) +\left( \sum _{i=1}^4\lambda _i+2\lambda _4\right) \frac{2(\lambda _1-\lambda _4)(\lambda _3-\lambda _4)K(m)}{(\lambda _1-\lambda _4)K(m)+(\lambda _3-\lambda _1)E(m)}, \end{aligned}$$

(57)

where E(m) is complete elliptic integral of the second kind.

2.3 Two-Phase Periodic Solution and Whitham Equations

In this subsection, Kamchatnov’s way (Kamchatnov 1997) is carried out to explore the two-phase periodic solution and the corresponding Whitham equations for the defocusing cmKdV equation (1). In doing so, take $P(\lambda )$ to be a polynomial of degree six in $\lambda$

$$\begin{aligned} f^2-gh&=P(\lambda )=\prod _{i=1}^6(\lambda -\lambda _i)\nonumber \\&=\lambda ^6-s_1\lambda ^5+s_2\lambda ^4-s_3\lambda ^3+s_4\lambda ^2 -s_5\lambda +s_6, \end{aligned}$$

(58)

where $\lambda _1>\lambda _2>\lambda _3>\lambda _4>\lambda _5>\lambda _6$ are six roots and $s_i~(i=1,2,\ldots ,6)$ are the elementary symmetric polynomials related to the six roots $\lambda _i$ $(i=1,2,\ldots ,6)$, furthermore $s_1$ and $s_2$ are

$$\begin{aligned} s_1=\sum _{j=1}^{6}\lambda _j,\quad s_2 =\sum _{i,j=1; i\ne j}^{6}\lambda _i\lambda _j. \end{aligned}$$

(59)

Recalling the Eqs. (14)–(15) for f, g and h, we have

$$\begin{aligned} f=\lambda ^3-f_1\lambda ^2+f_2\lambda -f_3,\quad g =q(\lambda -\mu _1)(\lambda -\mu _2),\quad h=q^{*} (\lambda -\mu _1^{*})(\lambda -\mu _2^{*}). \end{aligned}$$

(60)

Thus Eq. (58) along with Eq. (60) further gives

$$\begin{aligned} s_1=2f_1, \quad s_2=2f_2+f_1^2-|q|^2. \end{aligned}$$

(61)

In a similar way, the Dubrovin-type equation for functions $\mu _1(x,t)$ and $\mu _2(x,t)$ can be formulated from Eq. (17) for $N=2$. Substituting the f, g and h in Eq. (60) into equation $\varepsilon g_x=2iGf+2Fg$ and collecting the coefficients of $\lambda$ yields

$$\begin{aligned}&\varepsilon q_x=2iq(\mu _1+\mu _2-f_1), \quad \varepsilon (q\mu _1)_x +\varepsilon (q\mu _2)_x=2iq(\mu _1\mu _2-f_2), \\&\varepsilon (q\mu _1\mu _2)_x=-2iqf_3. \end{aligned}$$

The same substitution can be done for equation $\varepsilon g_t=2iBf+2Ag$ and one arrives at the “trace formula” for function q(x, t) from Eq. (18) with $N=2$, which finally gives rise to the two-phase periodic solution of the defocusing cmKdV equation (1) with phase functions

$$\begin{aligned} \theta _1=\kappa _1x+\omega _1t+\theta _{01}, \quad \theta _2=\kappa _2x+\omega _2t+\theta _{02}. \end{aligned}$$

(62)

Next we return to the construction of the Whitham equations for this two-phase periodic solution. Recalling that $g=q(\lambda -\mu _1)(\lambda -\mu _2)$ one has

$$\begin{aligned} \frac{G}{g}&=\frac{1}{(\lambda -\mu _1)(\lambda -\mu _2)},\nonumber \\ \frac{B}{g}&=\frac{1}{(\lambda -\mu _1)(\lambda -\mu _2)}\nonumber \\&\qquad \left[ 4\lambda ^2-4\lambda \left( \mu _1+\mu _2-\frac{1}{2}s_1\right) \right. \nonumber \\&\qquad \quad \left. -4\left( \frac{1}{2}\mu _1s_1 +\frac{1}{2}\mu _2s_1-\mu _1\mu _2\right) +\frac{3}{2}s_1^2-2s_2\right] . \end{aligned}$$

(63)

Therefore, following Kamchatnov’s way (Kamchatnov 1997) which is based on the general procedure of FFM approach (Flaschka et al. 1980), the Whitham equations for two-phase periodic solution can be derived as

$$\begin{aligned} \frac{\partial \lambda _i}{\partial t} +v_i(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6) \frac{\partial \lambda _i}{\partial x}=0,\quad i=1,2,\ldots , 6, \end{aligned}$$

(64)

where the characteristic velocities $v_i$ $(i=1,2,\ldots , 6)$ are given by

$$\begin{aligned} v_i(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6) =-\frac{I_2(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6)}{I_1(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6)}, \end{aligned}$$

(65)

where $I_1=I_1(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6)$ and $I_2=I_2(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6)$ are

$$\begin{aligned} I_1&=\int _{C_1}\frac{\hbox {d}\mu }{\sqrt{P(\mu )}}\int _{C_2} \frac{\hbox {d}\mu }{(\lambda _i-\mu )\sqrt{P(\mu )}}-\int _{C_2} \frac{\hbox {d}\mu }{\sqrt{P(\mu )}}\int _{C_1} \frac{\hbox {d}\mu }{(\lambda _i-\mu ) \sqrt{P(\mu )}},\\ I_2&=\left( \frac{3}{2}s_1^2-2s_2\right) I_1(\lambda _i)\\&\quad +\int _{C_1}\int _{C_2}\frac{\left[ 4\lambda ^2-4\lambda \left( \mu _1+\mu _2-\frac{1}{2}s_1\right) -4\left( \frac{1}{2}\mu _1s_1+\frac{1}{2}\mu _2s_1-\mu _1\mu _2\right) +\frac{3}{2}s_1^2-2s_2\right] (\mu _2-\mu _1)}{(\lambda _i-\mu _1)(\lambda _i-\mu _2)\sqrt{P(\mu _1)P(\mu _2)}}d\mu _1d\mu _2, \end{aligned}$$

where $P(\mu )=\prod _{i=1}^6(\mu -\lambda _i)$, and $C_1$ and $C_2$ are the cycles defining the solution of genus-2 Dubrovin-type equation (i.e., Eq. (18) with $N=2$) according to the Abel transform. In our case, $C_1$ is cycle from $\lambda _5$ to $\lambda _4$ and $C_2$ is cycle from $\lambda _3$ to $\lambda _2$. After tedious calculations, it is found that the characteristic velocities $v_i$ $(i=1,2,\ldots , 6)$ are rewritten as

$$\begin{aligned}&v_i(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6) =2s_2-\frac{3}{2}s_1^2-(4\lambda _i+2s_1)\nonumber \\&\quad \times \frac{[(\partial U_{11}/\partial \lambda _i)U_{21} -(\partial U_{21}/\partial \lambda _i)U_{11}] +4(\partial U_{12}/\partial \lambda _i)(\partial U_{23}/\partial \lambda _i) -4(\partial U_{13}/\partial \lambda _i)(\partial U_{22}/\partial \lambda _i)}{U_{11}(\partial U_{21}/\partial \lambda _i) -U_{21}(\partial U_{11}/\partial \lambda _i)}, \end{aligned}$$

(66)

where $U_{ij}$ are the hyperelliptic integrals

$$\begin{aligned} U_{ij}=\int _{C_i}\frac{\mu ^{j-1}}{\sqrt{P(\mu )}}d\mu , \quad i=1,2;\quad j=1,2,3. \end{aligned}$$

3 Basic Wave Structures

This section starts to study what kinds of basic wave structures will appear for the defocusing cmKdV equation (1) with the general step-like initial data

$$\begin{aligned} q(x,0)=\sqrt{\rho (x,0)} e^{i \phi (x,0)/\varepsilon }, \quad \phi _x(x,0)=v(x,0), \end{aligned}$$

(67)

where $\rho (x,0)$ and v(x, 0) are

$$\begin{aligned} \rho (x,0)=\left\{ \begin{array}{l} \rho ^r, ~x>0, \\ \rho ^l,~x<0, \end{array}\right. ~~\mathrm{and} ~~~v(x,0)=\left\{ \begin{array}{l} v^r, ~x>0, \\ v^l,~x<0, \end{array}\right. \end{aligned}$$

(68)

where $\rho ^r$, $\rho ^l$, $v^r$ and $v^l$ are four arbitrary real constants. The solution under the initial data (67) with (68) consisting of basic structures of rarefaction wave and dispersive shock wave are quite fruitful, which will be discussed in details below.

3.1 Rarefaction Waves

The genus-0 Whitham equation (39) corresponding to the zero-phase solution (31) can also be derived in a different way. To be specific, the rarefaction wave solution can be derived by taking the dispersionless limit as $\varepsilon \rightarrow 0$ for the dispersive hydrodynamics-like system (3) due to the property of smoothness itself. The system governing the rarefaction wave satisfies the following non-strictly hyperbolic system

$$\begin{aligned} \left\{ \begin{array}{l} \rho _t-(3\rho ^2+3v^2\rho )_x=0,\\ v_t-(6\rho v+v^3)_x=0. \end{array}\right. \end{aligned}$$

(69)

This limit provides the solution correctly up to the solution develops to a shock. A standard procedure shows that the system (69) can be transformed into diagonal form

$$\begin{aligned} \left( r_{\pm }\right) _t+V_\pm (r_+,r_-)\left( r_{\pm }\right) _x=0, \end{aligned}$$

(70)

which is the genus-0 Whitham equation equivalent to the Whitham equation (39) under the scale transformation $r_{+}=-\lambda _1, r_{-}=-\lambda _2,$ where the Riemann invariants $r_{+}$ and $r_{-}$ are

$$\begin{aligned} r_+=\frac{v}{2}+\sqrt{\rho },\quad r_-=\frac{v}{2}-\sqrt{\rho }, \end{aligned}$$

(71)

and the characteristic velocities in terms of the Riemann invariants are expressed by

$$\begin{aligned} V_+=-\left( \frac{15}{2}r_+^2+3r_+r_-+\frac{3}{2}r_-^2\right) , \quad V_- =-\left( \frac{15}{2}r_-^2+3r_+r_-+\frac{3}{2}r_+^2\right) . \end{aligned}$$

(72)

Thus the initial data (67) with (68) in physical variables can be converted into the forms of Riemann invariants with the aid of the transformation (71)

$$\begin{aligned} r_+(x,0)=\left\{ \begin{array}{l} r_+^r=\frac{v^r}{2}+\sqrt{\rho ^r}, ~x>0, \\ r_+^l=\frac{v^l}{2}+\sqrt{\rho ^l},~x<0, \end{array}\right. \quad r_-(x,0) =\left\{ \begin{array}{l} r_-^r=\frac{v^r}{2}-\sqrt{\rho ^r}, ~x>0, \\ r_-^l=\frac{v^l}{2}-\sqrt{\rho ^l},~x<0. \end{array}\right. \end{aligned}$$

(73)

Introducing the self-similar variable $\tau =x/t$, the Whitham equations (70) are rewritten as

$$\begin{aligned} (V_{+}-\tau )\frac{\hbox {d}r_{+}}{d\tau }=0,\quad (V_{-}-\tau ) \frac{\hbox {d}r_{-}}{\hbox {d}\tau }=0. \end{aligned}$$

(74)

Similar to the case of the defocusing NLS equation (El et al. 1995), the bi-directionality determines three cases of rarefaction waves, i.e., only $r_+$ is a constant, only $r_{-}$ is a constant, and both $r_+$ and $r_-$ are constants. However, different from the defocusing NLS case, the characteristic of the defocusing cmKdV equation (1) propagates along single direction and divides into five types basic rarefaction wave structures. The first two types are

$$\begin{aligned} r_+=r_+^0=\mathrm{constant},\quad r_-=-\frac{1}{5}r_+^0\pm \frac{1}{15}\sqrt{-36(r_+^{0})^2-30\cdot \frac{x}{t}}, \end{aligned}$$

(75)

and the characteristic velocity $V_-=V_-(r_+^0,r_-)$. The middle two types are

$$\begin{aligned} r_+=-\frac{1}{5}r_-^0\pm \frac{1}{15}\sqrt{-36(r_-^{0})^2 -30\cdot \frac{x}{t}},\quad r_-=r_-^0=\mathrm{constant}, \end{aligned}$$

(76)

with the velocity $V_+=V_+(r_+,r_-^0)$, and the fifth type is

$$\begin{aligned} r_+=\dfrac{1}{\sqrt{6}}\sqrt{-\dfrac{x}{t}}, \quad r_-=-\dfrac{1}{\sqrt{6}}\sqrt{-\dfrac{x}{t}}, \end{aligned}$$

(77)

with the velocities $V_+=V_+(r_+,r_-)=\tau =V_{-}$. It is observed that the evolution of Riemann invariants for any choice of the rarefaction waves will be on the parabola

$$\begin{aligned} \frac{15}{2}r^2+3rr^0+\frac{3}{2}(r^0)^2+\tau =0 \end{aligned}$$

(78)

for the cases in (75)–(76) and on the parabola

$$\begin{aligned} 6r^2+\tau =0 \end{aligned}$$

(79)

for the case in (77), which are displayed in blue dot lines in Fig. 1a–f. It is remarked that we denote r to represent the Riemann invariants $r_{\pm }$ and $\lambda _{i}$ $(i=1,2,\ldots ,2N)$ in all the figures in this work. As we shall see, there exists the case when two Riemann invariants collide, and the coalescence of the Riemann invariants results from the property of the defocusing cmKdV equation (1), i.e., not genuinely nonlinear [see Kodama et al. (2008)], which does not appear in the defocusing NLS case. In addition, we call the solution to be a plateau if both $r_+=$ and $r_-=$ are constants. The pure rarefaction wave or the plateau, excluding any type of dispersive shock wave, will occur in three situations, i.e., $r^l_+ \geqslant r^r_+\geqslant r^l_-\geqslant r^r_-\geqslant r^*$, $r^l_+ \geqslant r^r_+\geqslant r^*\geqslant r^r_-\geqslant r^l_-$ and $r^* \geqslant r^r_+ \geqslant r^l_+\geqslant r^r_-\geqslant r^l_-$, where $r^*$ represents the point on which $\frac{\partial V_\pm }{\partial r}$ changes sign. Otherwise, the oscillating region will appear that will be discussed in the next subsection. The distributions of Riemann invariants along with the basic structures of rarefaction waves are shown in Fig. 1, which are denoted as {RW-I}, {RW-II}, $\ldots$, {RW-VI}, where “RW” is the abbreviation of rarefaction wave. The formation of the two parabolas in the third type of rarefaction wave degenerates to linearity eventually. It is remarked that the rarefaction wave structure {RW-III} is a new basic wave structure in the defocusing cmKdV equation (1), which has not been proposed by Kodama et al. (2008) and Kong et al. (2019) and other studies before.

Fig. 1

(Color online) Sketches of Riemann invariants and five possible basic rarefaction waves structures: a $r_+^l=r_+^r=1,r_-^l=-1,r_-^r=-0.5;$ b $r_+^l=1,r_+^r=0.5,r_-^l=r_-^r=-1;$ c $r_+^l=1,r_+^r=0.5,r_-^l=-1,r_-^r=-0.5$; d $r_+^l=r_+^r=1.5,r_-^l=1,r_-^r=0.5$; e $r_+^l=-1,r_+^r=-0.5,r_-^l=r_-^r=-1.5$

Examples of the combination of rarefaction wave are demonstrated in Fig. 2, where the combined rarefaction wave consists of five regions, from left to right, which are plateau, {RW-II}, {RW-III}, {RW-I} and plateau again. The boundary velocity between each regions are given by analyzing the three cases of the rarefaction waves. In this example, they are separated by, from left to right, $x/t=-13.875,-6,-1.5,-0.3$, respectively. The combined rarefaction wave evolving from the initial condition $r_+^l=1.5,r_+^r=0.5,r_-^l=-1,r_-^r=-0.1$ displays excellent agreement with the direct numerical simulations; see Fig. 2b.

Fig. 2

(Color online) Examples of self-similar solution of the combined rarefaction wave at time $t=2$. The initial condition is $r_+^l=1.5,r_+^r=0.5,r_-^l=-1,r_-^r=-0.1$: a distributions of the Riemann invariants; b comparison of the analytical solution from Whitham modulation theory with direct numerical simulations, where the red dashed line indicates analytical solution and the blue solid line represents the numerical solution

3.2 Dispersive Shock Waves

From the analysis of the last subsection, it is clear that the rarefaction wave solutions are valid for the defocusing cmKdV equation (1) until the wave breaking appears. The solution of equation (1) is governed by smooth enough rarefaction wave in term of two Riemann invariants, but soon after the breaking, one of the Riemann invariants develops into three branches governed by the averaging Whitham equation (56) and (57). The corresponding multi-valued region is replaced by an oscillating region. However, in the case of self-similar solution, the oscillating region can be determined immediately once the initial data are given. In this subsection, all kinds of structures of basic DSWs that may appear in the defocusing cmKdV equation (1) will be discussed in details.

Let us now list the basic structures of DSWs possibly appearing in the defocusing cmKdV equation (1), in which part of them have been given in Kodama et al. (2008). The first four types of DSWs are similar to the defocusing NLS case, where three Riemann invariants are constants while the fourth one changes, either $\lambda _2$ or $\lambda _3$. The reason for existing four types of DSWs in equation (1) instead of only two of them in the defocusing NLS case lies in the fact that there still exists a parametric parabola determined by three constants. Thus the four basic structures of DSWs can be obtained by truncating the upper or lower part of the parabola on each Riemann invariant $\lambda _2$ and $\lambda _3$. The distributions of such kinds of Riemann invariants describing DSWs are displayed in Fig. 3. The second four types of DSW rising from the non-genuine nonlinear system (56) with (57) satisfy the solution in which two Riemann invariants are constants and the other two change, as shown in Fig. 4. For simplicity, denote the eight basic structures of DSWs as {DSW-I}, {DSW-II}, ..., {DSW-VIII}, respectively. The two black dotted lines in Figs. 3 and 4 represent two distinct speeds characterized the DSWs which is known as the leading and trailing speed of their edges. The leading and trailing speeds of two edges can be found from Eq. (57) by taking the limitation $m\rightarrow 1$ for $\lambda _3=\lambda _2$, and $m\rightarrow 0$ for $\lambda _3=\lambda _4$ or $\lambda _2=\lambda _1$, respectively. In the eight basic structures of DSWs listed in Figs. 3 and 4, the spatial structure is divided into three regions, i.e., the plateau for $\frac{x}{t}<v|_\mathrm{left}$, the DSW for $V|_\mathrm{left}<\frac{x}{t}<v|_\mathrm{right}$ and the plateau again for $\frac{x}{t}>v|_\mathrm{right}$. It is remarked that the distributions of DSWs shown in Figs. 3 and 4 are obtained numerically via the scheme of two-step variant of Lax-Wendroff with nonlinear filter for the step-like function (Engquist et al. 1989).

For simplicity, we only analyze the boundary speeds of two types of DSWs, the other types can be explained easily in a similar way. For the case of {DSW-I}, it is seen that

$$\begin{aligned} \tau =\frac{x}{t}=v_3\left( r_+^r,r_-^l,\lambda _3,r_-^r\right) , \end{aligned}$$

(80)

which indicates that the speeds of the right edge $\tau |_\mathrm{right}$ and the left edge $\tau |_\mathrm{left}$ can be expressed as

$$\begin{aligned} \tau |_\mathrm{right}=v_3(r_+^r,r_-^l,\lambda _3,r_-^r)|_{\lambda _3=r_-^l}, \quad \tau |_\mathrm{left}=v_3(r_+^r,r_-^l,\lambda _3,r_-^r)|_{\lambda _3=r_-^r}. \end{aligned}$$

For the case of {DSW-V}, one has

$$\begin{aligned}&\tau =\frac{x}{t}=v_2(r_{+}^l,\lambda _2,\lambda _3,r_-^r), \quad \tau =\frac{x}{t}=v_3(r_{+}^l,\lambda _2,\lambda _3,r_-^r),\nonumber \\&\tau |_\mathrm{left}=v_3(r_{+}^l,\lambda _2,\lambda _3, r_-^r)|_{\lambda _3=r_-^r},\quad \tau |_\mathrm{right} =v_2(r_{+}^l,\lambda _2,\lambda _3,r_-^r)|_{\lambda _3=\lambda _2}. \end{aligned}$$

(81)

Fig. 3

(Color online) Four typical distributions of the Riemann invariants $\lambda _1, \lambda _2, \lambda _3$, and $\lambda _4$ for the DSWs with only $\lambda _2$ or $\lambda _3$ varying. a Type I: $r_+^l=r_+^r=1,r_-^l=-0.5,r_-^r=-1$; b Type II: $r_+^l=0.5,r_+^r=1,r_-^l=r_-^r=-1$; c Type III: $r_+^l=r_+^r=1,r_-^l=0,r_-^r=0.5$; d Type IV: $r_+^l=0,r_+^r=-0.5,r_-^l=r_-^r=-1$

It is noted that all of above cases are obviously the extension of the basic structures of rarefaction waves. The vertex of parabola $r^*$, at which the signs of $\partial v_i/\partial r$ $(i=1,2,3,4)$ change, plays an important role in distinguishing the types of {DSW-I}-{DSW-IV} and the types of {DSW-V}-{DSW-VIII}. The regions of DSW in Fig. 4 include vertex of parabola $r^*$ connecting the two Riemann invariants such as $\lambda _2$ and $\lambda _3$ in Fig. 4a, b, $\lambda _3$ and $\lambda _4$ in Fig. 4c, $\lambda _1$ and $\lambda _2$ in Fig. 4d, and the vertex $r^*$ in Fig. 4a, b can be determined by

$$\begin{aligned} \frac{\partial v_3(\lambda _1,r^*,r^*,\lambda _4)}{\partial r^*}=0, \end{aligned}$$

(82)

from which we arrive at $r^*(\lambda _1,\lambda _4)=-\frac{1}{4}(\lambda _1+\lambda _4)$. The same way can be utilized to formulate the vertex $r^*$ in Fig. 4c, d, which do not display here because of their complexity.

Fig. 4

(Color online) Four typical distributions of the Riemann invariants $\lambda _1, \lambda _2, \lambda _3$, and $\lambda _4$ for the DSWs with two of them varying. a Type V: $r^*=-\frac{1}{8},r_+^l =r_+^r=1,r_-^l=-\frac{3}{2} +\frac{1}{2}\sqrt{13},r_-^r=-0.5$; b Type VI: $r^*=\frac{1}{8},r_+^l=\frac{3}{2}-\frac{1}{2} \sqrt{13},r_+^r=0.5,r_-^l=r_-^r=-1$; c Type VII: $r^*=-0.0378,r_+^l=r_+^r=1,r_-^l=-\frac{2}{3},r_-^r=0.5$; d Type VIII: $r^*=0.0378,r_+^l=\frac{2}{3},r_+^r=-0.5,r_-^l=r_-^r=-1$

The remaining two basic structures of DSWs, i.e., {DSW-IX} and {DSW-X}, are the one people have never seen before including the pioneering work on the defocusing cmKdV equation (1) in Kodama et al. (2008) and have not been appear neither in the strictly hyperbolic system such as the defocusing NLS equation nor in the non-strictly hyperbolic system for which $r_+$ or $r_-$ is constant. This kind of structure also requires two variables slowly varying, but importantly they must be on both sides of the parabola. Otherwise, it develops a combination of DSWs and rarefaction waves. For this case, we suffice to show that the velocity of leading edge in Fig. 5a (two variable on the one side) is

$$\begin{aligned} v_3|_{\lambda _3=\lambda _2}=-\left( \frac{3}{2}\lambda _1^2+4\lambda _2^2 +\frac{3}{2}\lambda _4^2 +2\lambda _1\lambda _2+2\lambda _2\lambda _4 +\lambda _1\lambda _4\right) . \end{aligned}$$

(83)

One may find at once that it exactly coincide with the trailing speed of rarefaction wave when taking $\lambda _2=\lambda _1$, thus we have

$$\begin{aligned} v_3|_{\lambda _3=\lambda _2, \lambda _2=\lambda _1} =-\left( \frac{15}{2}\lambda _1^2+3\lambda _1\lambda _4 +\frac{3}{2}\lambda _4^2\right) =v_1|_{\lambda _3=\lambda _2.} \end{aligned}$$

(84)

As followed from the last subsection, the velocity in (84) for the negative direction of the Riemann invariant $\lambda _1$ is an increasing function on the interval $[-\lambda _4/5,+\infty )$, i.e., the upper part of the parabola. The same procedure can be applied to Fig. 5b immediately.

Fig. 5

(Color online) Two typical distributions of the Riemann invariants $\lambda _1, \lambda _2, \lambda _3$, and $\lambda _4$ for the DSWs with two of them varying. a Type IX: $r_+^l=\sqrt{3}-\frac{1}{2},r_+^r=\frac{2}{3},r_-^l=-\frac{1}{2},r_-^r=-1$; b Type X: $r_+^l=\frac{1}{2},r_+^r=1,r_-^l=\frac{1}{2} -\sqrt{3},r_-^r=-\frac{2}{3}$

4 Complete Classification of the Solution to Riemann Problem

In the last section, five types of rarefaction waves and ten types of DSWs have been figured out, in which except for {RW-III} all basic wave structures have common feature with one of the initial data, i.e., either $r_+^{l}=r_+^{r}$ or $r_-^{l}=r_-^{r}.$ With the fifteen basic wave structures at hand, we are ready to carry out the classification of the solutions for the defocusing cmKdV equation (1) with step-like initial data (67) with (68). In doing so, recall that $r_+=\frac{v}{2}+\sqrt{\rho }$ and $r_-=\frac{v}{2}-\sqrt{\rho }$. Fixing the initial value (68) at the left-hand side $(v^{l}, \rho ^{l})$, the Riemann invariants $r_-^{l}$ and $r_+^{l}$ are determined. So we have two parabolas

$$\begin{aligned} \rho =\left( r_+^{l}-\frac{v}{2}\right) ^2 \quad \mathrm{and} \quad \rho =\left( r_-^{l}-\frac{v}{2}\right) ^2, \end{aligned}$$

(85)

with vertexes $2r_+^{l}>2r_-^{l}$ on the horizontal v-axis, which are shown in the solid lines in Fig. 6. It is observed that the two parabolas divide the $(v, \rho )$ plane into six regions. The order among $r_+^{l}, r_-^{l}, r_+^{r}, r_-^{r}$ can be determined easily in each region. Taking the region F as an example, for a point $(v_{F}, \rho _{F})=(v^{r}, \rho ^{r})$ in this region, similarly, two parabolas $\rho =(r_+^{r}-\frac{v}{2})^2$ and $\rho =(r_-^{r}-\frac{v}{2})^2$ with vertexes $2r_+^{r}>2r_-^{r}$ are obtained, which are displayed in the dotted lines in Fig. 6. It is clear that $r_+^{r}>r_-^{r}>r_+^{l}>r_-^{l}$ in region F. The orders of the left and right Riemann invariants for the other regions can be found in the same way:

$$\begin{aligned}&\mathbf{A.}~r_+^l>r_+^r> r_-^r>r_-^l, \quad \mathbf{B.}~r_+^l>r_+^r>r_-^l>r_-^r,\nonumber \\&\mathbf{C.}~r_+^l>r_-^l>r_+^r>r_-^r, \quad \mathbf{D.}~r_+^r>r_+^l>r_-^l>r_-^r,\nonumber \\&\mathbf{E.}~r_+^r>r_+^l>r_-^r>r_-^l,\quad \mathbf{F.}~r_+^r>r_-^r>r_+^l>r_-^l. \end{aligned}$$

(86)

In this section, the complete classification of the solution for the Riemann problem of the defocusing cmKdV equation (1) are given by analyzing the six possible cases above based on the Whitham equations proposed in Sect. 2 and the basic wave structures in Sect. 3.

Fig. 6

Regions in the $(v, \rho )$ plane corresponding to different classes of the solution for Riemann problem (67) with (68)

4.1 Case A. $r_+^l>r_+^r> r_-^r>r_-^l$

In each case, the vertexes of the parabolas (78) and (79) play a vital role in the classification of the solution for the Riemann problem of equation (1). In what follows, four subcases are discussed, which combine the complete classification of Case A.

4.1.1 Subcase $A_1$: $r_-^r>r^*$

Here the vertex of the parabola $r^*$ satisfies

$$\begin{aligned} \frac{\partial v_3(r_+^r,r_-^r,r^*,r^*)}{\partial r^*}=0. \end{aligned}$$

(87)

Note that putting $r_-^r=r^*$ into (87) yields $r^*=-r_+^r/5$, which is the critical condition followed from Sect. 3.1, and the $r_+^l>r_+^r$ located in the upper part of parabola will develop to a rarefaction wave. The whole scenarios under this condition are analyzed in eight more cases, which are $A_{1.1}-A_{1.8}$. See Figs. 7, 8 and Table 1 for details.

• $A_{1.1}:\quad r_+^l>r_+^r>r_-^r>r_-^l,\quad r_-^l\geqslant r^*.$

In this case, the solution of the Riemann invariants consists of five regions (see Fig. 7a), and in each region, the Riemann invariants can be formulated below.

• (1) For $x/t\leqslant v_1(r_+^l,r_-^r,r_-^r,r_-^l)$, the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^l,\quad r_-=r_-^l. \end{aligned}$$

• (2) For $v_1(r_+^l,r_-^r,r_-^r,r_-^l)<x/t<v_1(r_+^r,r_-^r, r_-^r,r_-^l)$, the solution is in the {RW-II} region with Riemann invariants

  $$\begin{aligned} r_+=-\frac{1}{5}r_-^l+\frac{1}{15}\sqrt{-36(r_-^{l})^2-30 \cdot \frac{x}{t}},\quad r_-=r_-^l. \end{aligned}$$

• (3) For $v_1(r_+^r,r_-^r,r_-^r,r_-^l)\leqslant x/t\leqslant v_3(r_+^r,r_-^r,r_-^r,r_-^l)$, the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^r,\quad r_-=r_-^l. \end{aligned}$$

• (4) For $v_3(r_+^r,r_-^r,r_-^r,r_-^l)<x/t<v_3 (r_+^r,r_-^r,r_-^l,r_-^l)$, the solution is in the {DSW-III} region with Riemann invariants

  $$\begin{aligned} \lambda _1=r_+^r,\quad \lambda _2=r_-^r,\quad \frac{x}{t} =v_3(r_+^r,r_-^r,\lambda _3,r_-^l),\quad \lambda _4=r_-^l. \end{aligned}$$

• (5) For $x/t\geqslant v_3(r_+^r,r_-^r,r_-^l,r_-^l)$, the solution is also in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^r,\quad r_-=r_-^l. \end{aligned}$$

  Figure 7b shows that the analytical result from Whitham modulation theory agrees well with the direct numerical simulations.

Fig. 7

(Color online) Example of the self-similar solution for case $A_{1.1}$ combined by {RW-II} and {DSW-III} with initial condition $r_+^l=1,r_+^r=0.8,r_-^l=0,r_-^r=0.5$ at time $t=2$: a distributions of the Riemann invariants in five regions; b comparison of the analytical solution from Whitham modulation theory (solid red line) with direct numerical simulations (dash blue line)

• $A_{1.2}:~r_+^l>r_+^r>r_-^r>r_-^l,~r_-^r>r^*,~r^*\geqslant r_-^l\geqslant r^{**}$.

Here the point $r^{**}$ is determined by

$$\begin{aligned} v_3(r_+^r,r_-^r,r_-^r,r^{**})=v_4(r_+^r,r_-^r,r_-^r,r^{**}). \end{aligned}$$

(88)

The solution regions in this case is very similar to the case $A_{1.1}$, but the {DSW-III} region in case $A_{1.1}$ is divided into two combined DSWs, i.e., {DSW-III} and {DSW-VII}. The Riemann invariants corresponding to the regions of the two combined DSWs are formulated as follows.

For $v_3(r_+^r,r_-^r,r_-^r,r_-^l)<x/t<v_4 (r_+^r,r_-^r,r_e^{*},r_-^l)$, where $r_e^{*}$ solves the implicit equation $v_3(r_+^r,r_-^r,r_e^{*}, r_-^l)=v_4(r_+^r,r_-^r,r_e^{*}, r_-^l)$, the solution is in the {DSW-III} region with Riemann invariants

$$\begin{aligned} \lambda _1=r_+^r,\quad \lambda _2=r_-^r,\quad \frac{x}{t} =v_3(r_+^r,r_-^r,\lambda _3,r_-^l),\quad \lambda _4=r_-^l. \end{aligned}$$

For $v_4(r_+^r,r_-^r,r_e^{*},r_-^l)\leqslant x/t\leqslant v_4(r_+^r,r_-^r,r^*,r^*)$, the solution is in the {DSW-VII} region with Riemann invariants

$$\begin{aligned} \lambda _1=r_+^r,\quad \lambda _2=r_-^r,\quad \frac{x}{t} =v_3(r_+^r,r_-^r,\lambda _3,\lambda _4),\quad \frac{x}{t}=v_4(r_+^r,r_-^r,\lambda _3,\lambda _4). \end{aligned}$$

• $A_{1.3}:~r_+^l>r_+^r>r_-^r>r_-^l,~r_-^r>r^*,~ -r_+^r \leqslant r_-^l\leqslant r^{**}$.

  This case emerges by dividing the intermediate plateau in case $A_{1.2}$ into a plateau and a {RW-I} rarefaction wave, where the other regions are similar to case $A_{1.2}$ except the {RW-I} region, whose Riemann invariants are expressed by

  $$\begin{aligned} r_+=r_+^r,\quad r_-=-\frac{1}{5}r_+^r-\frac{1}{15} \sqrt{-36(r_+^{r})^2-30\cdot \frac{x}{t}}, \end{aligned}$$

  for $v_1(r_+^r,r_-^r,r_-^r,r_-^l)<x/t<v_4(r_+^r,r_-^r,r_-^r,r^{**})$.

• $A_{1.4}:~r_+^l>r_+^r>r_-^r>r_-^l,\quad r_-^r>r^*,\quad -r_+^l \leqslant r_-^l\leqslant -r_+^r$.

  This case comes from case $A_{1.3}$ by dividing the {RW-II} region into two different regions of rarefaction waves, i.e., {RW-II} and {RW-III}. The solution of Riemann invariants for {RW-II} region is the same as case $A_{1.3}$ for $v_1(r_+^l,r_-^r.r_-^r,r_-^l)<x/t<v_4(-r_-^l,r_-^r,r_-^r,r_-^l)$, but for $v_4(-r_-^l,r_-^r,r_-^r,r_-^l)<x/t<v_4(r_+^r,r_-^r.r_-^r,r_-^l)$, the solution is in the {RW-III} region with Riemann invariants

  $$\begin{aligned} r_+=\dfrac{1}{\sqrt{6}}\sqrt{-\dfrac{x}{t}},\quad r_-=-\dfrac{1}{\sqrt{6}}\sqrt{-\dfrac{x}{t}}. \end{aligned}$$

• $A_{1.5}:~r_+^l>r_+^r>r_-^r>r_-^l,\quad r_-^r>r^*,\quad r_-^l< -r_+^l.$

  This case emerges by dividing the left plateau in case $A_{1.4}$ into a plateau and a {RW-I} rarefaction wave, in which for $v_4(r_+^l,r_-^r,r_-^r,r_-^l)<x/t<v_1(r_+^l,,r_-^r,r_-^r,-r_+^l)$, the solution is in the {RW-I} region with Riemann invariants

  $$\begin{aligned} r_+=r_+^l,\quad r_-=-\frac{1}{5}r_+^l-\frac{1}{15} \sqrt{-36(r_+^{l})^2-30\cdot \frac{x}{t}} \end{aligned}$$

  for $v_4(r+^l,r_-^r,r_-^r,r_-^l)<x/t<v_1(r_+^l,r_-^r,r_-^r,-r_+^l).$

Fig. 8

(Color online) Distributions of Riemann invariants for all the possible cases in Case $A_1$ under the condition $r_-^r>r^*$: a the characteristic velocity $r_-^l$ is chosen to divide the corresponding region into two smaller regions; b the characteristic velocity $r_-^l$ is chosen to equal to the boundary velocities

The cases $A_{1.6}-A_{1.8}$, where the characteristic velocity $r_-^l$ is chosen to equal to the boundary velocities, can also be analyzed in the same way. All these cases can be seen clearly in Fig. 8 and Table 1. It is remarked that all the descriptions of regions here and below are from left to right. For instance, the solution regions of case $A_{1.2}$ in Table 1 from left to right are Region 1 (plateau) to Region 6 (plateau).

Table 1 Classification of solutions and region distributions in eight subcases under condition $A_1$, where “Null” means there does not exist this region

4.1.2 Subcase $A_2$: $r_+^r<r_{A_{2}}^*$

Here the critical point $r_{A_{2}}^{*}$ satisfies

$$\begin{aligned} \frac{\partial v_2(r_{A_{2}}^{*}, r_{A_{2}}^{*}, r_+^r,r_-^r)}{\partial r_{A_{2}}^*}=0. \end{aligned}$$

(89)

Substituting $r_+^r=r_{A_{2}}^*$ into equation (89) yields $r_{A_{2}}^*=-1/r_-^r$. The distributions of the Riemann invariants under condition $A_2$ are much similar to that under condition $A_1$, moreover, the density $\rho$ in each case of $A_2$ are exactly the same with that in case $A_1$. Thus, we ignore the analysis and only list the results of the classification in this case. See Table 2 for details, where the other critical point $r_{A_{2}}^{**}$ solves the following implicit equation

$$\begin{aligned} v_1(r_{A_{2}}^{**},r_+^r,r_+^r,r_-^r)=v_2(r_{A_{2}}^{**}, r_+^r,r_+^r,r_-^r). \end{aligned}$$

(90)

In fact, the Riemann invariants in all the cases of $A_1$ and $A_2$ are symmetric with respect to x-axis. More specifically, {RW-II} in $A_1$ is symmetrical to {RW-I} in $A_2$, {DSW-III} in $A_1$ is symmetrical to {DSW-IV} in $A_2$, and {DSW-VII} in $A_1$ is symmetrical to {DSW-VIII} in $A_2$, which can also be understood through equation (57) and the equality below:

$$\begin{aligned}&v_1(\lambda _1^0,\lambda _2^0,\lambda _3^0,\lambda _4^0) =v_4(-\lambda _4^0,-\lambda _3^0,-\lambda _2^0,-\lambda _1^0),\\&v_2(\lambda _1^0,\lambda _2^0,\lambda _3^0,\lambda _4^0) =v_3(-\lambda _4^0,-\lambda _3^0,-\lambda _2^0,-\lambda _1^0). \end{aligned}$$

where $\lambda _i^0$ $(i=1,2,3,4)$ represent constants (see equation (96) for details).

It is seen that DSWs emerge in both case $A_1$ and $A_2$. In what follows, the case in which only plateau and rarefaction wave regions are produced is discussed under special conditions.

Table 2 Classification of solutions and region distributions in eight subcases under condition $A_2$

4.1.3 Subcase $A_3$: $r_-^r\leqslant r^*$ and $r_+^r\geqslant r_{A_{2}}^*$

In this case, the solution regions only evolve the combination of plateau and rarefaction waves, which makes the classification to be simpler than cases $A_1$ and $A_2$ above. Firstly, all the possible cases under condition $A_3$ are listed in Table 3.

Table 3 All of possible solution cases under condition $A_3$

The classification of solutions and region distributions for the subcases listed in Table 3 are given in Table 4. It is remarked that except case $A_{3.42}$, which is in fact one of the basic rarefaction wave structures of {RW-III}, all cases evolve at least four regions. In each case, the boundary velocities can also be derived in the similar way. For simplicity, we only analyze $A_{3.31}$ in details, which are found in Sect. 3.1 and Fig. 2.

• (1) For $x/t<v_+(r_+^l,r_-^l ),$ the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^l,\quad r_-=r_-^l. \end{aligned}$$

• (2) For $v_+(r_+^l,r_-^l)\leqslant x/t<v_+(-r_-^l,r_-^l),$ the solution is in the {RW-II} region with Riemann invariants

  $$\begin{aligned} r_+=-\frac{1}{5}r_-^l+\frac{1}{15}\sqrt{-36(r_-^{l})^2-30 \cdot \frac{x}{t}},\quad r_-=r_-^l. \end{aligned}$$

• (3) For $v_+(-r_-^l,r_-^l)\leqslant x/t<v_+(r_+^r,-r_+^r),$ the solution is in the {RW-III} region with Riemann invariants

  $$\begin{aligned} r_+=\dfrac{1}{\sqrt{6}}\sqrt{-\dfrac{x}{t}}, \quad r_-=-\dfrac{1}{\sqrt{6}}\sqrt{-\dfrac{x}{t}}. \end{aligned}$$

• (4) For $v_+(r_+^r,-r_+^r)\leqslant x/t<v_+(r_+^r,r_-^r),$ the solution is in the {RW-I} region with Riemann invariants

  $$\begin{aligned} r_+=r_+^r,\quad r_-=-\frac{1}{5}r_+^r-\frac{1}{15}\sqrt{-36(r_+^r)^2 -30\cdot \frac{x}{t}}. \end{aligned}$$

• (5) For $x/t\geqslant v_+(r_+^r,r_-^r ),$ the solution is again in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^r,\quad r_-=r_-^r. \end{aligned}$$

Table 4 Classification of solutions and region distributions for the subcases listed in Table 3 under condition $A_3$

4.2 Case B. $r_+^l>r_+^r>r_-^l>r_-^r$

This is a particular case in which certain exotic phenomena of wave breakings are demonstrated. The vertex $r^*_{BCD}$ of the parabolas connecting Riemann invariants $\lambda _2$ with $\lambda _3$ plays a vital role in the classification of the solution for the Riemann problem in Case B, where $r^*_{BCD}$ solves

$$\begin{aligned} \frac{\partial v_3(r_+^r,r^*_{BCD},r^*_{BCD},r_-^r)}{\partial r_B^*}=0, \end{aligned}$$

(91)

from which one has $r^*_{BCD}=-\frac{1}{4}(r_+^r+r_-^r).$ In viewing Fig. 4 and its descriptions, it is observed that there are several subcases in Case B (also in Cases C and D).

4.2.1 Subcase $B_1$: $r_-^l\leqslant r^*_{BCD}$ and $r_+^r>r^*_{BCD}$

Similar to case $A_1$, there also exist many subcases in $B_1$, where a new basic dispersive shock wave structure {DSW-IX} emerges. The classification of almost all subcases in $B_1$ are display in Table 5, in which $r_{B_{1}}^*=-\frac{1}{3}r_-^r-\frac{2}{3}r_-^l$ solves

$$\begin{aligned} v_1(r_{B_{1}}^*,r_-^l,r_-^l,r_-^r)=v_3(r_{B_{1}}^*,r_-^l,r_-^l,r_-^r), \end{aligned}$$

(92)

and $r_{B_{1}}^{**}=r_-^r-r_-^l+\sqrt{5(r_-^r)^2-4r_-^lr_-^r}$ solves

$$\begin{aligned} v_1(r_{B_{1}}^{**},r_-^l,r_-^r,r_-^r)=v_3(r_{B_{1}}^{**}, r_-^l,r_-^r,r_-^r). \end{aligned}$$

(93)

Table 5 All of possible solution cases under condition $B_1$

The classification of solutions and region distributions for the subcases listed in Table 5 are given in Table 6. It is remarked that except case $B_{1.42}$, which is in fact one of the basic rarefaction wave structures of {DSW-IX} displayed in Fig. 5 (and there exists only three regions), all cases evolve at least four regions, and in each case, the boundary velocities can be given easily.

Table 6 Classification of solutions and region distributions for the subcases listed in Table 5 under condition $B_1$

In what follows, only case $B_{1.33}$ is discussed in details, see also Fig. 9, where the initial conditions for Riemann invariants are chosen to be $r_+^l=1,r_+^r=0.8,r_-^l=-0.5,r_-^r=-1$. So one has $r_{B_{1}}^{*}=0.6$ and $r_{B_{1}}^{**}=\sqrt{3} -\frac{1}{2}.$ Figure 9 demonstrates the distributions of the Riemann invariants and the comparison of the analytical solution from Whitham modulation theory with direct numerical simulations. It is observed that the three middle regions form a larger genus-1 dispersive shock wave consisting of three genus-1 DSWs, in which an exotic DSW structure {DSW-IX} is evolved. The boundaries of those regions are separated by four black dotted lines, see Fig. 9a, which is quite different from the structures of wave breaking in the oscillation region observed before (Kodama et al. 2008). Moreover, Fig. 9b shows that the analytical result from Whitham modulation theory agrees well with the direct numerical simulations.

Fig. 9

(Color online) Example of the self-similar solution for case $B_{1.33}$ with initial condition $r_+^l=1,r_+^r=0.8,r_-^l=-0.5,r_-^r=-1$ at time $t=1$: a distributions of the Riemann invariants in five regions; b comparison of the analytical solution from Whitham modulation theory (solid red line) with direct numerical simulations (dash blue line)

In case $B_{1.33}$, the solution of the Riemann invariants consists of five regions (see Fig. 9a), and in each region, the Riemann invariants can be formulated below.

• (1) For $x/t\leqslant v_3(r_+^l,r_-^l,r_-^r,r_-^r)$, the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^l,\quad r_-=r_-^l. \end{aligned}$$

• (2) For $v_3(r_+^l,r_-^l,r_-^r,r_-^r)<x/t<v_1 (r_+^l,r_-^l,r_{Be}^*,r_-^r)$, where $r_{Be}^*$ solves $v_1(r_+^l,r_-^l,r_{Be}^*,r_-^r)=v_3(r_+^l,r_-^l,r_{Be}^*,r_-^r)$ and locates in the interval $[r_-^r,r_-^l]$, the solution is in the {DSW-I} region with Riemann invariants

  $$\begin{aligned} \lambda _1=r_+^l,\quad \lambda _2=r_-^l, \quad \frac{x}{t}=v_3(r_+^l,r_-^l,\lambda _3,r_-^r),\quad \lambda _4=r_-^r. \end{aligned}$$

• (3) For $v_1(r_+^l,r_-^l,r_{Be}^{**},r_-^r)\leqslant x/t<v_3(r_+^r,r_-^l,r_{Be}^{**},r_-^r)$, where $r_{Be}^{**}$ satisfies $v_1(r_+^r,r_-^l,r_{Be}^{**},r_-^r)=v_3(r_+^r,r_-^l,r_{Be}^{**},r_-^r)$ located in the interval $[r_-^r,r_-^l]$, the solution is in the {DSW-IX} region with Riemann invariants

  $$\begin{aligned} \frac{x}{t}=v_1(\lambda _1,r_-^l,\lambda _3,r_-^r), \quad \lambda _2=r_-^l, \quad \frac{x}{t}=v_3 (\lambda _1,r_-^l,\lambda _3,r_-^r), \quad \lambda _4=r_-^r. \end{aligned}$$

• (4) For $v_3(r_+^r,r_-^l,r_{Be}^{**},r_-^r)\leqslant x/t<v_3(r_+^r,r_-^l,r_-^l,r_-^r)$, the solution is in the {DSW-I} region with Riemann invariants

  $$\begin{aligned} \lambda _1=r_+^r,\quad \lambda _2=r_-^l,\quad \frac{x}{t} =v_3(r_+^r,r_-^l,\lambda _3,r_-^r), \quad \lambda _4=r_-^r. \end{aligned}$$

• (5) For $x/t\geqslant v_3(r_+^r,r_-^l,r_-^l,r_-^r)$, the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^r,\quad r_-=r_-^l. \end{aligned}$$

The boundary velocities $x/t= v_3(r_+^l,r_-^l,r_-^r,r_-^r)$ and $x/t=v_3(r_+^r,r_-^l,r_-^l,r_-^r)$ are known as trailing edge and leading edge, respectively. The Riemann invariants match at the boundaries of zero-phase solution region and one-phase solution region, see below.

At the trailing edge, i.e., harmonic front with $\lambda _3=\lambda _4$ and $m=0$ (see Eq. 55), we have

$$\begin{aligned} (\lambda _1,\lambda _2)=\text {the rarefaction wave solution outside the oscillation region}. \end{aligned}$$

At leading edge, i.e., soliton front with $\lambda _3=\lambda _2$ and $m=0$ (see Eq. 55), we have

$$\begin{aligned} (\lambda _1,\lambda _4)=\text {the rarefaction wave solution outside the oscillation region}. \end{aligned}$$

4.2.2 Subcase $B_2$: $r_-^l> r^*_{BCD}$ and $r_-^r<r^*_{BCD}$

Taking $r_{B_{2}}^*=r_-^r-r_-^l+\sqrt{5(r_-^r)^2-4r_-^lr_-^r}$, the complete classification under condition $B_2$ is given below:

For $r_-^l<r_{B_{2}}^{*}$, there are three cases as follows

• $B_{2.1}: ~r_+^r<r_{B_{2}}^*,r_+^l<r_{B_{2}}^*$.

• $B_{2.2}: ~r_+^r=r_{B_{2}}^*,r_+^l>r_{B_{2}}^*$.

• $B_{2.3}: ~r_+^r>r_{B_{2}}^*,r_+^l>r_{B_{2}}^*$.

  For $r_-^l=r_{B_{2}}^{*}$, there is only one case

• $B_{2.4}: ~r_+^r>r_{B_{2}}^*,r_+^l>r_{B_{2}}^*$.

  For $r_-^l>r_{B_{2}}^{*}$, there is also only one case

• $B_{2.5}: ~r_+^r>r_{B2}^*,r_+^l>r_{B2}^*$.

Table 7 Classification of solutions and region distributions for the subcases listed in Table 5 under condition $B_2$

The classification of solutions and region distributions for these subcases are given in Table 7. It is shown that all cases evolve at least five regions, in which the exotic DSW structure {DSW-IX} appears again.

Fig. 10

(Color online) Example of the self-similar solution for case $B_{2.1}$ with initial conditions $r_+^l=1,r_+^r=0.8,r_-^l=0.5,r_-^r=-1$ at time $t=1$: a distributions of the Riemann invariants in five regions; b comparison of the analytical solution from Whitham modulation theory (solid red line) with direct numerical simulations (dash blue line)

In the following, the case $B_{2.1}$ is taken as an example to exhibit the novel wave breaking appearing under the condition $B_2$. Figure 10 demonstrates the distributions of the Riemann invariants and the comparison of the analytical solution from Whitham modulation theory with direct numerical simulations. It is seen that the four middle regions form a larger genus-1 dispersive shock wave consisting of four genus-1 DSWs, which definitely does not appear under condition $B_1$. The boundaries of those regions are separated by five black dotted lines, see Fig. 10a, which has not been found before (Kodama et al. 2008). It is also found from Fig. 10b that the analytical result from Whitham modulation theory agrees well with the direct numerical simulations. In fact, the fourth and fifth regions for case $B_{2.1}$ shown in Fig. 10 is generated by dividing the fourth region in case $B_{1.33}$ displayed in Fig. 9. Thus only the boundary velocities of the fifth region, i.e., the {DSW-VI} region, is proposed below and the other regions are the same as case $B_{1.33}$.

For the {DSW-VI} region in Fig. 10, the Riemann invariants are

$$\begin{aligned} \lambda _1=r_+^r,\quad \frac{x}{t}=v_2(r_+^r,\lambda _2,\lambda _3,r_-^r), \quad \frac{x}{t}=v_3(r_+^r,\lambda _2,\lambda _3,r_-^r), \quad \lambda _4=r_-^r, \end{aligned}$$

(94)

for $v_2(r_+^r,r_-^l,r_{Be}^{**},r_-^r)<x/t\leqslant v_3(r_+^r,r_{Be}^*,r_{Be}^*,r_-^l)$, where $r_{Be}^*=r_B^*=-\frac{1}{4}(r_+^r+r_-^r)$ and $r_{Be}^{**}$ satisfies an implicit equation

$$\begin{aligned} v_2(r_+^r,r_-^l,r_{Be}^{**},r_-^r)=v_3(r_+^r,r_-^l,r_{Be}^{**},r_-^r). \end{aligned}$$

(95)

In particular, it can be calculated that $r_{Be}^*=r_B^*=0.05$ and $r_{Be}^{**}\thickapprox -0.4195$ in case $B_{2.1}$ with the initial conditions $r_+^l=1,r_+^r=0.8,r_-^l=0.5,r_-^r=-1$.

4.2.3 Subcase $B_3$: $r_-^r \ge r^*_{BCD}$

The condition $r_-^r \ge r_B^*$ indicates $r_-^r\geqslant -\frac{1}{5}r_+^r$ which only produces rarefaction waves, see Sect. 3.1 for details. Thus the solution regions in this case only consist of plateau, {RW-II}, plateau, {RW-I}, plateau. Figure 11 displays the distributions of the Riemann invariants and the comparison of the analytical solution from Whitham modulation theory with direct numerical simulations, which shows that the result of Whitham modulation theory agrees well with the direct numerical simulations.

Fig. 11

(Color online) Examples of self-similar solutions of of case $B_3$ with initial condition $r_+^l=2,r_+^r=1.5,r_-^l=1,r_-^r=-0.3$ at time $t=1$: a distributions of Riemann invariants; b comparison of numerical simulation (dash blue line) with analytical solution (solid red line) of the defocusing cmKdV equation (1)

4.2.4 Subcase $B_4$: $r_+^l < r^*_{BCD}$

In this case, the four initial values $r_+^l, r_+^r, r_-^l$ and $r_-^r$ are all below the critical point $r_B^*$. Define four critical points $r_{B_{4}}^*, r_{B_{4}}^{**}$, $rr_{B_{4}}^*$ and $rr_{B_{4}}^{**}$ in the following way:

• Let $r_{B_{4}}^*\in [r_+^r ,~r_{+}^l]$ solve the implicit equation $v_1(r_{B_{4}}^*,r_+^r,r_+^r,r_-^r)=v_2(r_{B_{4}}^*, r_+^r,r_+^r,r_-^r).$

• Let $r_{B_{4}}^{**}\in [r_+^r ,~r_{+}^l]$ satisfy equation $\partial v_1(r_{B_{4}}^{**},r_{B_{4}}^{**},r_+^r,r_-^r)/\partial r_{B_{4}}^{**}=0.$

• Let $rr_{B_{4}}^*\in [r_{B_{4}}^*, r_+^l]$ solve the implicit equation $v_1(rr_{B_{4}}^*,r_-^l,r_-^l,r_-^r) =v_3(rr_{B_{4}}^*,r_-^l,r_-^l,r_-^r).$

• Let $rr_{B_{4}}^{**}$ solve the implicit equation $v_1(rr_{B_{4}}^{**},r_-^l,r_-^r,r_-^r) =v_3(rr_{B_{4}}^{**},r_-^l,r_-^r,r_-^r).$

The complete classification of case $B_4$ can be given by adjusting the initial value $r_+^l$, which is, more or less, similar to case $A_1$ and is described in Fig. 12.

Fig. 12

(Color online) Distributions of Riemann invariants for all the possible cases in Case $B_4$: a the characteristic velocity $r_+^l$ is chosen to divide the corresponding region into two smaller regions; b the characteristic velocity $r_+^l$ is chosen to equal to the boundary velocities $r_{B_{4}}^*, r_{B_{4}}^{**}$, $rr_{B_{4}}^*$ and $rr_{B_{4}}^{**}$

The classification of solutions and region distributions for the subcases in Fig. 12 are given in Table 8, which shows that all cases evolve at least five regions. It is worth mentioning that three DSWs are separated by a plateau in case $B_{4.2}$, two DSWs are separated by a plateau and a rarefaction wave in case $B_{4.4}$, and three DSWs are separated by a rarefaction wave in case $B_{4.6}$. These are the main features in case $B_{4}$.

Table 8 Classification of solutions and region distributions for the subcases listed in Table 5 under condition $B_4$

We only take case $B_{4.4}$ as an example to exhibit the phenomena of wave breaking under the condition $B_2$. Figure 13 shows that the solution for case $B_{4.4}$ consists of six regions with two different patterns of the oscillating DSWs separated by a plateau and a rarefaction wave, in which the left oscillating pattern corresponding to {DSW-I} and the right pattern corresponding to {DSW-VIII}. Figure 13b indicates that the result of Whitham modulation theory agrees well with the direct numerical simulations. The boundary velocities in each region are given below.

Fig. 13

(Color online) Example of self-similar solution of case $B_{4.4}$ with initial condition $r_+^l=0.8,r_+^r=-0.3,r_-^l=-1,r_-^r=-1.2$ at time $t=1$: a distributions of Riemann invariants; b comparison of numerical simulation (dash blue line) with analytical solution (solid red line) of the defocusing cmKdV equation (1)

• (1) For $x/t\leqslant v_3(r_+^l,r_-^l,r_-^r,r_-^r)$, the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^l,\quad r_-=r_-^l. \end{aligned}$$

• (2) For $v_3(r_+^l,r_-^l,r_-^r,r_-^r)<x/t< v_3(r_+^l,r_-^l,r_-^l,r_-^r)$, the solution is in the {DSW-I} region with Riemann invariants

  $$\begin{aligned} \lambda _1=r_+^l,\quad \lambda _2=r_-^l, \quad \frac{x}{t}=v_3(r_+^l,r_-^l,\lambda _3,r_-^r),\quad \lambda _4=r_-^r. \end{aligned}$$

• (3) For $v_3(r_+^l,r_-^l,r_-^l,r_-^r)\leqslant x/t \leqslant v_1(r_+^l,r_+^r,r_+^r,r_-^r)$, the solution is in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^l,\quad r_-=r_-^r. \end{aligned}$$

• (4) For $v_1(r_+^l,r_+^r,r_+^r,r_-^r)<x/t<v_1 (r_{B4}^*,r_+^r,r_+^r,r_-^r)$, the solution is in the {RW-II} region with Riemann invariants

  $$\begin{aligned} r_+= -\frac{1}{5}r_-^r+\frac{1}{15}\sqrt{-36(r_-^{r})^2-30 \cdot \frac{x}{t}},\quad r_-=r_-^r. \end{aligned}$$

• (5) For $v_1(r_{B4}^*,r_+^r,r_+^r,r_-^r)\leqslant x/t \leqslant v_1(r_{B4}^{**},r_{B4}^{**},r_+^r,r_-^r)$, the solution is in the {DSW-VIII} region with Riemann invariants

  $$\begin{aligned} \frac{x}{t}=v_1(\lambda _1,\lambda _2,r_+^r,r_-^r), \quad \frac{x}{t}=v_2(\lambda _1,\lambda _2,r_+^r,r_-^r), \quad \lambda _3=r_+^r,\quad \lambda _4=r_-^r. \end{aligned}$$

• (6) For $x/t>v_1(r_{B4}^{**},r_{B4}^{**},r_+^r,r_-^r)$, the solution is again in the plateau region with Riemann invariants

  $$\begin{aligned} r_+=r_+^r,\quad r_-=r_-^r. \end{aligned}$$

  Notice that the directions of leading and trailing edges of two DSWs in Fig. 13 are opposite.

4.3 Case C. $r_+^l>r_-^l>r_+^r>r_-^r$

This is an special case, in which the vacuum region and genus-2 region appear. The vertex $r^*_{BCD}$ in equation (91) is a vital point in classifying the solution of the Riemann problem in Case C. In what follows, the subcase that only rarefaction wave emerges is discussed firstly.

4.3.1 Subcase $C_1$: $r_-^r \geqslant r^*_{BCD}$

This case only contains one class of solution, where one vacuum and two rarefaction waves are produced. Figure 14 shows that the solution for case $C_{1}$ consists of five regions, where two different patterns of rarefaction waves are separated by a vacuum, see also Table 10. Figure 14b indicates that the result of Whitham modulation theory agrees well with the direct numerical simulations. The formation of these profiles is much similar to the case under $B_3$ except the intermediate plateau connecting the two rarefaction waves is replaced with the vacuum region. This result can also be observed in the Riemann problem of the defocusing NLS equation (El et al. 1995). The boundary velocities in this case is omitted since it can also be easily derived in the same way.

Fig. 14

(Color online) Example of the self-similar solution of case $C_{1}$ with initial condition $r_+^l=2,r_+^r=1,r_-^l=1.5,r_-^r=0$ at time $t=1$: a distributions of Riemann invariants; b comparison of numerical simulation (dash blue line) with analytical solution (solid red line) of the defocusing cmKdV equation (1)

4.3.2 Subcase $C_2$: $r_+^r > r^*_{BCD}$ and $r_-^r < r^*_{BCD}$

The inequality $r_-^r < r^*_{BCD}$ indicates that $r_-^r<-\frac{1}{5}r_+^r$, which provides a condition of producing DSW. The complete classification under condition $C_2$ can be obtained by adjusting the value of $r_-^r$ in comparison with $r^*_{BCD}$ and $r_{C2}^{*}=r_+^r-r_-^r-\sqrt{5(r_+^r)^2 -4r_+^rr_-^r},$ see Table 9.

Table 9 All of possible solution cases under conditions $C_{1}$ and $C_{2}$

The classification of solutions and region distributions for the subcases listed in Table 9 are given in Table 10. It is observed that there are plateau, vacuum, rarefaction waves, and DSW in case $C_{2.3}$, which will be displayed in Fig. 15 below. Moreover, in cases $C_{2.5}-C_{2.7}$ there exist oscillatory regions, which are also genus-1 DSW regions but they do not belong to any basic structures of DSW shown in Sect. 3.2. Notice that oscillatory region is a bit similar to the region that found in El et al. (1995), in which the amplitude of the oscillation is a constant $(a=0)$ there. However, different from the results in El et al. (1995) the oscillation in this work is a variable . It seen in the oscillatory region that $a \rightarrow 0$ when $\lambda _3\rightarrow \lambda _4$ in the left hand side, or $\lambda _1 \rightarrow \lambda _2$ in the right hand side. This implies that the left half part of oscillatory region can be interpreted as the trailing edge of {DSW-IX} and the symmetry with respect to vertical axis of trailing edge of {DSW-X} for the right part.

Table 10 Classification of solutions and region distributions for the subcases listed in Table 5 under condition $C_2$

Figure 15 shows that the solution for case $C_{2.3}$ with initial condition $r_+^l=1.2,r_+^r=0.5,r_-^l=0.8,r_-^r=-0.7$ consists of six regions including plateau, vacuum, rarefaction waves and DSW. Figure 15b indicates that the analytical solution from Whitham modulation theory agrees well with the direct numerical simulations. From left to right, the fourth region is a rarefaction wave followed from equation (57), where

$$\begin{aligned} v_3|_{\lambda _1=\lambda _2}=-\frac{15}{2}r_3^2-3r_4r_3-\frac{3}{2}r_4^2, \quad v_4|_{\lambda _1=\lambda _2}=-\frac{3}{2}r_3^2-3r_4r_3 -\frac{15}{2}r_4^2. \end{aligned}$$

The fifth region, the dispersive shock wave region, can be regard as the limitation state of {DSW-VI} for $\lambda _2\rightarrow \lambda _3$.

Fig. 15

(Color online) Example of the self-similar solution of case $C_{2.3}$ with initial condition $r_+^l=1.2, r_+^r=0.5, r_-^l=0.8, r_-^r=-0.7$ at time $t=1$: a distributions of Riemann invariants; b comparison of numerical simulation (dash blue line) with analytical solution (solid red line) of the defocusing cmKdV equation (1)

4.3.3 Subcase $C_3$: $r_+^r\leqslant r^*_{BCD}$

The condition $r_+^r\leqslant r^*_{BCD}$ shows that $r_+^r\leqslant -\frac{1}{5}r_-^r$ which will result in the collision of two DSWs as seen in Fig. 13. Moreover, in this case, certain region can not be described by the genus-1 Whitham equations (56) with (57) definitely. Thus the genus-2 regions may emerge, which should be analyzed by the genus-2 Whitham equations (64) with (66). The initial condition of Fig. 16 is $r_+^l=0,r_+^r=-0.8,r_-^l=-0.5,r_-^r=-1.2$. It is shown that the solution for case $C_{3}$ consists of five regions with two different genus-1 DSWs separated by a genus-2 DSW. Fig. 16b demonstrates the result of direct numerical simulations, which displays that the middle genus-2 DSW region oscillates rapidly, which further verifies the result of Whitham modulation theory shown in Fig. 16a. The boundaries of the genus-2 DSW region are $x=s_1t$ and $x=s_2t$ with $s_1=-7.3$ and $s_2=-4.5$. In fact, $s_1$ and $s_2$ are characteristic velocities obtained by the genus-2 Whitham equations (64) with (66). The new feature of middle genus-2 DSW region can be explained as follows: Two genus-1 dispersive shock waves (genus-1 undular bores) move toward each other and overlap in region $[s_1, s_2]$ to generate a genus-2 dispersive shock wave that can be described by the two-phase solution of the defocusing cmKdV equation (1).

This work does not present the detailed description of the genus-2 DSW region but give the appropriate solution for the genus-2 averaged Whitham equation

$$\begin{aligned} \frac{\partial \lambda _i}{\partial t}+v_i(\lambda _1, \lambda _2, \lambda _3,\lambda _4, \lambda _5,\lambda _6) \frac{\partial \lambda _i}{\partial x}=0,\quad i=1,2,\ldots , 6, \end{aligned}$$

where $\lambda _1>\lambda _2>\lambda _3>\lambda _4>\lambda _5>\lambda _6$. The self-similar solution of this equation in the region $[s_1, s_2]$ is expressed by

$$\begin{aligned}&\lambda _1=r_+^l,\quad \lambda _2=r_-^l, \quad \lambda _5=r_+^r, \quad \lambda _6=r_-^r, \\&v_3(r_+^l, r_-^l,\lambda _3, \lambda _4, r_+^r,r_-^r)=\tau =v_4(r_+^l, r_-^l, \lambda _3, \lambda _4,r_+^r,r_-^r), \end{aligned}$$

from which the the boundary characteristic velocities $s_1$ and $s_2$ are

$$\begin{aligned} s_1=v_3(r_+^l, r_-^l,\lambda ^{*}_3, r_+^r,r_+^r,r_-^r), \quad s_2=v_4(r_+^l, r_-^l,r_-^l, \lambda ^{*}_4,r_+^r,r_-^r). \end{aligned}$$

Fig. 16

(Color online) Example of the self-similar solution of case $C_{3}$ with initial condition $r_+^l=0, r_+^r=-0.8, r_-^l=-0.5, r_-^r=-1.2$: a distributions of Riemann invariants; b direct numerical simulation, where the red lines represent the boundary of each region corresponding to the black dotted line in a

4.4 Case D. $r_+^r>r_+^l>r_-^l>r_-^r$

More genus-2 DSW regions emerge in this case because of the collisions of the DSWs. As before, the vertex $r^*_{BCD}$ in equation (91) is still an important point in classifying the solution of the Riemann problem for Case D.

4.4.1 Subcase $D_1$: $r_-^l > r^*_{BCD}$

The genus-2 DSW region also appears due to the collision of {DSW-II} and {DSW-V} under condition $D_1$. Since the genus-2 DSWs are more complicated than the genus-1 DSWs, this work only gives the classification cases under condition $D_1$ and lists the solution regions only including genus-1 DSWs; see Tables 11 and 12, respectively. Define three critical points $r_{D_{1}}^{*}, r_{D_{1}}^{**}$ and $r_{D_{1}}^{***}$ in the following way:

• Let $r_{D_{1}}^{*}$ solve the implicit equation $v_2(r_+^r,r_-^l,r_{D_{1}}^{*},r_-^r)=v_3(r_+^r,r_-^l, r_{D_{1}}^{*},r_-^r)$.

• Let $r_{D_{1}}^{**}$ solve the implicit equation $v_2(r_+^r,r_+^l,r_+^l,r_-^l)=v_3(r_+^r,r_-^l,r_{D_{1}}^{**}, r_{D_{1}}^{**})$ on the interval $(-\infty ,r_{D_{1}}^{*}).$

• Let $r_{D_{1}}^{***}$ solve the implicit equation $v_2(r_+^r,r_+^r,r_+^l,r_-^l)=v_3(r_+^r,r_+^l,r_{D_{1}}^{***}, r_{D_{1}}^{***})$ on the interval $(-\infty ,r_{D_{1}}^{**})$.

Table 11 All of possible solution cases under condition $D_1$

The complete classification of case $D_1$ can be given by adjusting the initial value $r_-^r$, which described in Table 11. The region remarked by $D_{1.g_{2}}$ is the place where the genus-2 region appears. The classification of solutions and region distributions for the subcases $D_{1.1}$-$D_{1.5}$ in Table 11 are given in Table 12.

Table 12 Classification of solutions and region distributions for the subcases $D_{1.1}$-$D_{1.5}$ listed in Table 11 under condition $D_1$

We only take case $D_{1.2}$ as an example to exhibit the phenomena of wave breaking under the condition $D_1$. Figure 17 shows that the solution for case $D_{1.2}$ consists of six regions with two different patterns of the oscillating DSWs separated by a plateau and a rarefaction wave. Figure 17b indicates that the analytical result of Whitham theory agrees well with the direct numerical simulations.

Fig. 17

(Color online) Example of self-similar solution of case $D_{1.2}$ with initial condition $r_+^l=0.8, r_+^r=1.2, r_-^l=0.5,r_-^r=-0.5$ at time $t=1$: a distributions of Riemann invariants; b comparison of numerical simulation (dash blue line) with analytical solution (solid red line) of the defocusing cmKdV equation (1)

4.4.2 Subcase $D_2$: $r_-^l \leqslant r^*_{BCD}$ and $r_+^l > r^*_{BCD}$

Two genus-1 DSWs and more genus-2 DSW are produced in this case. By adjusting the parameters $r_-^l$ and $r_-^r$ one can control the movement styles of the two genus-1 DSWs including the case when they collide with each other, where the genus-2 DSW appears. Define two critical points $r_{D_{2}}^*$ and $r_{D_{2}}^{**}$ as follows:

• Let $r_{D_{2}}^{*}$ solve the implicit equation $v_2(r_+^r,r_+^r,r_+^l,r_-^l)=v_3(r_+^l,r_-^l,, r_{D_{2}}^{*},r_{D_{2}}^{*})$ in the interval $(-\infty , r_{D_{2}}^{**})$, where $r_{D_{2}}^{**}$ can be formulated below.

• Let $r_{D_{2}}^{**}$ solve the implicit equation $v_2(r_+^r,r_+^l,r_+^l,r_-^l)=v_3(r_+^r,r_-^l,r_{D_{2}}^{**}, r_{D_{2}}^{**})$ in the interval $(-\infty , r_{-}^{l})$.

The complete classification of case $D_2$ can be given by classifying the initial values $r_-^l$ and $r_-^r$, which described in Table 13. Similar to Table 11, the region remarked by $D_{2.g_{2}}$ also represents the genus-2 region.

Table 13 All of possible solution cases under condition $D_2$

The classification of solutions and region distributions for the subcases $D_{2.11}, D_{2.21}, D_{2.54}$ and $D_{2.55}$ listed in Table 13 are given in Table 14. It is seen that there are two genus-1 DSWs in each case and the rarefaction waves don’t emerge here. The density profiles are very similar to case $D_1$ so we don’t display the Riemann invariants and the comparison of numerical simulation with Whitham modulation theory here.

Table 14 Classification of solutions and region distributions for the subcases $D_{2.11}, D_{2.21}, D_{2.54}$ and $D_{2.55}$ listed in Table 13 under condition $D_2$

4.4.3 Subcase $D_3$: $r_+^l < r^*_{BCD}$

Notice that the distribution of Riemann invariants in this case is symmetric with case $D_1$ and the density profile $\rho$ is exactly the same as case $D_1$. In fact, the symmetry of {DSW-II}, {DSW-VI}, {RW-IV}, {RW-I} in case $D_3$ is {DSW-I}, {DSW-VI}, {RW-VI}, {RW-II} in case $D_1$, respectively. Thus we omit the whole description here for simplicity.

4.5 Case E. $r_+^r>r_+^l>r_-^r>r_-^l$

The classification of solutions under condition E is the same as that in case B. Indeed, if setting $r_{+E}^r=-r_{-B}^r, r_{+E}^l=-r_{-B}^l, r_{-E}^r=-r_{+B}^r, r_{-E}^l=-r_{+B}^l$ where subscripts B and E represent the parameters under conditions B and E, respectively, we have immediately that $r_{+B}^l>r_{+B}^r>r_{-B}^l>r_{-B}^r$, which exactly coincides with the condition in case B. Moreover, the same symmetry can also be found in Whitham velocities as follows:

$$\begin{aligned} v_j(-\lambda _4,-\lambda _3,-\lambda _2,-\lambda _1)&=V-(-s_1+2\lambda _j)(\partial \ln L/\partial \lambda _j)^{-1}\nonumber \\&=V-(-s_1-2\lambda _i)(-\partial \ln L/\partial \lambda _i)^{-1}\nonumber \\&=V-(s_1+2\lambda _i)(\partial \ln L/\partial \lambda _i)^{-1}\nonumber \\&=v_i(\lambda _1,\lambda _2,\lambda _3,\lambda _4),\quad i=5-j. \end{aligned}$$

(96)

This proves our claim, so the detailed analysis for case E can omit for simplicity.

4.6 Case F. $r_+^r>r_-^r>r_+^l>r_-^l$

Similarly, the classification of solutions in this case is the same as that in case C. Setting $r_{+F}^r=-r_{-C}^r, r_{+F}^l=-r_{-C}^l, r_{-F}^r=-r_{+C}^r, r_{-F}^l=-r_{+C}^l$ yields $r_{+C}^l>r_{-C}^l>r_{+C}^r>r_{-C}^r$, which exactly coincides with the condition in case C.

5 Conclusion

The Riemann problem of the defocusing cmKdV equation has been investigated by Whitham modulation theory. The periodic solutions along with the Whitham equations in diagonal form are derived by means finite-gap integration approach. The complete classification for the general step-like initial data of the defocusing cmKdV equation have been discussed in six cases. Some new basic wave structures that do not appear in the NLS equation are found. The solution regions in the NLS equation consisted of five parts, while they can contain six regions in the defocusing cmKdV equation. The typical density profiles in each class have been compared with the direct numerical simulation with remarkable agreement.

During the analysis of the initial value problem of the defocusing cmKdV equation, an asymptotical description of the slowly modulated waves is given by solving the Whitham equations. However, the Whitham method is not the only way to investigate the asymptotics of the nonlinear integrable systems and the Riemann-Hilbert method, especially the Deift–Zhou nonlinear steepest descent approach (Deift and Zhou 1993) is a rigorous path to explore the asymptotics of various integrable systems, such as the nonlinear Schrödinger equation (Boutet de Monvel et al. 2020, 2021; Kotlyarov and Minakov 2019), KdV equation (Andreiev et al. 2016), real modified KdV equation (Grava and Minakov 2020), Camassa-Holm equation (Chang et al. 2016) and Toda lattice (Egorova et al. 2018). The results in this work will inspire further exploration of the long-time behaviors of the defocusing cmKdV equation based on Deift–Zhou nonlinear steepest descent approach (Deift and Zhou 1993).