Systematic generation of higher-order solitons and breathers of the Hirota equation on different backgrounds

Abstract

We investigate the systematic generation of higher-order solitons and breathers of the Hirota equation on different backgrounds. The Darboux transformation is used to construct proper initial conditions for dynamical generation of high-intensity solitons and breathers of different orders on a uniform background. We provide expressions for the Lax pair generating functions and the procedure for calculating higher-order solutions when Jacobi elliptic functions are the background seed solutions of the Hirota equation. We confirm that the peak height of each soliton or breather in the nonlinear Darboux superposition adds linearly, to form the intensity maximum of the final solution.

1 Introduction

In this work, we study the solitons and breathers of the Hirota equation (HE) [1,2,3]

$$\begin{aligned} i\frac{{\partial \psi }}{{\partial x}} + \frac{1}{2}\frac{{{\partial ^2}\psi }}{{\partial {t^2}}} + {{\left| \psi \right| }^2}\psi - i\alpha \left( \frac{{{\partial ^3}\psi }}{{\partial {t^3}}} + 6{{\left| \psi \right| }^2}\frac{{\partial \psi }}{{\partial t}}\right) = 0\nonumber \\ \end{aligned}$$

(1.1)

on various backgrounds. Here, x and t are the retarded time in the moving frame and the transverse spatial variable, respectively, while $\psi$ denotes the slowly varying wave envelope.

The fundamental solutions of HE are of great importance in many fields dealing with nonlinear propagating waves, ranging from oceanography to fiber-optics [4]. HE contains only one real parameter $\alpha$, related to the transverse velocity of the wave. Setting $\alpha =0$ reduces HE to the nonlinear Schrödinger equation (NLSE) [5,6,7,8]. The necessity of extending the NLSE to the HE and other higher-order members of the NLS hierarchy of equations arises from the need to understand propagation of ultrashort pulses through optical fibers [9, 10]. A more accurate description of the real physical systems requires additional terms, to account for self-steepening, self-frequency shift, and third-order dispersion [11,12,13,14,15]. It was shown that these third-order terms are of significant importance in pulse-deforming phenomena [16] and supercontinuum generation [17].

In this work, we show that the solitons and breathers of high intensity can be dynamically generated on zero and constant backgrounds by a direct numerical integration of an initial wave function extracted from the Darboux transformation (DT), with appropriate boundary conditions. We also find new Hirota solutions, derived by DT on an elliptic background. In all the cases, we demonstrate that the peak height formula [(PHF—Eq. (4.1)] gives the correct peak value of a higher-order solution of HE, as a linear sum of peak heights of its DT constituents [18].

Solitons, breathers and rational solutions of HE have been previously studied using DT in Refs. [2, 19,20,21,22]. Additional interest in NLSE and Hirota breathers arises from their relationship with the rogue waves (RWs) [23,24,25,26,27] that can be modeled by higher-order breathers concatenated from the first-order ones via DT iterations [24, 27,28,29]. An interesting finding in Hirota [30] and quintic NLSE [31] is the breather-to-soliton conversion, obtained for specific values of the breather eigenvalues—a feature not possible in the ordinary cubic NLSE. The study of NLSE breathers on a Jacobi elliptic function (JEF, see [32]) background was previously done in Refs. [33, 34]. A recent discovery of an infinite hierarchy of extended nonlinear Schrödinger equations [35,36,37,38] has also revealed that they may only have dnoidal and cnoidal backgrounds in their solutions. To the extent that HE can better describe light propagation in an optical fiber, this work should also contribute to a better understanding of optical RW generation.

The paper is organized as follows: In Sect. 2, we analyze the basic Hirota solutions and show the usefulness of DT for numerical simulations. In Sect. 3, we provide a derivation of the Lax pair generating functions and higher-order solutions when cnoidal and dnoidal functions are taken as seed solutions in DT. In Sect. 4, the PHF for HE is presented. In Sect. 5, we present analytical and numerical results concerning higher-order solitons and breathers on different backgrounds, utilizing the periodicity of ABs and breather-to-soliton conversion. In Sect. 6, we summarize the findings and importance of our results. The general DT scheme is provided in “Appendix.”

2 Solitons and breathers of the Hirota equation

Darboux transformation can be used to obtain higher-order soliton/breather and rogue wave solutions of the Hirota equation on different backgrounds. The key steps in obtaining these nonlinear structures dynamically, for discretized x and t intervals, are: 1. use DT to calculate the initial wave function at a particular value of the propagation variable $x=x_0$, i.e., to find $\psi \left( {x_0,t} \right)$, 2. analyze the wave function values at the edges of the t-interval in order to pick appropriate boundary conditions, and 3. apply a convenient numerical algorithm to calculate the wave function over the entire xt-grid (in this case, the fourth-order Runge–Kutta method).

To numerically generate higher-order solitons, we calculate the initial wave function from a zero seed: $\psi _0=0$. The appropriate boundary condition can be found from the first-order DT solution, characterized by a pure imaginary eigenvalue $\lambda =i\nu$ (without loss of generality, we can assume zero shifts along the x and t axes),

$$\begin{aligned} \psi \left( {x,t} \right) = \frac{{2\nu {e^{2i{\nu ^2}x}}}}{{\cosh \left[ {2\nu \left( {t + 4\alpha {\nu ^2}x} \right) } \right] }}. \end{aligned}$$

(2.1)

From (2.1), it follows that the first-order soliton has a single peak with the amplitude $\left| {\psi \left( {0,0} \right) } \right| = 2\nu$, and its intensity decays exponentially, following the $\cosh$ function. Consequently, the Neumann boundary condition $\frac{{\partial \psi }}{{\partial t}}=0$ can be used in numerical evaluations of higher-order Hirota solitons. It should be noted that the Hirota coefficient $\alpha$ appears only in the denominator, giving a tilt to the soliton propagation.

A higher-order Akhmediev breather (AB) of HE is constructed by DT when a plane-wave seed, $\psi _0=e^{ix}$, is utilized. The first-order AB is given by [19]

$$\begin{aligned} \psi (x,t) = \left[ {\frac{{2(1 - 2a)\cosh (\beta x) + i\beta \sinh (\beta x)}}{{\cosh (\beta x) - {\sqrt{2a}} \cos \left[ {\omega (t + vx)} \right] }} - 1} \right] {e^{ix}}, \end{aligned}$$

(2.2)

where a is an arbitrary parameter in the range $0< a < 1/2$. The frequency $\omega$, the growth factor $\beta$, and the parameter v determining the tilt of the breather in the xt-plane are given solely in terms of a:

$$\begin{aligned} \omega = 2{\sqrt{1 - 2a}},\quad \beta = \omega {\sqrt{2a}},\quad v = 2\alpha (1 + 4a).\nonumber \\ \end{aligned}$$

(2.3)

The difference between NLSE and Hirota breathers originates only from the parameter v. NLSE solutions are obtained when $v=0$ ($\alpha =0$).

From Eq. (2.2), one can see that Hirota’s AB is periodic along t-axis, with the period

$$\begin{aligned} L = \frac{2\pi }{\omega }=\frac{\pi }{{{\sqrt{1 - 2a}}}}. \end{aligned}$$

(2.4)

Consequently, in contrast to solitons, periodic boundary conditions are required for dynamical generation of higher-order ABs. Briefly, one assumes that each of N eigenvalues is purely imaginary $\lambda _j=i\nu _j$. The parameter $a_j$ of each breather constituent is connected to the imaginary part of its eigenvalue via the relation

$$\begin{aligned} \nu _j=\sqrt{2a_j}. \end{aligned}$$

(2.5)

Due to periodicity, it is natural to choose higher-order breathers that retain the same periodic length L of the first-order breather, having $\omega _j=j\omega$ $\left( a_1\equiv a\text {, }\omega _1\equiv \omega \right)$. From Eqs. (2.3) to (2.5), it follows

$$\begin{aligned} \lambda _j=i\sqrt{2\left( j^2 a-\frac{j^2-1}{2}\right) }. \end{aligned}$$

(2.6)

If, for any j, $\omega _j$ is not an integer multiple of $\omega$, then the periodicity of the original breather is destroyed. For more details, see [18]. The procedure for generating higher-order ABs is to take an initial a, calculate other eigenvalues according to Eq. (2.6), and apply the DT.

3 New solutions of the Hirota equation on dnoidal/cnoidal backgrounds

In recent articles [35, 36], new exact solutions of HE that include JEFs are presented:

$$\begin{aligned} {\psi _{\text {dn}}}(x,t)= & {} c e^{i\phi x} \text {dn}\left( {ct + vx,m} \right) , \end{aligned}$$

(3.1)

$$\begin{aligned} {\psi _{\text {cn}}}(x,t)= & {} \frac{c}{{\sqrt{2} }}\sqrt{s + 1} e^{i\phi x} \cdot \nonumber \\&\quad \cdot \text {cn}\left( \sqrt{s} ct + \sqrt{s} vx,m = \frac{1}{2} + \frac{1}{{2s}}\right) .\nonumber \\ \end{aligned}$$

(3.2)

Here, m is the elliptic modulus squared, c, s, $\phi$ are constants, and v is the speed of the wave that depends on $\alpha$. By setting $c=2\nu$ and $m=1$, we reproduce the soliton solution of Eq. (2.1). Any of these functions may be used as a seed for the Darboux transformation. This motivated us to examine how higher-order solutions can be obtained on dnoidal/cnoidal backgrounds. All results in this section are derived using the general Darboux transformation scheme for Hirota equation given in “Appendix.”

3.1 Derivation of the DT Lax pair generating functions and higher-order solutions for $\psi _0=\psi _{\text {dn}}$

Here the seed function $\psi _0$ is given by Eq. (3.1) with $\phi = \left( {1 - \frac{m}{2}} \right) {c^2}$ and $v = \left( {2 - m} \right) \alpha {c^3}$ [35]. The Lax pair equations consist of a system of four coupled linear differential equations for $r_t$, $s_t$, $r_x$, and $s_x$. Here, the subscripts stand for partial derivatives. To solve these equations, we introduce the traveling variable

$$\begin{aligned} u=ct+xv \end{aligned}$$

(3.3)

and continue the analysis with x and u. Now, the t-derivatives are replaced by the u-derivatives: $\frac{{\partial \psi }}{{\partial t}} = c^{2}{e^{i\phi x}}\frac{{\partial dn(u)}}{{\partial u}}$ and $\frac{{{\partial ^2}\psi }}{{\partial {t^2}}} = {c^3}{e^{i\phi x}}\frac{{{\partial ^2}dn(u)}}{{\partial {u^2}}}$. We take the ansatz:

$$\begin{aligned}&r(x,u) = g(x,u){e^{ - i\phi x/2}} \text { and }\nonumber \\&\quad s(x,u) = h(x,u){e^{ + i\phi x/2}}. \end{aligned}$$

(3.4)

The matrix elements of U and V from Eq. (7.4) in “Appendix” can be calculated by substituting $\psi _0$ into Eq. (7.5). From Eq. (7.4), the u-evolutions are obtained

$$\begin{aligned} g_u= & {} i\frac{\lambda }{c}g + i \cdot \text {dn}(u)h, \end{aligned}$$

(3.5)

$$\begin{aligned} h_u= & {} i \cdot \text {dn}(u) g - i \frac{\lambda }{c} h. \end{aligned}$$

(3.6)

The equations for the x-derivative are a bit more complicated, but actually not needed for numerical calculations. It is only necessary to obtain the x-derivative at $u=0$, to initiate the numerical integration of r and s. Evolution along the x-axis at $u=0$ follows from Eq. (7.3)

$$\begin{aligned} {g_x}= & {} iAg + iBh, \end{aligned}$$

(3.7)

$$\begin{aligned} {h_x}= & {} iBg - iAh, \end{aligned}$$

(3.8)

where constants A and B are

$$\begin{aligned} A&= \left( {{\lambda ^2} - \frac{mc^2}{4}} \right) ( 1-4 \alpha \lambda ), \end{aligned}$$

(3.9)

$$\begin{aligned} B&= c\lambda (1-4\alpha \lambda ). \end{aligned}$$

(3.10)

By differentiating and combining Eqs. (3.7) and (3.8) one obtains:

$$\begin{aligned} {g_{xx}} = - ({A^2} + {B^2})g \end{aligned}$$

(3.11)

and

$$\begin{aligned} {h_{xx}} = - ({A^2} + {B^2})h. \end{aligned}$$

(3.12)

Thus, the exact solution for g or h is $h(x,u = 0) = {C_1}{e^{i\kappa \lambda x}} + {C_2}{e^{ - i\kappa \lambda x}}$. Here, $\kappa = \frac{{\sqrt{{A^2} + {B^2}} }}{\lambda }$, while $C_1$ and $C_2$ are arbitrary constants. We can embed the phase shift of $\pi /4$ into $h(x,u=0)$, to center the solution at the origin. Next, we take ${C_1} = {e^{ - i\pi /4 - i\chi }}$ and ${C_2} = {e^{ + i\pi /4 + i\chi }}$. Therefore: $h(x,u = 0) = 2\cos ( - \chi + \kappa \lambda x - \pi /4)$. If we substitute $h(x,u = 0)$ in (3.8), we will get the solution $g(x,u = 0) = 2i\sin ( - \chi + \kappa \lambda x - \pi /4 + \varphi )$. The idea is to make the solutions for g and h symmetrical and in agreement with the published articles [27, 28, 33]. We can choose $-\chi +\varphi =+\chi$, and therefore $\chi = \frac{1}{2}\varphi = \frac{1}{2}\arccos \frac{{\kappa \lambda }}{B}$. The solution for g is $g(x,u = 0) = 2i\sin ( + \chi + \kappa \lambda x - \pi /4)$.

Now, we go back to the Lax pair generating functions, introducing N eigenvalues and x-shifts (t-axis shifts are ignored since they provide negligible advance in the understanding, alongside with numerical complications). The quantities $\kappa$ and $\chi$ are labeled with the j index. Hence, the Lax pair generating functions for eigenvalue $\lambda _j$ and shift $x_{0j}$ $\left( 1 \le j \le n\right)$ are:

$$\begin{aligned}&{r_{1j}}(x,u = 0)\nonumber \\&\quad = 2i{e^{ - i{\phi }x/2}}\sin [ + {\chi _j} + {\kappa _j}{\lambda _j}(x - {x_{0j}}) - \pi /4] \nonumber \\\end{aligned}$$

(3.13)

$$\begin{aligned}&{s_{1j}}(x,u = 0) \nonumber \\&\quad = 2{e^{ + i{\phi }x/2}}\cos [ - {\chi _j} + {\kappa _j}{\lambda _j}(x - {x_{0j}}) - \pi /4],\nonumber \\ \end{aligned}$$

(3.14)

where

$$\begin{aligned}&{\kappa _j} = \left( {1 - 4\alpha {\lambda _j}} \right) \sqrt{c^{2}+\left( \lambda _{j}-\frac{mc^{2}}{4\lambda _{j}}\right) ^{2}} \end{aligned}$$

(3.15)

$$\begin{aligned}&{\chi _j} = \frac{1}{2}\arccos \frac{\kappa _j}{c\left( 1- 4\alpha \lambda _{j} \right) }.\nonumber \\ \end{aligned}$$

(3.16)

The remaining task is to find the $r_{1,j}$ and $s_{1,j}$ functions for any u. The evolution equations are given by expressions (3.5) and (3.6). We can solve them only numerically, for instance using the fourth-order Runge–Kutta method. We write $u = ct + xv = 0$ and for a given x, we find $t=-\frac{v}{c}x$. We start from this t value and numerically evolve $du = cdt$ (in both directions) to find $r_{1,j}(x,u)$ and $s_{1,j}(x,u)$. Next, we shift these values from the (x, u) to the (x, t) frame, to get all values on the grid: r(x, t) and s(x, t). Finally, by employing the well-known recursive relations in Eq. (7.2) and iterative relations in Eq. (7.1), the Nth-order solution, $\psi _N(x,t)$, can be calculated. In the special case of $\alpha =0$ and $c=1$, the Hirota Lax pair generating functions and the final Nth-order wave function are equivalent to the NLSE solutions described in Section 2.2 of [33].

3.2 Derivation of the DT Lax pair generating functions and higher-order solutions for $\psi _0=\psi _{\text {cn}}$

The analysis is analogous to the previous case. The seed function $\psi _0$ is given by Eq. (3.2), where $\phi = \frac{1}{2}{c^2}$ and $v = \alpha {c^3}$. The substitution

$$\begin{aligned} u = \sqrt{s} \left( {ct + xv} \right) \end{aligned}$$

(3.17)

gives $\frac{{\partial \psi }}{{\partial t}} = {c^2}\sqrt{\frac{{s\left( {s + 1} \right) }}{2}} {e^{i\phi x}}\frac{{\partial \text {cn}(u)}}{{\partial u}}$ and $\frac{{{\partial ^2}\psi }}{{\partial {t^2}}} = {c^3}s\sqrt{\frac{{s + 1}}{2}} {e^{i\phi x}}\frac{{{\partial ^2}\text {cn}(u)}}{{\partial {u^2}}}$. The ansatz is now:

Table 1 Seed solutions for Hirota equation and the corresponding Lax pair generating functions ($r_{1,j}\equiv r$ and $s_{1,j}\equiv s$)

$$\begin{aligned}&r(x,u) = g(x,u){e^{ - i\phi x/2}} \text { and } \nonumber \\&\quad s(x,u) = h(x,u){e^{ + i\phi x/2}}. \end{aligned}$$

(3.18)

The evolution along u-axis (from the U matrix) is

$$\begin{aligned} {g_u}= & {} \frac{{i\lambda }}{{c\sqrt{s} }}g + \frac{{i\sqrt{s + 1} }}{{\sqrt{2s} }} \text {cn}(u) h, \end{aligned}$$

(3.19)

$$\begin{aligned} {h_u}= & {} \frac{{i\sqrt{s + 1} }}{{\sqrt{2s} }}\text {cn}(u)g - \frac{{i\lambda }}{{c\sqrt{s} }}h. \end{aligned}$$

(3.20)

Again, we restrict further analysis only to $u=0$. The evolution along the x-axis is:

$$\begin{aligned} {g_x}= & {} iAg + iBh, \end{aligned}$$

(3.21)

$$\begin{aligned} {h_x}= & {} iBg - iAh, \end{aligned}$$

(3.22)

where

$$\begin{aligned} A= & {} \left( 1-4\alpha \lambda \right) \left( \lambda ^2-\frac{1}{4}c^2 s\right) , \end{aligned}$$

(3.23)

$$\begin{aligned} B= & {} \frac{{c\lambda \sqrt{m}}}{{\sqrt{2m - 1} }}\left( {1 - 4\alpha \lambda } \right) . \end{aligned}$$

(3.24)

By differentiating the last two equations, one gets

$$\begin{aligned} {g_{xx}}= & {} - ({A^2} + {B^2})g, \end{aligned}$$

(3.25)

$$\begin{aligned} {h_{xx}}= & {} - ({A^2} + {B^2})h. \end{aligned}$$

(3.26)

The exact solution for h is $h(x,u = 0) = {C_1}{e^{i\kappa \lambda x}} + {C_2}{e^{ - i\kappa \lambda x}}$, with $\kappa = \frac{{\sqrt{{A^2} + {B^2}} }}{\lambda }$. We choose $C_1$ and $C_2$ constants as in the previous case. The final solution for Lax pair generating functions is obtained after the complete sets of eigenvalues and x-shifts are introduced,

$$\begin{aligned}&{r_{1j}}(x,u = 0) \nonumber \\&\quad = 2i{e^{ - i{\phi }x/2}}\sin [ + {\chi _j} + {\kappa _j}{\lambda _j}(x - {x_{0j}}) - \pi /4], \nonumber \\\end{aligned}$$

(3.27)

$$\begin{aligned}&{s_{1j}}(x,u = 0) \nonumber \\&\quad = 2{e^{ + i{\phi }x/2}}\cos [ - {\chi _j} + {\kappa _j}{\lambda _j}(x - {x_{0j}}) - \pi /4],\nonumber \\ \end{aligned}$$

(3.28)

where

$$\begin{aligned} {\kappa _j}= & {} \left( {1 - 4\alpha {\lambda _j}} \right) \cdot \nonumber \\&\cdot \sqrt{m} \sqrt{\frac{{{c^2}}}{{2m - 1}} + \frac{1}{m}{{\left( {\lambda - \frac{{{c^2}}}{{2m - 1}} \cdot \frac{1}{{4\lambda }}} \right) }^2}}, \nonumber \\\end{aligned}$$

(3.29)

$$\begin{aligned} {\chi _j}= & {} \frac{1}{2}\arccos \frac{{{\kappa _j}\sqrt{2m - 1} }}{{c\sqrt{m}\left( {1 - 4\alpha {\lambda _j}} \right) }}. \end{aligned}$$

(3.30)

The Lax pairs generating functions for both $\text {dn}$ and $\text {cn}$ seedings have the same form. However, parameters $\kappa _j$ and $\chi _j$ differ in the two cases, so the r and s functions and the final solutions differ too. An equivalent numerical scheme, proposed in the previous subsection, can be used to calculate r and s functions for any x and u, and then over the entire xt-grid. Again, in the special case of $\alpha =0$ and $c = \sqrt{2m - 1}$, the Hirota Lax pair generating functions and the final Nth-order solution are equivalent to the NLSE solutions described in Section 2.3 of [33].

4 Peak height formula for the Hirota equation

We have stated that the maximum amplitude of the Nth order DT solution is just a linear sum of peak heights of the individual solitons or breathers in the nonlinear superposition. In [34], we have proven that the peak height formula (PHF) holds regardless of the choice of the seed function in DT,

$$\begin{aligned} \psi _N(0,0) = \psi _0(0,0) + \sum \limits _{n = 1}^N {2{\nu _n}}. \end{aligned}$$

(4.1)

Again, the assumption is that all N eigenvalues are purely imaginary $\lambda _j=i\nu _j$ (the real part would not change the PHF). It is proven in [18] that PHF holds when the Lax pair generating functions satisfy the following equation, assuming zero x and t shifts for all eigenvalues:

$$\begin{aligned} s_{1,j}(0,0)=ir_{1,j}(0,0). \end{aligned}$$

(4.2)

In Table 1, explicit expressions for the seed and Lax pair generating functions of HE are given. For brevity, we omit “j” index in N eigenvalues.

Fig. 1

Propagation of the soliton with $\alpha =1/8$: a 2D intensity plot of the first-order DT solution with $\psi _0=0$ and $\lambda =1+0.9i$; b numerical evolution of the initial condition taken at $x=0$ from (a); c 2D intensity plot of the second-order DT solution with $\psi _0=0$, $\lambda _1=1+0.9i$, and $\lambda _2=1+i$; and d numerical evolution of the initial condition taken at $x=-2.5$ from (c). Maximum intensity value $\left| \psi \right| ^2$ is 3.24 in (a) and (b), and 14.44 in (c) and (d), in agreement with PHF

It is easy to check that the relation in Eq. (4.2) applies to all cases given in Table 1.

For the potential applications of soliton propagation in optical fibers modeled by HE, a high soliton intensity is an advantage. As explained in detail in [30], a breather can be converted to a soliton for a specific value of the real part $\mathfrak {R}\{\lambda _j\}$ of all eigenvalues,

$$\begin{aligned} \mathfrak {R}\{\lambda _j\}=\frac{1}{8\alpha }. \end{aligned}$$

(4.3)

According to the PHF, this means that such a “converted” soliton (single or higher order) would have greater amplitude compared to the one obtained with zero seed, having the same set of eigenvalues.

Fig. 2

Dynamically generated third-order breathers. Initial wave function is extracted from DT solution, having a plane-wave seed, at $x=-2.5$. The fundamental period is characterized by $a=0.464$ from Eq. (2.3). Pure imaginary eigenvalues $\lambda _j$, with $1\le j \le 3$, are calculated using Eq. (2.6). The peak intensity is $\left| \psi \right| ^2_{\mathrm{max}}=33.64$, in agreement with PHF. a $\alpha =-0.1$, b $\alpha =+0.1$

Fig. 3

2D intensity plot of dynamically generated fifth-order breather. Initial wave function is extracted from the DT solution, having a plane-wave seed, at $x=-2.5$. The fundamental period is characterized by $a=0.485$ from Eq. (2.3). Pure imaginary eigenvalues $\lambda _j$, with $1\le j \le 5$, are calculated using Eq. (2.6). The peak intensity is $\left| \psi \right| ^2_{\mathrm{max}}=81$, in agreement with PHF. Hirota parameter is $\alpha =0.1$

5 Numerical verification

Here we present numerical simulations of higher-order solitons and breathers on uniform and nonuniform backgrounds and introduce a simple criterion to estimate the solution peak value using the PHF.

5.1 Solutions of Hirota equation on uniform backgrounds

First, we show that the single-soliton solutions, obtained by DT (Fig. 1a) and by dynamical integration (Fig. 1b), are in excellent agreement if one chooses the Neumann boundary conditions. The maximum intensity in both figures is 7.84, in agreement with PHF ($\lambda =1+0.9i$), $7.84=\left( 0+2\cdot 0.9\right) ^2$. The second-order soliton is shown in Fig. 1c (DT) and 1d (direct integration). It is generated from the two single solitons, characterized by eigenvalues $\lambda _1=1+0.9i$ and $\lambda _2=1+i$. The maximum intensity in this case is $14.44=\left( 0+2\cdot 0.9+2\cdot 1\right) ^2$. The tilt in the soliton propagation in all figures mainly comes from the Hirota parameter $\alpha =1/8$, but also from the real part of eigenvalues.

In Fig. 2, we demonstrate that higher-order breathers can numerically evolve from an initial DT wave function by applying a periodic boundary condition. First, we choose $a_1=a=0.464$ for the first AB in a nonlinear superposition and calculate its eigenvalue according to Eq. (2.5). As stated in Sect. 2, the second- and third-order breathers should have their frequencies exactly two and three times higher than the frequency of the first-order one. We show the third-order breather in Fig. 2a for $\alpha =-0.1$, and in Fig. 2b for $\alpha =0.1$. The tilt and a slight stretching of the solution are determined by the parameter $\alpha$ solely. The PHF holds: $\left| \psi \right| ^2_{\mathrm{max}}=33.64=\left( 1+2\nu _1+2\nu _2+2\nu _3 \right) ^2$.

In Fig. 3, we show a fifth-order breather, numerically obtained in the same way as the third-order breather from the previous figure. We start from $a_1=a=0.485$, calculate other eigenvalues accordingly (Eq. 2.6), and set the domain for numerics assuming periodic boundary conditions. The peak intensity is $\left| \psi \right| ^2_{\mathrm{max}}=81=\left( 1+2\nu _1+2\nu _2+2\nu _3+2\nu _4+2\nu _5 \right) ^2$.

Next, we utilize a breather-to-soliton conversion of HE, to further exemplify the utility of PHF. Even when the plane-wave seeding is used to generate breathers, the solitons of arbitrary orders are generated under the condition given by Eq. (4.3). One can immediately see the benefits of PHF: The “converted” soliton will have the amplitude maximum of $1+\sum _{i}{2\nu _i}$, instead of $\sum _{i}{2\nu _i}$, obtained from the usual zero seed. We demonstrate this by using the same eigenvalues as in Fig. 1, but having real parts equal to $\frac{1}{8\alpha }$. The intensity peak heights of single and double converted solitons are 7.84 and 23.04, respectively, which are higher than those of the solitons of the same shape shown in Fig. 1. Real and imaginary parts of the wave function have a discontinuity at the edges of the t interval. Furthermore, $\mathop {\lim }\limits _{t \rightarrow \pm \infty } \left| \psi \right| ^2 = 1$ and $\mathop {\lim }\limits _{t \rightarrow \pm \infty } \frac{{\partial \psi }}{{\partial t}} = 0$, so the Neumann boundary condition can still be applied in this case (Fig. 4).

Fig. 4

Dynamically generated soliton solutions of the a first and b second order, for $\alpha =1/8$. A breather-to-soliton conversion in Hirota equation is utilized to obtain higher-intensity solitons. Initial wave functions are extracted from the DT solutions having a plane-wave seed. The real parts of all eigenvalues satisfy Eq. (4.3). Parameters are: a $\lambda _1=1+0.9i$, $\left| \psi \right| ^2_{\mathrm{max}}=\left( 1+2\cdot 0.9\right) ^2=7.84$, and b $\lambda _1=1+0.9i$ , $\lambda _2=1+i$, $\left| \psi \right| ^2_{\mathrm{max}}=\left( 1+2\cdot 0.9+2\cdot 1\right) ^2=23.04$, both in agreement with the PHF for breathers

Fig. 5

Breather-to-soliton conversion of the Hirota equation on dnoidal background. Wave functions are calculated using the DT with the seed given by Eq. (3.1). Parameters are: $\alpha =1/6$, $m=0.5^2$, $\lambda =0.75+0.9i$, with a $c=1$ and b $c=1.5$

Fig. 6

First-order Akhmediev breathers obtained by the DT, with dnoidal seed from Eq. (3.1). Parameters are: $\alpha =1/6$ and $m=0.5^2$. The eigenvalue is $\lambda =0.75i$, with a $c=1$ and b $c=1.2$. Kuznetsov–Ma soliton is obtained for $\alpha =0.03, m=0.5^{2}$, and $\lambda =1.25i$, with c $c=1$ and d $c=1.2$

Fig. 7

a The dim and b the bright rogue waves obtained for $\alpha =0.3$, $m=0.5$, and $\nu \approx \frac{1}{2} \mp \frac{1}{2}\sqrt{1-m}$. c The second-order solution formed by the DT from RWs of (a) and (b). The peak intensity is 9, precisely equal to the sum of its constituents’ intensities

5.2 Solutions of Hirota equation on nonuniform backgrounds

We now switch to the solutions generated by the DT when the seed is taken from dn or cn family of JEFs, given by Eqs. (3.1) and (3.2). First, we examine if the breather-to-soliton conversion holds for nonuniform backgrounds. In Fig. 5a, b, we show the first-order converted solitons, differing only in the parameter c of Eq. (3.1). For $c=1$, the background oscillations have longer periods and smaller amplitudes, as compared to the $c=1.5$ case. We conclude that a weaker oscillation causes less perturbation in the soliton propagation. Therefore, the converted soliton is slightly affected, as shown in Fig. 5a, while in Fig. 5b a considerable distortion is observed.

Akhmediev breathers and Kuznetsov–Ma solitons of HE can be also produced on elliptic backgrounds. Similar to the NLSE case, ABs are produced when the imaginary part of an eigenvalue is $\nu <1$, while the KM solitons are obtained in the $\nu >1$ case. It was shown in [33] that the NLSE ABs are stretching out and KM solitons are compressing, as the elliptic modulus $\sqrt{m}$ or $\nu$ is increasing. Our calculations for Hirota equation produced the same results (not shown). Here we examine the influence of parameter c in the seed function $\psi _{\text {dn}}$ on the period of ABs and KM solitons. We set $\alpha =1/6$, $m=0.5^2$, and $\lambda =0.75i$ for AB, and $\alpha =0.03, m=0.5^{2}$, and $\lambda =1.25i$ for KM soliton. When $c=1$ (Fig. 6a), five peaks are visible, compared to seven in the $c=1.2$ case (Fig. 6b). Results differ when the KM soliton is produced: More crests are visible for lower c, as shown in Fig. 6c, d. It is clear that a higher c yields shorter periods of ABs and longer periods of KM solitons.

Next, we generate the first-order breathers, denoted as the dim and bright RWs, on a dnoidal background ($c=1$). These results are similar to those in [33] and [34]. In general, the dim RW is obtained when $\nu \approx \frac{1}{2}-\frac{1}{2}\sqrt{1-m}$, while the bright RW is generated for $\nu \approx \frac{1}{2}+\frac{1}{2}\sqrt{1-m}$. In Fig. 7a, b, the dim and bright RWs are shown, respectively, for $\alpha =0.3$ and $m=0.5$. The intensities of dim (bright) RWs are 1.6716 (7.3284). These two RWs can be combined into a second-order solution using DT, as shown in Fig. 7c. Its intensity is always 9 (in agreement with PHF), independent of the background parametrized by m. In the special case of $m=0.5$, the intensity of this second-order rogue wave is equal to the sum of intensities of the dim and bright RWs. In this manner, we have generated a “Pythagorean triplet” of rogue waves for the Hirota equation. A comparison of these results with the same solutions of NLSE (Figs. 3, 4, and 5 in [34]) reveals a similar structure, except for the tilt introduced by the parameter $\alpha$.

In Fig. 8a, b, we show the third-order breathers generated by DT for both dnoidal and cnoidal seeds. The backgrounds differ in the two cases, showing distorted oscillatory patterns. However, very sharp and strong central peaks appear similarly in both figures, suggesting that these maxima can be recognized as RWs of HE on nonuniform backgrounds.

Fig. 8

Third-order breathers calculated using the DT on a nonuniform background. Parameters are $\lambda _1=0.91i$, $\lambda _2=0.81i$, $\lambda _3=0.61i$, and $\alpha =0.05$. Seed functions: a $\psi _0=\psi _{\text {dn}}$, $m=1/9$, $c=1$, and b $\psi _0=\psi _{\text {cn}}$, $m=0.98^2$, $c={\sqrt{2m-1}}$. Both intensity maxima are in agreement with the PHF: a $\left| \psi \right| ^2_{\mathrm{max}}=32.036=\left[ c+2(0.91+0.81+0.61)\right] ^2$ and b $\left| \psi \right| ^2_{\mathrm{max}}=31.810=\left[ c\sqrt{\frac{m}{2m-1}}+2(0.91+0.81+0.61)\right] ^2$. Insets in both figures show 2D intensity distributions, in order to depict strong and sharp central peaks imposed on irregular oscillatory patterns of thebackground

6 Conclusion

In this work, we have presented a procedure for dynamical generation of higher-order solitons and breathers of the Hirota equation on uniform background, using the Darboux transformation. Once the wave function at a particular value of the evolution variable is found using DT, it can be used as an initial condition for numerical evaluation, with appropriate boundary conditions. This dynamical evolution toward higher-order solitons and breathers is important in the situations when the existence of such solutions is questionable in the presence of modulation instability. Thus, the DT may provide analytical higher-order solutions that might not exist, owing to modulation instability, which—as a rule—exists in these solutions. We have shown that the breather-to-soliton conversion can be used to produce solitons of higher amplitude and that the periodicity of Akhmediev breathers can be utilized for dynamical generation of rogue waves.

We also presented a derivation of the Lax pair generating functions and a procedure for calculating higher-order solutions of Hirota equation when dnoidal and cnoidal JEFs are used as seed solutions for the DT. We demonstrated that higher-order ABs and Kuznetsov–Ma solitons can be generated on these nonuniform backgrounds. For particular sets of parameters, we found that the higher-order DT solutions can be treated as rogue waves of the Hirota equation.

Finally, we generalized our peak height formula for the Hirota equation, proving that the peak height of a higher-order solution is a simple linear sum of the constituent peak heights. This formula may be useful in guiding the design and production of breathers of maximal intensity in physical systems modeled by the Hirota equation.

Appendix: The general Darboux transformation scheme

A higher-order soliton (breather) solution of the Nth order is a nonlinear superposition of N independent solitons (breathers), each determined by a complex eigenvalue $\lambda _j$, where $1 \le j \le N$ (the real part of the eigenvalue determines the angle between the localized solution and x-axis, while the imaginary part characterizes the periodic modulation frequency [27]). The Nth-order wave function given by the DT is:

$$\begin{aligned} {\psi _n} = {\psi _{n - 1}} + \frac{{2\left( {\lambda _n^* - {\lambda _n}} \right) {s_{n,1}}r_{n,1}^*}}{{{{\left| {{r_{n,1}}} \right| }^2} + {{\left| {{s_{n,1}}} \right| }^2}}}. \end{aligned}$$

(7.1)

In order to find $r_{n,1}$ and $s_{n,1}$, one has to analyze recursive relations between $r_{n,p}(x,t)$ and $s_{n,p}(x,t)$ functions of the Lax pair equations in the general form:

$$\begin{aligned} \begin{aligned} {r_{n,p}} =&\,\, [\left( {\lambda _{n - 1}^* - {\lambda _{n - 1}}} \right) s_{n - 1,1}^*{r_{n - 1,1}}{s_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - {\lambda _{n - 1}}} \right) {\left| {{r_{n - 1,1}}} \right| ^2}{r_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - \lambda _{n - 1}^*} \right) {\left| {{s_{n - 1,1}}} \right| ^2}{r_{n - 1,p + 1}}] \\&/\left( {{{\left| {{r_{n - 1,1}}} \right| }^2} + {{\left| {{s_{n - 1,1}}} \right| }^2}} \right) \\ {s_{n,p}} =&\,\, [\left( {\lambda _{n - 1}^* - {\lambda _{n - 1}}} \right) {s_{n - 1,1}}r_{n - 1,1}^*{r_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - {\lambda _{n - 1}}} \right) {\left| {{s_{n - 1,1}}} \right| ^2}{s_{n - 1,p + 1}} \\&+ \left( {{\lambda _{p + n - 1}} - \lambda _{n - 1}^*} \right) {\left| {{r_{n - 1,1}}} \right| ^2}{s_{n - 1,p + 1}}] \\&/\left( {{{\left| {{r_{n - 1,1}}} \right| }^2} + {{\left| {{s_{n - 1,1}}} \right| }^2}} \right) . \\ \end{aligned} \end{aligned}$$

(7.2)

From the last equation, it can be deduced that all $r_{n,p}$ and $s_{n,p}$ can be calculated just from $r_{1,j}$ and $s_{1,j}$, with $1 \le j \le N$. The functions $r_{1,j}(x,t)$ and $s_{1,j}(x,t)$, forming the Lax pair $R = \left( {\begin{array}{*{20}{c}} r \\ s \\ \end{array}} \right) \equiv \left( {\begin{array}{*{20}{c}} {{r_{1,j}}} \\ {{s_{1,j}}} \\ \end{array}} \right) ,$ are determined by the eigenvalue $\lambda \equiv \lambda _j$, an embedded arbitrary center of the solution $\left( x_{0j},t_{0j}\right)$, and a system of linear differential equations:

$$\begin{aligned} \frac{{\partial R}}{{\partial t}} = U \cdot R,\quad \frac{{\partial R}}{{\partial x}} = V \cdot R. \end{aligned}$$

(7.3)

Particularly for the Hirota equation, matrices U and V are defined as ($\psi \equiv \psi _0$) [30]:

$$\begin{aligned} U= & {} i\left[ {\begin{array}{*{20}{c}} \lambda &{} {\psi {{(x,t)}^*}} \\ {\psi (x,t)} &{} { - \lambda } \\ \end{array}} \right] ,\nonumber \\ V= & {} \sum \limits _{k = 0}^3 {{\lambda ^k} \cdot i\left[ {\begin{array}{*{20}{c}} {{A_k}} &{} {B_k^*} \\ {{B_k}} &{} { - {A_k}} \\ \end{array}} \right] }, \end{aligned}$$

(7.4)

where the coefficients $A_k$ and $B_k$ are

$$\begin{aligned} {A_0}= & {} - \frac{1}{2}|\psi {|^2} - i\alpha \left( {\psi _t^*\psi - {\psi _t}{\psi ^*}} \right) ,\nonumber \\ {B_0}= & {} 2\alpha {\left| \psi \right| ^2}\psi + \frac{1}{2}i{\psi _t} + \alpha {\psi _{tt}}, \nonumber \\ {A_1}= & {} 2\alpha {\left| \psi \right| ^2},\quad {B_1} = \psi - 2i\alpha {\psi _t} \nonumber \\ {A_2}= & {} 1,\quad {B_2} = - 4\alpha \psi ,\quad {A_3} = - 4\alpha ,\quad {B_3} = 0.\nonumber \\ \end{aligned}$$

(7.5)