Near-Ordinary Periodic Waves of a Generalized Reaction–Convection–Diffusion Equation

Abstract

We investigate near-ordinary periodic traveling wave solutions bifurcated from a family of ordinary periodic traveling wave solutions for a generalized reaction–convection–diffusion equation, especially its dependence on the nonlinear reaction. Using the Abelian integral method and the Chebyshev criteria, we find conditions for the existence and number of near-ordinary periodic wave solutions not only for the monotone case of the ratio of the Abelian integral as previous publications, but also for the non-monotone case. In a parameter region we provide a conjecture about the uniqueness of near-ordinary periodic traveling wave solutions for any degree of the nonlinear reaction and prove it up to degree four. The final simulations illustrate theoretical results numerically.

1 Introduction and Main Results

Reaction–convection–diffusion models play a vital role in physics, chromatography, engineering, population dynamics. As in [8], the general reaction–convection–diffusion equation is of form

$$\begin{aligned} u_t=\left( D(u)u_x\right) _x+F(u)u_x+R(u), \end{aligned}$$

(1.1)

where u(x, t) is nonnegative and continuous, D(u) is the diffusion coefficient term, F(u) is the convection term and R(u) is the reaction term. In the flow theory of an unsaturated porous medium, the effect of gravity is reflected in the convection term. In recent decades, investigations on the exact solutions of reaction–convection–diffusion equations have attracted more and more researchers’ attention.

In the case of $F(u)\equiv 0$, reaction–convection–diffusion Eq. (1.1) is a reaction–diffusion equation. When R(u) is a quadratic or cubic polynomial, Clarkson and Mansfield [4, 5] obtained the exact solutions by using the nonclassical reductions method. When R(u) is a cubic polynomial with three distinct zeros, Chen and Guo [3] presented the exact solutions by using a truncated Painlevé expansion. Li and Chen [19] investigated its bifurcations and obtained traveling wave solutions by the bifurcation theory of dynamical systems. When R(u) is more general, Herrera, Minzoni and Ondarza [15] found traveling wave solutions in the phase space. Alikakos, Bates and Chen in [1] established periodic traveling wave solutions for bistable R(u) by using the Schauders Fixed Point Theorem and the singular perturbation method. When $D(u) \equiv 1$, $R(u)=u(1-u)$, Eq. (1.1) is the well-known Fisher equation $u_t=u_{xx}+u(1-u)$, which was proposed as a model not only for the propagation of a mutant gene but also in flame propagation [7]. When $D(u) \equiv 1$, $R(u)=u(1-u)(u-a)$ with $-1\le a\le 1$, as in [16] Eq. (1.1) is the Nagumo’s equation $u_t=u_{xx}+u(1-u)(u-a)$, which is an important simplification model of the FitzHugh-Nagumo system modeling impulse propagation along a nerve fiber. When $D(u)\equiv d$, $R(u)=pu(1-u^\alpha )(u^\alpha +q)$, as in [30] Eq. (1.1) reduces to a generalized Fisher equation $u_t=du_{xx}+pu(1-u^\alpha )(u^\alpha +q)$, whose explicit traveling wave front solutions were obtained by an algebraic method with boundary condition $u(-\infty ,t)=0, \ u(+\infty ,t)=1$. When $D(u)=a_0+u^p$, $R(u)=bu(1-u^q)(1+u^q)$, in [6] the existence of bounded traveling wave solutions was proved by the qualitative theory of planar dynamical systems and explicit forms are given under certain conditions by the Lie symmetry method.

Now we introduce the case of $F(u)\not \equiv 0$. When $D(u)=u^m$, $F(u)=bnu^{n-1},$ $R(u)=cu^p$, as in [9] Eq. (1.1) becomes a full power-law reaction–convection–diffusion equation $u_t=(u^m)_{xx}+b(u^n)_x+cu^p$, which is usually to describe the flow of liquids in porous media and the transmission of thermal energy in plasma. A complete analysis of semi-wave front solutions decreasing to 0 was conducted in [9]. When $D(u)=u$, $F(u)=\alpha u^{n+1},$ $R(u)=u(1-u^n)(u^n-\beta )$, as in [2] Eq. (1.1) reduces to the well-known Burgers-Huxley equation $u_t=u_{xx}-\alpha u^nu_x+u(1-u^n)(u^n-\beta )$, which is applicable in traffic flows, turbulence theory, acoustics, hydrodynamics. When $D(u)=mu^{m-1}$, $F(u)=u(b_0+b_1u^p),$ $R(u)=u^{2-m}(1-u^p)(c_0+c_1u^p)$, traveling wave solutions were investigated in [14] for Eq. (1.1) in cases of density-independent diffusion and density-dependent diffusion. When $D(u)=1$, $F(u)=\alpha u^m,$ $R(u)=\gamma u (1-u^m)$, as in [31] Eq. (1.1) reduces the generalized Burgers-Fisher equation, which admits a unique periodic wave solution based on the monotonicity of the ratio of two Abelian integrals. In [23], an analytical solution of the Burgers-Huxley equation was given in the form of a Taylor multivariate series by the proposed Elzaki homotopy transformation perturbation method. Some singular perturbations of the reaction–convection–diffusion equation are investigated in [25, 28].

When $F(u)\equiv a_0\ne 0$, periodic traveling wave solution $u(x,t)=\phi (x+a_0t)$ of Eq. (1.1) is called an ordinary periodic wave solution. When $D(u)\equiv 1, F(u)\equiv a_0, R(u)=\delta _1u(u^q+\delta _2)(u^q+\beta )$ with $\delta _1=\pm 1, \delta _2=\pm 1$, Eq. (1.1) takes form

$$\begin{aligned} u_t=u_{xx}+a_0u_x+\delta _1u(u^q+\delta _2)(u^q+\beta ). \end{aligned}$$

(1.2)

It is not hard to check that in the case of $q=1, \delta _1=1, \delta _1=-1, \beta >0$, Eq. (1.2) has a family of ordinary periodic wave solutions

$$\begin{aligned} u(x,t)=\frac{\phi _4(\phi _3-\phi _1)+\phi _1(\phi _4-\phi _3)sn^2(\omega (x+a_0t),k)}{\phi _3-\phi _1+(\phi _4-\phi _3)sn^2(\omega (x+a_0t),k)}, \end{aligned}$$

where $sn(\cdot ,\cdot )$ is the Jacobian elliptic function and

$$\begin{aligned} \omega :=\frac{\sqrt{(\phi _4-\phi _2)(\phi _3-\phi _1)}}{4}, ~~~~k:=\sqrt{\frac{(\phi _4-\phi _3)(\phi _2-\phi _1)}{(\phi _4-\phi _2)(\phi _3-\phi _1)}} \end{aligned}$$

when $\phi _i$ ($i=1,2,3,4$) satisfying $\phi _1<\phi _2<0<\phi _3<\phi _4$ are real roots of $3\phi _i^4+4(\beta -1)\phi _i^3-6 \beta \phi _i^2+h=0$ for some h. Moreover, ordinary periodic wave solution u(x, t) oscillates between $\phi _3$ and $\phi _4$. When F(u) changes from constant $a_0$ to a non-constant function, usually this family of ordinary periodic wave solutions disappear and some new periodic traveling wave solutions may appear, as a bifurcation of ordinary periodic wave solutions. For instance, when $F(u)=a_0+a_1 u$ with sufficiently small $a_1$, some new periodic traveling wave solutions may appear and their wave speed c is sufficiently close to the wave speed $a_0$ of ordinary periodic wave solutions, i.e., $c\rightarrow a_0^+$ as $a_1\rightarrow 0^+$. We call such appearing periodic traveling wave solution as a near-ordinary periodic wave solution.

Consider a perturbation of Eq. (1.2) as

$$\begin{aligned} u_t=u_{xx}+(a_0+a_1u)u_x+\delta _1u(u^q+\delta _2)(u^q+\beta ), \end{aligned}$$

(1.3)

where $0<|a_1|\ll 1$ and $\beta \ne 0$. The existence of unique near-ordinary periodic wave solution of Eq. (1.3) is investigated in [32] by the monotonicity of the ratio of the Abelian integral and the bifurcation theory of dynamical systems when $q=2, \delta _1=1, \delta _2=-1, \beta \in (0,1)$. There are two interesting problems for Eq. (1.3): Are there more near-ordinary periodic wave solutions than one? How much do near-ordinary periodic wave solutions depend on the nonlinear reaction term, i.e., how about the influence of q greater than 2? Motivated by these two problems, we investigate the number of near-ordinary periodic wave solutions of Eq. (1.3) not only for the monotone case of the ratio of the Abelian integral as [32] but also for the non-monotone case.

This paper is organized as follows. In Sect. 2, Eq. (1.3) is reduced to its traveling system and the conditions for the uniqueness of near-ordinary periodic wave solutions are obtained by the monotonicity of the ratio of the Abelian integral. We investigate near-ordinary periodic wave solutions by analyzing the non-monotonicity of the ratio of the Abelian integral in Sect. 3. In Sect. 4, some simulations are provided to show the consistence with theoretical results and a conjecture about the uniqueness of near-ordinary periodic traveling wave solutions for any degree of the nonlinear reaction is proposed.

2 System Reduction and Result of Uniqueness

Consider $u(x,t)=u(x+ct)=\phi (\xi )$. Equation (1.3) becomes a traveling wave system

$$\begin{aligned} \left\{ \begin{aligned}&\dot{\phi }=y, \\&\dot{y}=-\delta _1\phi (\phi ^q+\delta _2)(\phi ^q+\beta )+(c-a_0-a_1\phi )y, \end{aligned} \right. \end{aligned}$$

(2.1)

where $\dot{\phi }={d\phi }/{d\xi }$. As in [18], a periodic traveling wave solution of Eq. (1.3) corresponds to a periodic orbit of system (2.1). Since we investigate near-ordinary periodic wave solutions, both $c-a_0$ and $a_1$ are sufficiently small. Denote $\varepsilon \alpha _0=c-a_0$ and $\varepsilon \alpha _1= -a_1$ with $0<\varepsilon \ll 1$. System (2.1) is rewritten as

$$\begin{aligned} \left\{ \begin{aligned}&\dot{\phi }=y, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\&\dot{y}=-\delta _1\phi (\phi ^q+\delta _2)(\phi ^q+\beta )+\varepsilon (\alpha _0+\alpha _1\phi )y. \end{aligned} \right. \end{aligned}$$

(2.2)

System (2.2)$|_{\varepsilon =0}$ is Hamiltonian and has Hamiltonian function

$$\begin{aligned} H(\phi ,y)=\frac{1}{2}y^2+\delta _1 \left( \frac{\phi ^{2q+2}}{2q+2}+\frac{(\beta +\delta _2)\phi ^{q+2}}{q+2} +\frac{\beta \delta _2\phi ^2}{2}\right) . \end{aligned}$$

(2.3)

Let $\Gamma _h:= \{(\phi , y): H(\phi , y) = h\}$, which corresponds closed orbits of (2.2)$|_{\varepsilon =0}$ for each h in some interval. For a general Hamiltonian system, the zero problem of the Abelian integral is investigated in [17, 20,21,22, 26, 27, 29] when $\Gamma _h$ is an elliptic curve. For system (2.2)$|_{\varepsilon =0}$, closed curve $\Gamma _h$ is elliptic when $q=1$, but hyperelliptic when $q\ge 2$.

Consider the case $\delta _2=-1$. Clearly, system (2.2)$|_{\varepsilon =0}$ has three equilibria $E_0(0,0)$, $E_1(1,0)$, $E_2(\root 3 \of {-\beta },0)$ when $q=3$, three equilibria $E_0(0,0), E_1(1,0), E_2(-1,0)$ when $q=4$ and $\beta >0$, five equilibria $E_0(0,0)$, $E_1(1,0)$, $E_2(\root 4 \of {-\beta },0)$, $E_3(-1,0)$, $E_4(-\root 4 \of {-\beta },0)$ when $q=4$ and $\beta <0$. By the Jacobian matrix of vector field of (2.2)$|_{\varepsilon =0}$ at each equilibrium, one gets the type of each equilibrium. On the other hand, we focus on periodic orbits of (2.2)$|_{\varepsilon =0}$ in the right-hand phase plane because our aim is to find near-ordinary periodic wave solutions of Eq. (1.3). Similar analysis can be done for the case $\delta _2=1$ and we give all phase portraits of (2.2)$|_{\varepsilon =0}$ having periodic orbits in the right-hand phase plane as shown in Figs. 1, 2, 3, 4, 5.

Suppose that there exists a closed orbit $\Gamma _{h_0}$ of (2.2)$|_{\varepsilon =0}$ surrounding a center, $A(h_0)\in \Gamma _{h_0}$ is the rightmost point on the positive $\phi$-axis. Let $\Gamma _{h_\varepsilon }$ be a piece of the orbit for (2.2) starting from $A(h_0)$ to the next intersection point $B(h_\varepsilon )$ with the positive $\phi$-axis for $0<|h_\varepsilon -h_0|\ll 1$. By [13], the displacement function is given by $H(\phi ,y)$ as

$$\begin{aligned} d(h,\varepsilon )=\int _{\widehat{AB}}dH=\varepsilon (I(h,\delta )+O(\varepsilon )), \end{aligned}$$

where $\delta =(\alpha _0,\alpha _1)$ and $I(h,\delta )=\alpha _0I_0(h)+\alpha _1I_1(h)$. Here $I_i(h)=\oint _{\Gamma _{h}}\phi ^iyd\phi , i=0,1$. By the Poincaré bifurcation theory [11], the isolated zeros of $d(h,\varepsilon )$ corresponds to limit cycles of (2.2). Since $I_0=\oint _{\Gamma _h}yd\phi =\iint _Dd\phi dy\ne 0$, we write $I(h,\delta )$ as $I(h)=\alpha _1I_0(h)\left( {\alpha _0}/{\alpha _1}+P(h)\right)$, where D is the region enclosed by $\Gamma _h$ and $P(h):={I_1(h)}/{I_0(h)}$. Thus, if P(h) is monotonic strictly, there exists a unique $h^*$ such that $P(h^*)=-{\alpha _0}/{\alpha _1}$, implying that I(h) has a unique zero $h^*$. Therefore, there exists a unique near-ordinary periodic wave solution for (1.3). This is the idea used in [21, 26, 32] and it is also used to prove the following theorem.

Theorem 2.1

When $q=3$ (resp. $q=4$), Eq. (1.3) has at most one near-ordinary periodic wave solutions and it is reachable, if one of (i), (ii), (iii) (resp. (iv), (v), (vi)) holds, where

• (i) $\delta _1=1, \delta _2=-1$, $\beta \in [-4,-1)\cup (-1,0)\cup (0,+\infty )$;

• (ii) $\delta _1=\delta _2=1$, $\beta \in [-1,0)$;

• (iii) $\delta _1=\delta _2=-1$, $\beta \in [-5/2,-1)\cup (-1,-2/5]\cup (-1/4,0)$;

• (iv) $\delta _1=1, \delta _2=-1$, $\beta \in [-2,-1)\cup (-1,0)\cup (0,1)$;

• (v) $\delta _1=\delta _2=1$, $\beta \in [-5/13,0)$;

• (vi) $\delta _1=\delta _2=-1$, $\beta \in [-13/5,-1)\cup (-1,-5/13]\cup (-1/5,0)$.

Fig. 1

Phase portraits of (2.2)$|_{\varepsilon =0}$ with $q=3$, $\delta _1=1$, $\delta _2=-1$. a $-4\le \beta <-1$; b $-1<\beta <0$; c $\beta >0$

Fig. 2

Phase portraits of (2.2)$|_{\varepsilon =0}$ with $q=4$, $\delta _1=1$, $\delta _2=-1$. a $-2\le \beta <-1$; b $\beta >0$; c $-1<\beta <0$

Fig. 3

Phase portraits of (2.2)$|_{\varepsilon =0}$. a $q=3$, $\delta _1=1$, $\delta _2=1$, $-1\le \beta <0$; b $q=4$, $\delta _1=1$, $\delta _2=1$, $-5/13\le \beta <0$

Fig. 4

Phase portraits of (2.2)$|_{\varepsilon =0}$ with $q=3$, $\delta _1=-1$, $\delta _2=-1$. a $-1/4<\beta <0$; b $\beta =-1/4$; c $-1<\beta <-1/4$; d $-4<\beta <-1$; e $\beta =-4$

Fig. 5

Phase portraits of (2.2)$|_{\varepsilon =0}$ with $q=4$, $\delta _1=-1$, $\delta _2=-1$. a $\beta =-1/5$; b $-1/5<\beta <0$; c $-5<\beta <-1$; d $-1<\beta <-1/5$; e $\beta =-5$

Proof

In order to prove this theorem, we only need to give the conditions for the monotonicity of function P(h). That is, we prove that for system (2.2) function P(h) is monotonic strictly when $\beta \in [-4,-1)\cup (-1,0)\cup (0,+\infty )$ (resp. $\beta \in [-2,-1)\cup (-1,0)\cup (0,1)$), $\delta _1=1, \delta _2=-1$; $\beta \in [-1,0)$ (resp. $\beta \in [-5/13,0)$), $\delta _1=\delta _2=1$; $\beta \in [-5/2,-1)\cup (-1,-2/5]\cup (-1/4,0)$ (resp.$\beta \in [-13/5,-1)\cup (-1,-5/13]\cup (-1/5,0)$), $\delta _1=\delta _2=-1$, for the case $q=3$ (resp. $q=4$).

In the case $q=3$, $\beta \in [-4,-1)\cup (-1,0)\cup (0,+\infty )$, $\delta _1=1, \delta _2=-1$, system (2.2) is of form

$$\begin{aligned} \left\{ \begin{aligned}&\dot{\phi }=y, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\&\dot{y}=-\phi (\phi ^3-1)(\phi ^3+\beta )+\varepsilon (\alpha _0+\alpha _1\phi )y. \end{aligned} \right. \end{aligned}$$

(2.4)

The Hamiltonian function of corresponding unperturbed system is

$$\begin{aligned} H_1(\phi ,y)=\frac{1}{2}y^2+\frac{1}{8}\phi ^{8}+\frac{\beta -1}{5}\phi ^{5}-\frac{\beta }{2}\phi ^2. \end{aligned}$$

(2.5)

Clearly, when $\beta <0$ (resp. $\beta >0$), $H_1(\phi ,y)=h$ corresponds a family of closed orbits of system (2.4) in the right-hand phase plane for $h\in K_{11}:=(\min \{h_{11},h_{1a}\},\max \{h_{11},h_{1a}\})$ (resp. $h\in K_{12}:=(h_{11},0)$), where $h_{11}:=-3(1+4\beta )/40, h_{1a}:=-3(4+\beta ) \beta ^{5/3}/40$. That is, P(h) is defined for $h \in K_{11}$ (resp. $h\in K_{12}$) when $\beta <0$ (resp. $\beta >0$).

When $-4\le \beta <-1$, by Fig. 1a there exists a homoclinic orbit connecting saddle $E_1(1,0)$. The family of closed orbits surrounds center $E_2(-\root 3 \of {\beta },0)$ for all $h\in K_{11}$. Let

$$\begin{aligned} F_1(\phi ):=H_1(\phi ,0)=\frac{1}{8}\phi ^{8}+\frac{\beta -1}{5}\phi ^{5}-\frac{\beta }{2}\phi ^2 \end{aligned}$$

and $m_1$ be the abscissa of the intersection point between the homoclinic orbit and $\phi$-axis. Thus, $F_1(1)=F_1(m_1)$ and

$$\begin{aligned} F_1^\prime (\phi )\left( \phi +\root 3 \of {\beta }\right) =\phi \left( \phi +\root 3 \of {\beta }\right) ^2\left( \phi ^3-1\right) \left( \phi ^2-\root 3 \of {\beta }\phi +\root 3 \of {\beta ^2}\right) >0 \end{aligned}$$

for $\phi \in (1,m_1)\setminus \{-\root 3 \of {\beta }\}$, implying that $F_1$ has a minimum at $-\root 3 \of {\beta }$, and monotonic strictly on $(1,\root 3 \of {-\beta })$ and $(\root 3 \of {-\beta },m_1)$ respectively. Let $\mu (h)$ and $\nu (h)$ be inverse functions of $F_1$ on these two intervals and $1<\mu (h)<-\root 3 \of {\beta }<\nu (h)<m_1$ for $h\in (h_{1a},h_{11})$.

By (2.5),

$$\begin{aligned}{} & {} \left\{ I_1^\prime (h)I_0(h)-I_0^\prime (h)I_1(h)\right\} /4 \nonumber \\{} & {} \quad =\int _{\mu (h)}^{\nu (h)}\frac{\phi }{y}d\phi \int _{\mu (h)}^{\nu (h)}yd\phi -\int _{\mu (h)}^{\nu (h)}\frac{1}{y}d\phi \int _{\mu (h)}^{\nu (h)}\phi yd\phi \nonumber \\{} & {} \quad =\int _{\mu (h)}^{\nu (h)}\frac{\phi -w(h)}{y}d\phi \int _{\mu (h)}^{\nu (h)}yd\phi -\int _{\mu (h)}^{\nu (h)}\frac{1}{y}d\phi \int _{\mu (h)}^{\nu (h)}(\phi -w(h)) yd\phi \nonumber \\{} & {} \quad =\int _{-z(h)}^{z(h)}\frac{s}{y(w(h)+s)}ds\int _{-z(h)}^{z(h)}y(w(h)+s)ds -\int _{-z(h)}^{z(h)}\frac{1}{y(w(h)+s)}ds\nonumber \\{} & {} \qquad \int _{-z(h)}^{z(h)}s y(w(h)+s)ds \nonumber \\{} & {} \quad =\int _{0}^{z(h)}\frac{s(y(w(h)-s)-y(w(h)+s))}{y(w(h)+s)y(w(h)-s)}ds\int _{0}^{z(h)}[y(w(h)+s)+y(w(h)-s)]ds \nonumber \\{} & {} \qquad -\int _{0}^{z(h)}\frac{y(w(h)+s)+y(w(h)-s)}{y(w(h)+s)y(w(h)-s)}ds\int _{0}^{z(h)}s [y(w(h)+s)-y(w(h)-s)]ds, \nonumber \\ \end{aligned}$$

(2.6)

where $w(h):=({\mu (h)+\nu (h)})/{2}, z(h):=({\nu (h)-\mu (h)})/{2}$. Since $F_1(\mu (h))=F_1(\nu (h))$, we get

$$\begin{aligned} \int _{-\root 3 \of {\beta }}^{\mu (h)}\phi (\phi ^3-1)(\phi ^3+\beta )d\phi =\int _{-\root 3 \of {\beta }}^{\nu (h)}\phi (\phi ^3-1)(\phi ^3+\beta )d\phi . \end{aligned}$$

(2.7)

By (2.7),

$$\begin{aligned} \int _0^{-\root 3 \of {\beta }-\mu (h)}f_1(\phi )d\phi =\int _0^{\nu (h)+\root 3 \of {\beta }}f_2(\phi )d\phi , \end{aligned}$$

(2.8)

where $-\root 3 \of {\beta }-\mu (h)>0, \nu (h)+\root 3 \of {\beta }>0$ and

$$\begin{aligned} \begin{aligned} f_1(\phi )&:=(\phi +\root 3 \of {\beta })\left[ (\root 3 \of {\beta }+\phi )^3+1\right] \left[ (\root 3 \of {\beta }+\phi )^3-\beta \right] ,\\ f_2(\phi )&:=(\phi -\root 3 \of {\beta })\left[ (\phi -\root 3 \of {\beta })^3-1\right] \left[ (\phi -\root 3 \of {\beta })^3+\beta \right] . \end{aligned} \end{aligned}$$

Clearly, both $f_1, f_2$ are increasing and $f_1(0)=f_2(0)=0$, $f_2(\phi )>f_1(\phi )$ for all $\phi >0$. Thus, from (2.8) we get $\nu (h)+\root 3 \of {\beta }<-\root 3 \of {\beta }-\mu (h)$, that is $0<w(h)<-\root 3 \of {\beta }$. Define the criterion function in $s\in [0,z(h)]$ as the form

$$\begin{aligned} G(s):=F_1(w(h)+s)-F_1(w(h)-s) =\frac{2s}{5}g(s^2), \end{aligned}$$

(2.9)

where $g(\tau )=5w(h)\tau ^3+\![35w(h)^3+\beta -1]\tau ^2\!+5w(h)^2[7w(h)^3\!+2\beta -2]\tau \!+5w(h)(w(h)^3\!-1)(\!w(h)^3+\beta )$. It is not difficult to find that 0, z(h) are real zeros of G(s). $\tau _1=z(h)^2$ is a zero of $g(\tau )$ and assume that $\tau _2, \tau _3$ are the other two zeros of $g(\tau )$. Then, $\tau _1\tau _2\tau _3=-(w(h)^3-1)(w(h)^3+\beta )>0$ and $\tau _1+\tau _2+\tau _3=-(35w(h)^3+\beta -1)/5w(h)$. Since $-4\le \beta <-1$, $1<\mu (h)<w(h)<-\root 3 \of {\beta }$, we obtain

$$\begin{aligned}&\tau _2+\tau _3=-\frac{35w(h)^3+\beta -1}{5w(h)}-\tau _1<-\frac{35w(h)^3+\beta -1}{5w(h)}\\&\quad<-\frac{35\mu (h)^2+\beta -1}{5}<0. \end{aligned}$$

Therefore, $\tau _2<0$ and $\tau _3<0$, implying

$$\begin{aligned} G(s)=2w(h)s(s^2-z(h)^2)(s^2-\tau _2)(s^2-\tau _3)<0 \end{aligned}$$

for $s\in (0,z(h))$. Associated with (2.9), $F_1(w(h)+s)<F_1(w(h)-s)$ for $s\in (0,z(h))$. Since for $s\in (0,z(h))$ points $(w(h)+s, y(w(h)+s))$ and $(w(h)-s, y(w(h)-s))$ lie on the same closed orbit, we get $y^2(w(h)+s)/2+F_1(w(h)+s)=y^2(w(h)-s)/2+F_1(w(h)-s)$. Thus, $y(w(h)+s)>y(w(h)-s)$. Associated with (2.6), $I_1^\prime (h)I_0(h)-I_0^\prime (h) I_1(h)<0$, implying that $P'(h)<0$ when $-4\le \beta <-1$. The monotonicity of P(h) can be proved similarly when either $-1<\beta <0$ or $\beta >0$.

Similarly to case of $q=3$, $\delta _1=1, \delta _2=-1$ we prove that in case of $q=4$ function P(h) is monotonic strictly when $\delta _1=1, \delta _2=-1$, $\beta \in [-2,-1)\cup (-1,0)\cup (0,1)$; $\delta _1=\delta _2=1$, $\beta \in [-5/13,0)$; $\delta _1=\delta _2=-1$, $\beta \in [-13/5,-1)\cup (-1,-5/13]\cup (-1/5,0)$. This theorem is proved. $\square$

3 Result of Non-uniqueness

In this section we consider the case of non-monotonicity of function P(h), which may lead to more near-ordinary periodic wave solutions for Eq. (1.3).

Theorem 3.1

When $q=3$, there exists $a_1$ such that the number of near-ordinary periodic wave solutions of Eq. (1.3) is at least 2 if $\delta _1=\delta _2=-1$, $\beta \in [-4, -5/2) \cup (-2/5,-1/4]$, and exactly 2 if $\beta =-3$ additionally. When $q=4$, there exists $a_1$ such that the number of near-ordinary periodic wave solutions of Eq. (1.3) is at least 2 if $\delta _1=\delta _2=-1$, $\beta \in [-5, -13/5)\cup (-5/13,-1/5]$.

In order to prove Theorem 3.1, we give the conditions for the non-monotonicity of function P(h) and get the non-uniqueness of near-ordinary periodic wave solutions.

Lemma 3.1

For system (2.2), P(h) is non-monotonic when $\beta \in [-4, -5/2)\cup (-2/5, -1/4]$ (resp. $\beta \in [-5, -13/5)\cup (-5/13, -1/5]$) for the case $q=3$ (resp. $q=4$), $\delta _1=\delta _2=-1$.

Proof

In the case that $q=3, \delta _1=\delta _2=-1$, system (2.2) is of form

$$\begin{aligned} \dot{\phi }=y, ~~~~ \dot{y}=\phi (\phi ^3-1)(\phi ^3+\beta )+\varepsilon (\alpha _0+\alpha _1\phi )y. \end{aligned}$$

(3.1)

The Hamiltonian function of corresponding unperturbed system is

$$\begin{aligned} H_{3}(\phi ,y)=y^2/2-\phi ^8/8+(1-\beta )\phi ^5/5 +\beta \phi ^2/2. \end{aligned}$$

$h=H_3(\phi ,y)$ ($h\in K_3$) corresponds a family of closed orbits of system (3.1)$|_{\varepsilon =0}$, where

$$\begin{aligned}{} & {} K_3=(\min \{h_{31},h_{3a}\},\max \{h_{31},h_{3a}\}),~~ h_{31}=3(1+4\beta )/40, \\{} & {} h_{3a} =3\beta ^{5/3}(4+\beta )/40. \end{aligned}$$

When $\beta =-4$, system (3.1) has Hamiltonian function $H_{31}(\phi ,y)=y^2/2-\phi ^{8}/8+\phi ^{5}-2\phi ^2.$ By Fig. 4e, when $H_{31}(\phi ,y)=0$, there is a heteroclinic orbit connecting saddles (0, 0) and $(\root 3 \of {4},0)$. When $H_{31}(\phi ,y)=h\in (h_{31},0)$, there is a family of closed orbits $\Gamma _{h}$ surrounding center (1, 0). $\Gamma _{h}$ approaches to heteroclinic orbit as $h\rightarrow 0^-$, to the center as $h\rightarrow h_{31}^+$. The projection of $\Gamma _{h}$ on the $\phi$-axis is $(0, \root 3 \of {4})$. Let $F_3(\phi ):=H_{31}(\phi ,0)=-{\phi ^8}/{8}+\phi ^5-2\phi ^2$ and $z(\phi )$: $(0,1)\rightarrow (1,\root 3 \of {4})$ by $F_3(\phi )=F_3(z(\phi ))$. By the expression of $F_3$, we get that

$$\begin{aligned} F_3(\phi )-F_3(z)=(z-\phi )(\phi ^4+z^4-4\phi -4z)\bar{g}(\phi ,z)/8, \end{aligned}$$

where $\bar{g}(\phi ,z):=\phi ^3+\phi ^2z+\phi z^2+z^3-4$. Since $z-\phi \ne 0$ and $\phi ^4+z^4-4\phi -4z=\phi (\phi ^3-4)+z(z^3-4)\ne 0$ for $\phi \in (0,1)$ and $z\in (1,\root 3 \of {4})$, we get that $z(\phi )$ satisfies $\bar{g}(\phi ,z(\phi ))=0$.

From [24, Lemma 4.1], $2\,h I_i(h) =\oint _{\Gamma _h}f_i(\phi )y^3d\phi$ for $i=0, 1$, where $f_i(\phi )=G_i(\phi )+\phi ^i$, $G_i(\phi )={\phi ^i(i\phi ^6\!+\!\phi ^6\!-\!5i\phi ^3\!+\!4\phi ^3\!+\!4i\!+\!4)(\phi -1)^{-2}(\phi ^2+\phi +1)^{-2}}/{12}$. Then, the Wronskians $W[\ell _0(\phi )]$, $W[\ell _0(\phi ),\ell _1(\phi )]$ of $\{\ell _0(\phi ), \ell _1(\phi )\}$ can be computed as

$$\begin{aligned} W[\ell _0(\phi )]=\left. \frac{(\phi -z)w_1(\phi ,z)}{P(\phi ,z)}\right| _{z=z(\phi )}, ~~ W[\ell _0(\phi ),\ell _1(\phi )]=\left. \frac{(z-\phi )^3w_2(\phi ,z)}{Q(\phi ,z)}\right| _{z=z(\phi )}, \end{aligned}$$

where $\ell _i(\phi ):=f_i(\phi )/F'_3(\phi )-f_i(z(\phi ))/F'_3(z(\phi ))$, $P(\phi ,z):=12\phi z(\phi -1)^3(\phi ^2+\phi +1)^3(\phi ^3-4)(z-1)^3(z^2+z+1)^3(z^3-4), Q(\phi ,z):=144\phi ^2z^2(z^2+z+1)^5(z-1)^5(\phi ^3-4)^2(z^3-4)^2(\phi ^2+\phi +1)^5(\phi -1)^5(\phi ^2+2\phi z+3z^2)$, $w_1(\phi ,z)$, $w_2(\phi ,z)$ are polynomials in $(\phi ,z)$ of degree 18 and 31, respectively. It is not hard to check that, for $\phi \in (0,1)$ and $z\in (1,\root 3 \of {4})$, both $P(\phi ,z)$ and $Q(\phi ,z)$ are nonzero. On one hand, the resultants with respect to z between $\bar{g}(\phi ,z)$ and $w_1(\phi ,z)$, $w_2(\phi ,z)$ are computed as $R(\bar{g},w_1,z)=\phi ^3(\phi ^3-4)^5 (\phi -1)^2(\phi ^2+\phi +1)^2\varphi _1(\phi ), R(\bar{g},w_2,z)=-32\phi ^6(\phi ^3-4)^6(\phi -1)^2(\phi ^2+\phi +1)^2\varphi _2(\phi ),$ where $\varphi _1(\phi ), \varphi _2(\phi )$ are polynomials of degree 30 and 60 in $\phi$. In (0, 1) we find that $\varphi _1(\phi )$ has no zeros and $\varphi _2(\phi )$ has a unique zero $\phi ^*\approx 0.8271$. Thus, $\bar{g}(\phi , z)$ and $w_1(\phi , z)$ have no common roots for $\phi \in (0,1)$, $z\in (1,\root 3 \of {4})$. Therefore, $w_1(\phi ,z(\phi ))\ne 0$ for $\phi \in (0,1)$, implying $W[\ell _0(\phi )]\ne 0$.

To solve $\bar{g}(\phi ,z)=w_2(\phi ,v)=0$, we firstly get its regular chains $[ [\phi ^2 +\phi z+z^2,z^3-4], [\phi , z^3-4], [\phi ^3-4, z], [\phi -1, z-1],[(z+1)\phi +1, z^2+z+1],[p_{11}(z)\phi +p_{12}(z), p_{2}(\phi )]]$, where $p_{11}$ are polynomials of degree 44, 45 and 60, respectively. It is not hard to check that each of the first five regular chains has no common zeros for $\phi \in (0,1)$, $z\in (1,\root 3 \of {4})$. For the last one, we get one pair of root intervals for $\phi \in (0,1)$,$z\in (1,\root 3 \of {4})$. Thus, there exists a unique $(\phi ^*, z(\phi ^*))$ such that $\bar{g}(\phi ^*,z(\phi ^*))=0$ and $w_2(\phi ^*, z(\phi ^*))=0$. Moreover, $\phi ^*\in [111008675/134217728, 55504341/67108864]$. That is, $W[\ell _0(\phi )]\ne 0$ and $W[\ell _0(\phi ),\ell _1(\phi )]$ has a unique zero $\phi ^*$ in (0, 1). In order to determine the multiplicity of this zero, we compute

$$\begin{aligned} \left. \frac{dW[\ell _0(\phi ),\ell _1(\phi )]}{d\phi } =\frac{(\phi -z)^2w_3(\phi ,z)}{\tilde{R}(\phi ,z)}\right| _{z=z(\phi )}, \end{aligned}$$

where $w_3(\phi ,z)$ is a polynomial of degree 49 in $\phi$, $\tilde{R}(\phi ,z):=72\phi ^3z^3(\phi ^2+2\phi z+3z^2)^3(z^3-4)^3(z^2+z+1)^6(z-1)^6(\phi ^3- 4)^3(\phi ^2+\phi +1)^6(\phi -1)^6$. It is not hard to check that the resultant with respect to z between $\bar{g}(\phi ,z)$ and $w_3(\phi ,z)$ has no zeros in $\phi ^*\in [111008675/134217728, 55504341/67108864]$, implying $w_3(\phi ^*,z(\phi ^*))\ne 0$. Thus, $\phi ^*$ is a simple zero of $W[\ell _0(\phi ), \ell _1(\phi )]$. That is, $\phi _0$ is a simple zero of $W[\ell _0(\phi ), \ell _1(\phi )]$. By [10], $\{I_0(h), I_1(h)\}$ is a complete Chebyshev system with accuracy 1. Therefore, in $(h_{31}, 0)$ the Wronskian $W[I_0(h), I_1(h)]$ of $\{I_0(h), I_1(h)\}$ has a unique zero, which is simple. This implies that P(h) is non-monotonic.

When $-4<\beta <-5/2$, by Fig. 4d, there is a homoclinic orbit connecting the saddle $E_2(-\root 3 \of {\beta },0)$ when $h=h_{3a}$. By $X=\phi -1$, $Y=y$ system (3.1)$|_{\varepsilon =0}$ is changed into

$$\begin{aligned} \dot{X}=Y, ~~~~ \dot{Y} =(X+1)[(X+1)^3-1][(X+1)^3+\beta ], \end{aligned}$$

(3.2)

which has Hamiltonian function

$$\begin{aligned} \begin{aligned} H_{32}(X,Y)\!\!=\frac{1}{2}Y^2\! -\!\frac{1}{8}X^8\!-X^7\!-\!\frac{7}{2}X^6\!\! -\!\!\frac{34+\beta }{5}X^5\!-\!\frac{31+4\beta }{4}X^4\!\!-\!(5+2\beta )X^3\!-\!\frac{3(\beta +1)}{2}X^2. \end{aligned} \end{aligned}$$

For system (3.2), (0, 0) is a center and $(-\root 3 \of {\beta }-1,0)$ is a saddle. $\gamma _{l}:= \{(X, Y)\in \mathbb {R}^2:~ H_{32}(X, Y) = l, l\in (0,h_{3a}-h_{31})\}$ is a family of closed orbits and $l=h-h_{31}$. Hence,

$$\begin{aligned} \begin{aligned} I_{0}(h)=\oint _{\gamma _{l}}YdX=J_{10}(l), ~~ I_{1}(h)=\oint _{\gamma _{l}}(X+1)YdX=J_{11}(l)+J_{10}(l), \end{aligned} \end{aligned}$$

implying $P(h)=1+J_{11}(l)/J_{10}(l)$. In a small neighborhood of the origin, let

$$\begin{aligned} \begin{aligned} \chi&:=\!X\left[ \!-\frac{1}{8}X^6-X^5\!-\frac{7}{2}X^4 \!-\frac{34+\beta }{5}X^3\!-\frac{31+4\beta }{4}X^2\!-(5+2\beta )X-\!\frac{3(\beta +1)}{2}\right] ^\frac{1}{2},\\&~\eta :=\frac{Y}{\sqrt{2}}, \end{aligned} \end{aligned}$$

which has an inverse transformation $X=\psi _0(\chi )$, $Y=\sqrt{2}\eta$. Here

$$\begin{aligned} \psi _0(\chi ):=\frac{\sqrt{2}}{\sqrt{-3(\beta +1)}}\chi +\frac{2(2\beta +5)}{9(\beta +1)^2}\chi ^2+O(\chi ^3). \end{aligned}$$

Thus, the closed orbits near the origin are expressed as $\chi ^2+\eta ^2=l$ with $0<l\ll 1$. Then,

$$\begin{aligned} \begin{aligned} J_{10}(l)&=\!-\sqrt{2}\int _0^{2\pi }\!\! l\sin ^2\theta \left[ \frac{\sqrt{2}}{\sqrt{-3(\beta \!+\!1)}}\!+\!\frac{4(4\beta \!+\!5)}{9(\beta \!+\!1)^2}\sqrt{l}\cos \theta \right] d\theta \!+\!O(l^2)\\&=\frac{-2\pi l}{\sqrt{-3(\beta \!+\!1)}}\!+\!O(l^2),\\ J_{11}(l)&=\!-\sqrt{2}\int _0^{2\pi }\!\! l^2\sin ^2\theta \psi _0(\sqrt{l}\cos \theta )\psi _0^\prime (\sqrt{l}\cos \theta )d\theta =\frac{-2\pi l^2(5\!+\!2\beta )}{9(\beta \!+\!1)^2\sqrt{-3(\beta \!+\!1)}}\!+\!O(l^3), \end{aligned} \end{aligned}$$

implying

$$\begin{aligned} \lim _{h\rightarrow h_{31}^+}P'(h)=\lim _{l\rightarrow 0^+}\left( \frac{J_{11}(l)}{J_{10}(l)}\right) '_l= \frac{5+2\beta }{9(\beta +1)^2}<0. \end{aligned}$$

(3.3)

On the other hand, by transformation $X=-\root 3 \of {\beta }-\phi , Y=-y$ system (3.1)$|_{\varepsilon =0}$ is changed into

$$\begin{aligned} \dot{X}=Y, ~~~~ \dot{Y}=\left( -\root 3 \of {\beta }-X\right) \left[ (-\root 3 \of {\beta }-X)^3-1\right] \left[ (-\root 3 \of {\beta }-X)^3+\beta \right] , \end{aligned}$$

(3.4)

which has Hamiltonian function

$$\begin{aligned} H_{33}(X,Y)= & {} \frac{1}{2}Y^2 +\frac{1}{8}X^8 + \beta ^\frac{1}{3}X^7\!\!+\!\frac{7}{2}\beta ^\frac{2}{3}X^6 + \frac{34\beta +1}{5}X^5 + \frac{31\beta + 4}{4}\beta ^\frac{1}{3}X^4\\{} & {} + \beta ^\frac{2}{3}(5\beta \!+\!2)X^3+ \frac{3\beta (1 + \beta )}{2}X^2. \end{aligned}$$

For system (3.4), $(1-\root 3 \of {\beta },0)$ is a center and (0, 0) is a saddle. $\hat{\gamma }_{l}:= \{(X, Y)\in R^2:~ H_{33}(X, Y) = l, l\in (h_{31}-h_{3a}, 0)\}$ is a family of closed orbits and $l=h-h_{3a}$. Hence,

$$\begin{aligned} I_{0}(h)=\oint _{\hat{\gamma }_{l}}YdX=J_{20}(l),~~ I_{1}(h)=\oint _{\hat{\gamma }_{l}}\left( -\root 3 \of {\beta }-X\right) YdX=-\root 3 \of {\beta }J_{20}(l)-J_{21}(l), \end{aligned}$$

implying $P(h)=-\root 3 \of {\beta }-J_{21}(l)/J_{20}(l)$. For $0<-l\ll 1$, by [12] we obtain

$$\begin{aligned}{} & {} J_{20}(l)=\oint _{\hat{\gamma }_{0}}YdX-\frac{1}{\sqrt{3\beta (1+\beta )}}l\ln (-l)+l \oint _{\hat{\gamma }_{0}}\frac{1}{Y}dX +O(l^2\ln (-l)),\\{} & {} J_{21}(l)=\oint _{\hat{\gamma }_{0}}XYdX+l \oint _{\hat{\gamma }_{0}}\frac{X}{Y}dX +O(l^2\ln (-l)), \end{aligned}$$

implying

$$\begin{aligned} \lim _{h\rightarrow h_{3a}^-}P'(h)=\lim _{l\rightarrow 0^-}\left( -\frac{J_{21}(l)}{J_{20}(l)}\right) '_l=\lim _{l\rightarrow 0^-}\frac{- (\ln (-l)+1) \oint _{\gamma _{0}}XYdX}{\sqrt{3\beta (1+\beta )}J_{20}(l)^2}=+\infty . \end{aligned}$$

(3.5)

From (3.3), (3.5), we get $P^\prime (h)<0$ (resp. $>0$) for h near $h_{31}$ (resp. $h_{3a}$), implying that P(h) is non-monotonic in $K_3$. For $-2/5<\beta \le -1/4$, we get the non-monotonicity of P(h) in $K_3$ similarly.

Similarly to case of $q=3$, we prove that in case of $q=4$ function P(h) is non-monotonic when $\beta \in [-5, -13/5)\cup (-5/13, -1/5]$. This lemma is proved. $\square$

From the non-monotonicity of function P(h), we get the existence of at least two near-ordinary periodic wave solutions. In the following we finish the proof of Theorem 3.1.

Proof of Theorem 3.1

In order to prove that there are exactly two near-ordinary periodic wave solutions, we consider (3.1) with $q=3$, $\delta _1=-1$, $\delta _2=-1$, $\beta =-3$, i.e.,

$$\begin{aligned} \left\{ \begin{aligned}&\dot{\phi }=y, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\&\dot{y}=\phi (\phi ^3-1)(\phi ^3-3)+\varepsilon (\alpha _0+\alpha _1\phi )y. \end{aligned} \right. \end{aligned}$$

(3.6)

The corresponding Hamiltonian function of (3.6)$|_{\varepsilon =0}$ is

$$\begin{aligned} H_{34}(\phi ,y)=\frac{1}{2}y^2-\frac{1}{8}\phi ^8+\frac{4}{5}\phi ^5-\frac{3}{2}\phi ^2. \end{aligned}$$

By Fig. 4d, $H_{34}(\phi ,y)=-9\root 3 \of {9}/40$ corresponds to a homoclinic orbit connecting saddle $(\root 3 \of {3},0)$. When $H_{34}(\phi ,y)=h\in (-33/40,-9\root 3 \of {9}/40)$, there is a family of closed orbits $\Gamma _{h}$ surrounding center (1, 0). Moreover, $\Gamma _{h}$ approaches to this homoclinic orbit as $h\rightarrow -9\root 3 \of {9}/40^-$, to the center as $h\rightarrow -33/40^+$. The projection of $\Gamma _h$ on the $\phi$-axis is $(m_2,\root 3 \of {3})$, $m_2$ be the abscissa of the intersection point between the homoclinic orbit and $\phi$-axis. Let $F_4(\phi ):=-\phi ^8 /8+4\phi ^5/5-3\phi ^2/2$. Then we define function $v(\phi )$: $(m_2,1)\rightarrow (1,\root 3 \of {3})$ by $F_4(\phi )=F_4(v(\phi ))$. Since $F_4(\phi )-F_4(v)=(v-\phi )\tilde{g}(\phi ,v)/40$, where

$$\begin{aligned} \begin{aligned} \tilde{g}(\phi ,v)&:=5\phi ^7+5\phi ^6v+5\phi ^5v^2+5\phi ^4v^3+5\phi ^3v^4+5\phi ^2v^5+5\phi v^6+5v^7\\&\quad -32\phi ^4 -32\phi ^3v-32\phi ^2v^2-32\phi v^3-32v^4+60\phi +60v, \end{aligned} \end{aligned}$$

we get $\tilde{g}(\phi ,v(\phi ))=0$.

Similar to Lemma 3.1, we have

$$\begin{aligned} I_i(h) =\frac{1}{4h^2}\oint _{\Gamma _h}\tilde{f}_i(\phi )y^5d\phi , \end{aligned}$$

where $\tilde{f}_i(\phi )=\tilde{H}_i(\phi )+\tilde{G}_i(\phi )+\phi ^i$ and

$$\begin{aligned}{} & {} \tilde{H}_i(\phi )=\frac{\phi ^i\tilde{h}_i(\phi )}{600(\phi -1)^4(\phi ^2+\phi +1)^4(\phi ^3-3)^4},\\{} & {} \tilde{G}_i(\phi )=\frac{\phi ^ir_i(\phi )}{60(\phi -1)^2(\phi ^2+\phi +1)^2(\phi ^3-3)^2}. \end{aligned}$$

Here $\tilde{h}_i(\phi )=25i^2\phi ^{24}+350i\phi ^{24}+325\phi ^{24}-520i^2\phi ^{21}-6020i\phi ^{21}-3880\phi ^{21}+4734i^2\phi ^{18}+43542i\phi ^{18}+ 16020\phi ^{18}-24472i^2\phi ^{15}-171860i\phi ^{15}-22864\phi ^{15}+77953i^2\phi ^{12}+405416i\phi ^{12}-17921\phi ^{12} -155136i^2\phi ^9-605904i\phi ^9+128544\phi ^9+185976i^2\phi ^6+621756i\phi ^6250452\phi ^6-120960i^2\phi ^3-449280i\phi ^3+138240\phi ^3+32400i^2+162000i+ 12960$ and $r_i(\phi )=5i\phi ^{12}+5\phi ^{12}-52i\phi ^9-16\phi ^9+203i\phi ^6-67\phi ^6 - 336i\phi ^3+96\phi ^3+180i+180$. Let $\tilde{\ell }_i(\phi ):=\tilde{f}_i(\phi )/F'_4(\phi )-\tilde{f}_i(v(\phi ))/F'_4(v(\phi ))$, $i=0,1$. The Wronskians $\widetilde{W}[\tilde{\ell }_1(\phi )], \widetilde{W}[\tilde{\ell }_0(\phi ),\tilde{\ell }_1(\phi )]$ of $\{\tilde{\ell }_0(\phi ), \tilde{\ell }_1(\phi )\}$ can be computed as

$$\begin{aligned} \widetilde{W}[\tilde{\ell }_1(\phi )]=\left. \frac{(\phi -v)\tilde{w}_1(\phi ,v)}{\widetilde{P}(\phi ,v)}\right| _{v=v(\phi )},~~~~~ ~~ \widetilde{W}[\tilde{\ell }_0(\phi ),\tilde{\ell }_1(\phi )]=\left. \frac{(v-\phi )^3\tilde{w}_2(\phi ,v)}{\widetilde{Q}(\phi ,v)p(\phi ,v)}\right| _{v=v(\phi )}, \end{aligned}$$

where $p(\phi ,v):=5\phi ^6+10\phi ^5v+15\phi ^4v^2+20\phi ^3v^3+25\phi ^2v^4+30\phi v^5+35v^6-32\phi ^3-64\phi ^2v-96\phi v^2-128v^3+60$, $\widetilde{P}(\phi ,v):=1500(\phi -1)^5(\phi ^3-3)^5(\phi ^2+\phi +1)^5(v-1)^5(v^3-3)^5(v^2+v+1)^5, \widetilde{Q}(\phi ,v):=3000000\phi ^2v^2(v^3\!-3)^9(v^2+v+1)^9 (v-1)^9(\phi ^3-3)^9(\phi ^2+\phi +1)^9(\phi -1)^9$, $\tilde{w}_1(\phi ,v)$, $\tilde{w}_2(\phi ,v)$ are polynomials in $(\phi ,v)$ of degrees 53 and 101, respectively. For $\phi \in (m_2,1)$ and $v\in (1,\root 3 \of {3})$, both $\widetilde{P}(\phi ,v)$ and $\widetilde{Q}(\phi ,v)$ are nonzero. On one hand, the resultant with respect to v between $\tilde{g}(\phi ,v)$ and $p(\phi ,v)$ is computed as $R(\tilde{g},p,v)= 4096000000(5\phi ^6-32\phi ^3+60)\varphi _3(\phi ),$ where $\varphi _3(\phi )$ is a polynomial of degree 36. We check that $\varphi _3(\phi )\ne 0$ for all $\phi \in (m_2,1)$. Associated with $5\phi ^6-32\phi ^3+60=5(\phi ^3-4/\sqrt{5})^2+44\ne 0$, we get $R(\tilde{g},p,v)\ne 0$, that is, $\tilde{g}(\phi ,v)$ and $p(\phi ,v)$ have no common roots for $\phi \in (m_2,1)$, $v\in (1,\root 3 \of {3})$, implying $p(\phi ,v(\phi ))\ne 0$ for $\phi \in (m_2,1)$.

To solve $\tilde{g}(\phi ,v)=\tilde{w}_1(\phi ,v)=0$, we firstly get its regular chains $[\phi ^2 +\phi v+v^2,5v^6-32v^3+60], [\phi -v, v^3-4], [\phi -v,v^2+v+1], [\phi ^3,v], [\phi -1, v-1],[p_{11}(v)\phi +p_{12}(v), p_2(v)]$, where $p_{11}$, $p_{12}$ and $p_2$ are polynomials of degrees 306, 307 and 330, respectively. It is not hard to check that each regular chain has no common zeros for $\phi \in (m_2,1)$, $v\in (1,\root 3 \of {3})$. Thus, $\tilde{g}(\phi , v)$ and $\tilde{w}_1(\phi , v)$ have no common roots for $\phi \in (m_2,1)$, $v\in (1,\root 3 \of {3})$. Therefore, $\tilde{w}_1(\phi ,v(\phi ))\ne 0$ for $\phi \in (m_2,1)$, implying $\widetilde{W}[\tilde{\ell }_1(\phi )]\ne 0$.

To solve $\tilde{g}(\phi ,v)=\tilde{w}_2(\phi ,v)=0$, we firstly get its regular chains $[\phi -v, v^3-3], [\phi -v, v^2+v+1], [\phi -1,v-1], [\phi ,v],[\eta _{11}^*(v)\phi +\eta _{12}^*(v), \eta _2^*(\phi ,v)]$, where $\eta _{11}^*$, $\eta _{12}^*$ and $\eta _2^*$ are polynomials of degrees 600, 601 and 666, respectively. It is not hard to check that each of the first four regular chains has no common zeros for $\phi \in (m_2,1)$, $v\in (1,\root 3 \of {3})$. For the last one, we get one pair of root intervals for $\phi \in (m_2,1)$,$v\in (1,\root 3 \of {3})$. Thus, there exists a unique $(\phi ^*, v(\phi ^*))$ such that $\tilde{g}(\phi ^*,v(\phi ^*))=0$ and $\tilde{w}_2(\phi ^*, v(\phi ^*))=0$. Moreover, $\phi ^*\in [19926445/33554432, 9963355/16777216]$. That is, $\widetilde{W}[\tilde{\ell }_1(\phi )]\ne 0$ and $\widetilde{W}[\tilde{\ell }_0(\phi ),\tilde{\ell }_1(\phi )]$ has a unique zero $\phi ^*$ in $(m_2,1)$. In order to determine the multiplicity of this zero, we compute

$$\begin{aligned} \left. \frac{d\widetilde{W}[\tilde{\ell }_0(\phi ),\tilde{\ell }_1(\phi )]}{d\phi } =\frac{(\phi -v)^2\tilde{w}_3(\phi ,v)}{\tilde{R}(\phi ,v)p^3(\phi ,v)}\right| _{v=v(\phi )}, \end{aligned}$$

where $\tilde{w}_3(\phi ,v)$ is a polynomial of degree 79 in $\phi$, $\tilde{R}(\phi ,v):=1500000\phi ^3v^3(v^3-3)^{10}(v^2+v+1)^{10}(v-1)^{10}(\phi ^3-3)^{10}(\phi ^2+\phi +1)^{10} (\phi -1)^{10}$. It is not hard to check that the resultant with respect to v between $\tilde{g}(\phi ,v)$ and $\tilde{w}_3(\phi ,v)$ has no zeros in [19926445/33554432, 9963355/16777216], implying $\tilde{w}_3(\phi ^*,v(\phi ^*))\ne 0$. Thus, $\phi ^*$ is a simple zero of $\widetilde{W}[\tilde{\ell }_0(\phi ), \tilde{\ell }_1(\phi )]$.

By [10, 24], $\{I_0(h),I_1(h)\}$ is a T-system with accuracy 1 for $\phi \in (m_2,1)$. Thus, $I(h,\delta )$ has at most two zeros in $(-33/40,-9\root 3 \of {9}/40)$. By the non-monotonicity of P(h) given in Lemma 3.1, $I(h,\delta )$ has at least two zeros in $(-33/40,-9\root 3 \of {9}/40)$. Therefore, $I(h,\delta )$ has exactly two zeros in $(-33/40,-9\root 3 \of {9}/40)$, i.e., there are exactly two near-ordinary periodic wave solutions. $\square$

4 Simulations and Discussions

In this section, we present some simulations to verify the theoretical results of near-ordinary periodic wave solutions given in Theorems 2.1 and 3.1 for Eq. (1.3) and discuss our results further, special for the degree of the nonlinear reaction.

For the case $q=3, \delta _1=1, \delta _2=-1$, taking $\beta =1/2$ we get a family of closed orbits $\Gamma _h$ for the unperturbed system and obtain the monotonic strictly function P(h) for $h\in (-9/40, 0)$ as shown in Fig. 6. By straight computation, $P(-0.1)\approx 0.9269240824$. Further, we take $\alpha _0=-0.9269240824, \varepsilon =0.001, a_1=0.001$ and then get $\alpha _1=1$, $\alpha _0/\alpha _1 \approx -P(-0.1)$. Thus, for the near-Hamiltonian system (2.2) we get a unique periodic orbit as shown in Fig. 7a. Then for Eq. (1.3), the corresponding near-ordinary periodic wave solution $u(x,t)=\phi (x+ct)$ is obtained as shown in Fig. 7b with wave speed $c=a_0-0.0009269240824$, which is sufficiently close to the ordinary wave speed $a_0$. This is consistent with Theorem 2.1.

Fig. 6

Monotonicity of P(h) with $q=3, \delta _1=1, \delta _2=-1, \beta =1/2$

Fig. 7

Uniqueness. a A unique periodic orbit; b a unique near-ordinary periodic wave solution

For the case $q=3, \delta _1=-1, \delta _2=-1$, taking $\beta =-3$ and $\varepsilon =0.001$, the graph of P(h) is shown in Fig. 8, then we get that the traveling system has a family of closed orbits $\Gamma _h$ for $h\in (-33/40, -9\root 3 \of {9}/40)$. Choosing three different points $h_1=-0.48$, $h_2=-0.55$, $h_3=-0.65$ in the interval $(-0.7634279698,-9\root 3 \of {9}/40)$, we get $P(h_1)\approx 0.9948070928$, $P(h_2)\approx 0.9918030434$ and $P(h_3)\approx 0.9935019279$. Therefore, we can find two points $\hat{h}=-0.5\in (h_2,h_1)$ and $\tilde{h}\approx -0.6327407090501\in (h_3,h_2)$ such that $P(\hat{h})=P(\tilde{h})\approx 0.99302409\in (P(h_2),P(h_3))$. Consequently, taking $\alpha _0=-0.99302409, a_1=0.001$ we get $\alpha _0/\alpha _1=\alpha _0\varepsilon /a_1=-0.99302409\approx P(\hat{h})$ and hence obtain two periodic orbits for system (2.2), see Fig. 9a. Then for Eq. (1.3), two corresponding near-ordinary periodic wave solutions are obtained with the wave speed $c=a_0-0.00099302409$. The profile is shown in Fig. 9b. Although P(h) is not monotonic strictly for $h\in (-33/40, -9\root 3 \of {9}/40)$, P(h) is one to one for $h\in (-33/40, -0.7634279698]$. Fixing $\alpha _0=-0.99302409$ and choosing $a_1=0.0009940505919$, we get a unique $\bar{h}=-0.8\in (-33/40, -0.7634279698)$ such that $P(\bar{h})\approx 0.9989673545=-\alpha _0/\alpha _1$. It implies that there is a unique near-ordinary periodic wave solution for Eq. (1.3), which has wave speed $c=a_0-0.00099302409$. Therefore, we see that there exists $a_1$ such that the number of near-ordinary periodic wave solutions of Eq. (1.3) is at least 2 as stated in Theorem 3.1 but, it is possible that there exist some other values of $a_1$ such that Eq. (1.3) has a unique near-ordinary periodic wave solution.

Fig. 8

Non-monotonicity of P(h) with $q=3, \delta _1=-1, \delta _2=-1, \beta =-3$

Fig. 9

Non-uniqueness. a Two periodic orbits; b two near-ordinary periodic wave solutions

It is worth to point out that in [31] the reaction–convection–diffusion Eq. (1.1) with $D(u)=1$, $F(u)=\alpha u^m,$ $R(u)=\gamma u (1-u^m)$ is investigated and the monotonicity of the ratio of two Abelian integrals is given when $m=1, 3, 2^k-2$, where $\gamma >0$ and $k>1$ is an integer. On this parametric condition, the corresponding traveling wave system has a center at the origin, which leads to $\phi =-\tilde{\phi }$. Here $\phi$ and $\tilde{\phi }$ denote the abscissas of the intersection points between a closed orbit and $\phi$-axis. Additionally, the Hamiltonian function has a unique term including m, which provides advantage to analyze the criterion function. Subsequently, the sign of the ratio of two Abelian integrals along with [20, Corollary 2] is derived and then the existence of unique periodic wave solution for this reaction–convection–diffusion equation is proved. In this paper, in the case $\delta _1=1, \delta _2=-1, \beta \in (0,1)$ we obtain the uniqueness of near-ordinary periodic traveling wave solutions for reaction–convection–diffusion Eq. (1.3) with $q=3, 4$ in Theorem 2.1. Associated with the uniqueness given in [32] for $q=2$, we provide the following conjecture about the uniqueness of near-ordinary periodic traveling wave solutions for any degree of the nonlinear reaction.

Conjecture 4.1

For any positive integer q, Eq. (1.3) has at most one near-ordinary periodic wave solutions when $\delta _1=1, \delta _2=-1, \beta \in (0,1)$ and it is reachable.

In the case $q=1$, $\delta _1=1, \delta _2=-1$ and general $\beta \in (0,1)$, we need to determine the sign of the criterion function G(s) for $s\in [0,z(h)]$ defined in (2.9), where

$$\begin{aligned} \begin{aligned} G(s)&=\left. \left( \frac{1}{4}\phi ^4+\frac{\beta -1}{3}\phi ^3-\frac{\beta }{2}\phi ^2\right) \right| _{\phi =w(h)+s} -\left. \left( \frac{1}{4}\phi ^4+\frac{\beta -1}{3}\phi ^3-\frac{\beta }{2}\phi ^2\right) \right| _{\phi =w(h)-s}\\&=2s\left( w(h)+\frac{\beta -1}{3}\right) \left( s^2-z(h)^2\right) . \end{aligned} \end{aligned}$$

Here $w(h)=(\mu (h)+\nu (h))/2$ as given below (2.6). Since the traveling wave system has a center at (1, 0), we get $w(h)>1/2$. Then $G(s)<0$ for all $s\in (0,z(h))$. By the method used in the proof of Theorem 2.1, function P(h) is monotonic strictly and then Conjecture 4.1 holds for $q=1$.

By Theorem 2.1 and [32], Conjecture 4.1 also holds for $q=2,3,4$. We tried to prove it for $q\ge 5$ but failed. In fact, it suffices to prove that ratio function P(h) is monotonic strictly for all $h\in \left( q(\beta +1-\beta q)(2q+2)^{-1}(q+2)^{-1}, 0\right)$ and uniformly for all $\beta \in (0,1)$. Here the interval of h corresponds to a family of periodic orbits of the unperturbed traveling wave system (2.2)$|_{\varepsilon =0}$. This traveling wave system admits a center (1, 0) and a homoclinic orbit connecting saddle (0, 0). Then the criterion function

$$\begin{aligned} \begin{aligned} G(s)\!&=\!\frac{1}{2q+2}[(w(h)\!+s)^{2q+2}\!-\!(w(h)\!-\!s)^{2q+2}]+\!\frac{\beta -1}{q+2}\! \,\, [(w(h)+s)^{q+2}\!\\&-\!(w(h)\!-s)^{q+2}]\!-\!2\beta w(h)s\\&=\frac{1}{q+1}\sum _{i=0}^{q}C_{2q+2}^{2i+1}w(h)^{2(q-i)+1}s^{2i+1} +\frac{2(\beta -1)}{q+2}\sum _{i=0}^{q}C_{2q+2}^{i+1}w(h)^{q-i+1}s^{i+1}\\&\quad -\!2\beta w(h)s. \end{aligned} \end{aligned}$$

Unfortunately, for general $q\ge 5$ and $\beta \in (0,1)$ we are not able to verify the sign of G(s). So we failed to prove Conjecture 4.1 for $q\ge 5$ because the Hamiltonian function $H(\phi ,y)$ is too complicated to provide an effective help.

Trying our best to the quest for Conjecture 4.1, we choose some certain $q\ge 5$ and $\beta \in (0,1)$ as examples. Taking $q=7$, $\beta =1/2$, we can prove that P(h) is monotonic strictly by showing $\{I_0(h), I_1(h)\}$ is an extended complete Chebyshev system (see[10]), which is effective for P(h) without parameters. Taking $q=10$, $\beta =1/7$ as another example, we also prove the strict monotonicity of P(h) as above. To observe the strict monotonicity of P(h) clearly, we give the numerical simulations in Fig. 10a for $q=7$, $\beta =1/2$ and in Fig. 10b for $q=10$, $\beta =1/7$.

Fig. 10

Monotonicity of P(h) with $\delta _1=1, \delta _2=-1$. a $q=7$, $\beta =1/2$; b $q=10$, $\beta =1/7$