Exact explicit nonlinear wave solutions to a modified cKdV equation

Abstract: In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.


Introduction
As is well known, the Korteweg-de Vries (KdV) equation has attracted much attention due to its significant nature in physical contexts, stratified internal waves, ion-acoustic wave, plasma physics [1]. Many generalized forms of KdV equation have been introduced, such as modified KdV (mKdV) equation and high-order KdV equation. Besides, There has been considerate studies on time-delayed KdV-related equation. Zhao and Xu [2] dealt with the existence of solitary waves for KdV equation with time delay. Liu et al. [3] studied the KdV-Burgers-Kuramoto chaotic system with distributed delay feedback and analyzed the conditions under which Hopf bifurcation occurs. Baudouin et al. [4] employed two approaches to study the stability of the nonlinear KdV equation with boundary time-delay feedback. Komornik and Pignotti [5] considered well-posedness and exponential decay estimates for a KdV-Burgers equation with time-delay.
In this paper, we focus on the following modified coupled Korteweg-de Vries (cKdV) equation, where α is a constant, which was introduced in [6]. As suggested in [6][7][8][9], Eq (1.1) is a general example of N-component systems, energy dependent schrödinger operators and bi-Hamiltonian structures for multi-component systems. The authors in [6] studied the soliton solutions to Eq (1.1), and demonstrated the soliton fission effect, kink to anti-kink transitions, and multipeaked solitons by using a class of commuting Hamiltonian systems on Riemann surfaces. Additionally, they indicated that many important equations that model physical phenomena in fluid dynamics and nonlinear optics, such as the generalized Kaup equation, the classical Boussinesq equation and the systems governing second harmonic generation (SHG), are connected to the cKdV equation through nonsingular transformations [6], which potentially enables solutions of cKdV equations to be interpreted in the context of these related equations. Besides, Wen and Wang [10] constructed some exact explicit solutions to Eq (1.1) by employing the three forms of (ω/g)-expansion method, i.e., (g /g 2 )-expansion method, (g /g)-expansion method and (g )-expansion method.
The remaining paper is organized as follows. In section 2, we outline the procedure of identifying the bifurcation conditions and obtaining the phase portraits of the corresponding planar system in the different regions of the parametric space. In section 3, we present the main results about exact explicit of nonlinear wave solutions to Eq (1.1), especially, the V-shaped kink (antikink) wave solutions, Wshaped solitary wave solutions, and W-shaped periodic wave solutions, and the proof follows. Section 4 is devoted to numerical simulations of solutions. The paper is ended with the conclusion.

Bifurcation conditions and phase portraits
In this section, we identify the bifurcation conditions and derive the phase portraits corresponding to Eq (1.1).
For given constant c, substituting u( where the prime stands for the derivative with respect to ξ. Integrating the first equation of (2.1) once leads to where g 1 is integral constant. Substituting (2.2) into the second equation of (2.1) and integrating the equation, it follows that where g 2 is integral constant. Letting y = u , we obtain a planar system with Hamiltonian To study the singular points of system (2.5), let and Obviously, if g 1 < 2(c + α) 2 , then f 0 (ϕ) has three zero points, where ϕ 1 = 1 2 2(c + α) 2 − g 1 . In addition, it is easy to obtain the two extreme points of f (ϕ) as follows which denotes the absolute value of extreme values of f 0 (ϕ). Now we can easily give the profiles of f (ϕ) in Figure 1. Figure 1. The profiles of f (ϕ).
Based on the above analysis, we obtain the phase portraits of system (2.4) in Figure 2. Note that in the phase portraits ϕ 2 = −2 1 6 2(c + α) 2 − g 1 and the other ϕ i s are given in Section 3. Remark 1. In the above analysis, we have supposed that g 1 < 2(c + α) 2 . In fact, when g 1 ≥ 2(c + α) 2 , system (2.4) has only one saddle and the phase portrait is similar with Figure 2(a) or 2(g). Therefore, we omit the case when Figure 2. The phase portraits of system (2.4).

Main results and the theoretic derivations of the main results
In this section, we state our results about solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions for the component u(x, t) of Eq (1.1), and especially, we emphasize the results of the V-shaped kink (antikink) wave solution, W-shaped solitary wave solution, and W-shaped periodic wave solution for the component v(x, t) of Eq (1.1). Note that the relation between the solutions of Eq (1.1) and the solutions of system (2.4) can be derived through the transformations u(x, t) = u(ξ) and in the following theorems.

20)
where Substituting (3.20) into the first equation of system (2.4), and integrating along these two special orbits Γ ± 4 , it follows that From (3.21), we get the periodic singular wave solution (3.15).
(iii) Third, from Figure 2(c) or more specifically Figure 3(b), there exist one family of periodic orbits defined by H(ϕ, y) = h, h ∈ (H(ϕ 5 , 0), H(ϕ 6 , 0)), the expressions of which are given by Substituting (3.22) into the first equation of (2.4) and integrating along the family of periodic orbits, it follows that From (3.23), we derive the family of periodic wave solutions (3.16) by the elliptic integral formula 254.00 in [29].
Remark 3. In the above theorems, we just list the results when g 2 ≤ g 1 (c + α), since the results when g 2 > g 1 (c + α) can be derived similarly. Here, we deduce that the profiles of the solutions when g 2 > g 1 (c + α) will be completely symmetric to the corresponding profiles of the solutions when g 2 < g 1 (c + α) about the ξ-axis.

Conclusions
In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the nonlinear wave solutions to the modified cKdV equation (1.1), and obtain exact explicit expressions of the various types of nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Among these solutions, of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found previously. These solutions may be interpreted in the context of some related equations, such as the generalized Kaup equation, the classical Boussinesq equation and the systems governing second harmonic generation (SHG), which are connected to the cKdV equation (1.1) through nonsingular transformations [6]. Additionally, in the Theorems 1 and 2, we see that if the parameters c, α, g 1 and g 2 satisfies certain conditions, the V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions are found. This potentially provide a way to control the appearance of these interesting solutions. Finally, we know that time delay and perturbation play an important in modeling mathematics physics problems [30], so we may further study the solutions and their properties of the time delayed or perturbed version of Eq (1.1).