Anti-kink Wave Solutions Research Articles

Bifurcations and Exact Traveling Wave Solutions of Degenerate Coupled Multi-KdV Equations

In this paper, we consider the degenerate coupled multi-KdV equations. Depending on the coupled multiplicity [Formula: see text], the study of the traveling wave solutions for this model derives a series of planar dynamical systems. We consider the cases of [Formula: see text] On the basis of the investigation on the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions) under different parameter conditions.

Exact traveling wave solutions and bifurcations of the Biswas–Milovic equation

In this paper, we consider the Biswas–Milovic equation. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and compactons) under different parameter conditions.

THE EXACT SOLUTIONS OF GENERALIZED ZAKHAROV EQUATIONS WITH HIGH ORDER SINGULAR POINTS AND ARBITRARY POWER NONLINEARITIES

In this paper, we using bifurcation theory method of dynamical systems to find the exact solutions of generalized Zakharov equations with high order singular points and arbitrary power nonlinearities. Under different parameter conditions, we obtain exact solitary wave solutions, periodic wave solutions as well as kink and anti-kink wave solutions.

Nonlinear Dynamics and Exact Traveling Wave Solutions of the Higher-Order Nonlinear Schrödinger Equation with Derivative Non-Kerr Nonlinear Terms

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.

Qualitative Analysis and Explicit Exact Solitary, Kink and Anti-Kink Wave Solutions of the Generalized Nonlinear Schrödinger Equation with Parabolic Law Nonlinearity**Supported by the National Natural Science Foundation of China under Grant No. 11461022 and the Major Natural Science Foundation of Yunnan Province under Grant No. 2014FA037

The generalized nonlinear Schrödinger equation with parabolic law nonlinearity is studied by using the factorization technique and the method of dynamical systems. From a dynamic point of view, the existence of smooth solitary wave, kink and anti-kink wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible explicit exact parametric representations of the waves are presented.

Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equationut+ux+νuxxt+βuxxx+αuux+1/3να(uuxxx+2uxuxx)+3μα2u2ux+νμα2(u2uxxx+ux3+4uuxuxx)+ν2μα2(ux2uxxx+2uxuxx2)=0is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.

Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy

Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the bifurcation method of planar dynamical systems. The bifurcation regions in different subsets of the parameters space are obtained. According to the different phase portraits in different regions, we obtain kink (antikink) wave solutions, solitary wave solutions, and periodic wave solutions for the third of these models by dynamical system method. Furthermore, the explicit exact expressions of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.

Dust ion acoustic travelling waves in the framework of a modified Kadomtsev-Petviashvili equation in a magnetized dusty plasma with superthermal electrons

For the critical values of the parameters q and V, the work (Samanta et al. in Phys. Plasma 20:022111, 2013b) is unable to describe the nonlinear wave features in magnetized dusty plasma with superthermal electrons. To describe the nonlinear wave features for critical values of the parameters q and V, we extend the work (Samanta et al. in Phys. Plasma 20:022111, 2013b). To extend the work, we derive the modified Kadomtsev-Petviashvili (MKP) equation for dust ion acoustic waves in a magnetized dusty plasma with q-nonextensive velocity distributed electrons by considering higher order coefficients of ϵ. By applying the bifurcation theory of planar dynamical systems to this MKP equation, the existence of solitary wave solutions of both types rarefactive and compressive, periodic travelling wave solutions and kink and anti-kink wave solutions is proved. Three exact solutions of these above waves are determined. The present study could be helpful for understanding the nonlinear travelling waves propagating in mercury, solar wind, Saturn and in magnetosphere of the Earth.

Loop solutions, breaking kink (or anti-kink) wave solutions, solitary wave solutions and periodic wave solutions for the two-component Degasperis–Procesi equation

In this paper, the method of bifurcation of planar dynamic systems is used to investigate bounded traveling wave solutions for the two-component Degasperis–Procesi shallow water equation. By using the phase portrait bifurcation of the traveling wave system, solitary wave solutions, periodic wave solutions and breaking kink (or anti-kink) wave solutions are considered under different parameter conditions. And the loop soliton solution is also discussed. By the use of numerical simulation, the graphs of some solutions are obtained under specific parameter conditions.

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED CAMASSA–HOLM EQUATION

In this paper, we study all possible traveling wave solutions of an integrable system with both quadratic and cubic nonlinearities: [Formula: see text], m = u-uxx, where b, k1 and k2 are arbitrary constants. We call this model a generalized Camassa–Holm equation since it is kind of a cubic generalization of the Camassa–Holm (CH) equation: mt + mxu + 2mux = 0. In the paper, we show that the traveling wave system of this generalized Camassa–Holm equation is actually a singular dynamical system of the second class. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. Some exact solutions such as smooth soliton solutions, kink and anti-kink wave solutions, M-shape and W-shape wave profiles of the breaking wave solutions are obtained. To guarantee the existence of those solutions, some constraint parameter conditions are given.

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE GENERALIZED TWO-COMPONENT CAMASSA–HOLM EQUATION

In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.

BIFURCATION OF PHASE AND EXACT TRAVELING WAVE SOLUTIONS OF A HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION

The dynamical behavior of a higher-order nonlinear Schrödinger equation is studied by using the bifurcation theory method of dynamical systems. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. Moreover, some new traveling wave solutions corresponding to the orbits on phase portraits are given, which include solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions.

Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K(m, n) Equation

By using the integral bifurcation method, we study the nonlinear K(m, n) equation for all possible values of m and n. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton‐like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of K(n, n), K(2n − 1, n), K(3n − 2, n), K(4n − 3, n), and K(m, 1) equations are chosen to illustrate with the concrete features.

Travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation

The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut −uxxt + (1+b)umux = bux + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also given.

Extended Mixed Function Method and Its Application for Solving Two Classic Toda Lattice Equations

The mixed function method is extended from the (1 + 1)‐dimensional space to the (2 + 1)‐dimensional one, even those forms of exact solution do not exist in (1 + 1)‐dimensional NDDEs. By using this extended method, the Toda lattice and (2 + 1)‐dimensional Toda lattice equations are studied. Some new solutions such as discrete solitary wave solutions, discrete kink and antikink wave solutions, and discrete breather soliton solutions are obtained, and their dynamic properties are discussed.

Exact traveling discrete kink-soliton solutions for the discrete nonlinear electrical transmission lines

We investigate exact soliton solutions for the discrete nonlinear electrical transmission line by performing the simplest equation method from a trivial seed solution. Starting from the nonlinear propagation of signals in electrical transmission lines, we derive exact traveling kink and antikink solitary wave solutions. It is shown that under a safe range of parameter, the shape of kink soliton can be controlled well by adjusting the parameter of the line. The analytical solutions for the kink and antikink solitary waves are tested in direct simulations.

Traveling wave solutions to the(n+1)-dimensional sinh–cosh–Gordon equation

Traveling wave solutions for a generalized sinh–cosh–Gordon equation are studied. The equation is transformed into an auxiliary partial differential equation without any hyperbolic functions. By using the theory of planar dynamical system, the existence of different kinds of traveling wave solutions of the auxiliary equation is obtained, including smooth solitary wave, periodic wave, kink and antikink wave solutions. Some explicit expressions of the blow-up solution, kink-like solution, antikink-like solution and periodic wave solution to the generalized sinh–cosh–Gordon equation are given. Planar portraits of the solutions are shown.

Bifurcation of traveling wave solutions for the generalized compound KDV equations

By using the theory of bifurcations of dynamical systems to the generalized compound KDV equations, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Symmetric and non-symmetric waves in the osmosis K(2, 2) equation

We give an improved qualitative method to solve the osmosis K(2, 2) equation. This method combines several characteristics of other methods. Using this method, the existence of symmetric and non-symmetric wave solutions of the osmosis K(2, 2) equation is studied. Besides abundant symmetric forms such as smooth wave solutions, peaked waves, cusped waves, looped waves, stumpons and fractal-like waves, this equation also admits non-symmetric ones including breaking kink wave solutions, breaking anti-kink wave solutions and rampons. As regards this equation most of those solutions, either symmetric or non-symmetric solutions, have not appeared in the literature. We also study the limiting behavior of all periodic solutions as the parameters tend to some special values.

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.