Soliton molecules and mixed solutions of the(2+1)-dimensional bidirectional Sawada-Kotera equation

In this paper, we investigate some new interesting solution structures of the(2+1)-dimensional bidirectional Sawada–Kotera(bSK) equation. We obtain soliton molecules by introducing velocity resonance. On the basis of soliton molecules, asymmetric solitons are obtained by changing the distance between two solitons of molecules. Based on the N-soliton solutions,several novel types of mixed solutions are generated, which include the mixed breather-soliton molecule solution by the module resonance of the wave number and partial velocity resonance,the mixed lump-soliton molecule solution obtained by partial velocity resonance and partial long wave limits, and the mixed solutions composed of soliton molecules(asymmetric solitons), lump waves, and breather waves. Some plots are presented to clearly illustrate the dynamic features of these solutions.     

Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications

In the present study, we are concerned with the generalized Boussinesq equation including the singular sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional sixth-order nonlinear Boussinesq equation with constant coefficients. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions is discussed.     

Breather molecules and localized interaction solutions in the (2+1)-dimensional BLMP equation

The (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is an important integrable model. In this paper, we obtain the breather molecule, the breather-soliton molecule and some localized interaction solutions to the BLMP equation. In particular, by employing a compound method consisting of the velocity resonance, partial module resonance and degeneration of the breather techniques, we derive some interesting hybrid solutions mixed by a breather-soliton molecule/breather molecule and a lump, as well as a bell-shaped soliton and lump. Due to the lack of the long wave limit, it is the first time using the compound degeneration method to construct the hybrid solutions involving a lump. The dynamical behaviors and mathematical features of the solutions are analyzed theoretically and graphically. The method introduced can be effectively used to study the wave solutions of other nonlinear partial differential equations.     

Kink degeneracy and rogue potential solution for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, we obtained the exact breather-type kink soliton and breather-type periodic soliton solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation using the extended homoclinic test technique. Some new nonlinear phenomena, such as kink and periodic degeneracies, are investigated. Using the homoclinic breather limit method, some new rational breather solutions are found as well. Meanwhile, we also obtained the rational potential solution which is found to be just a rogue wave. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.     

Dynamics of D’Alembert wave and soliton molecule for a (2+1)-dimensional generalized breaking soliton equation

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.     

Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation

New exact wave solutions including homoclinic wave, kink wave and soliton solutions for the 2D Ginzburg-Landau equation are obtained using the auxiliary function method, generalized Hirota method and the ansatz function technique under the certain constraint conditions of coefficients in equation, respectively. The result shows that there exists a kink-wave solution which tends to one and the same periodic wave solution as time tends to infinite.     

Dynamics of a D’Alembert wave and a soliton molecule for an extended BLMP equation

The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory.The study of the D’Alembert wave is worthy of deep consideration in nonlinear partial differential systems.In this paper,we construct a(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli(eBLMP)equation which fails to pass the Painleve property.The D’Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable.The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation.The dynamics of the three-soliton molecule,the three-kink soliton molecule,the soliton molecule bound by an asymmetry soliton and a one-soliton,and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.     

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and(2+1)-dimensional breaking soliton equations via a generalized exponential rational function(GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons,double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons,elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.     

一类非线性演化方程新的显式行波解

借助Mathematica软件,采用三角函数法和吴文俊消元法,获得了一类非线性演化方程u_tt+au_xx+bu+cu³=0的三组行波解,其中包括新的行波解、扭状孤波解和钟状孤波解.从而作为该方程的特例,如Duffing方,Klein-Gordon方程、Landau-Ginburg-Higgs方程和⁴方程等也都获得了相应的若干行波解.这种方法也适用于其他非线性方程. 关键词：     

Exact solutions of the nonlocal Gerdjikov-Ivanov equation

The nonlocal nonlinear Gerdjikov-Ivanov (GI) equation is one of the most important integrable equations, which can be reduced from the third generic deformation of the derivative nonlinear Schrödinger equation. The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation. As applications, we obtain the bright-dark soliton, breather, rogue wave, kink, W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation. These solutions show rich wave structures for selections of different parameters. In all these instances we practically show that these solutions have different properties than the ones for local case.     

非线性延时修正光纤孤子方程的稳态解问题Ⅰ.静态扭结孤波解

严格求解含非线性延时修正光纤孤立子方程,得到一类完全不同于光纤中已知的亮孤子和暗孤子的新型光学孤波解,并讨论了其物理含义及在光纤实验中观察这种扭结孤波的可能性。     

非线性波方程尖峰孤子解的一种简便求法及其应用

根据尖峰孤子解的特点,提出了一种待定系数法求非线性波方程尖峰孤子解的思路和方法,并利用该方法求解了5个非线性波方程,即CH（Camassa-Holm)方程、五阶KdV-like 方程、广义Ostrovsky方程、组合KdV-mKdV方程和Klein-Gordon方程,比较简便地得到了这些方程的尖峰孤子解.文献中关于CH方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.简要说明了非线性波方程存在尖峰孤子解所须满足的特定条件.该方法也适用于求其他非线性波方程的尖峰孤子解. 关键词： 非线性波方程 尖峰孤子解 待定系数法     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

Solitons molecules,lump and interaction solutions to a (2+1)-dimensional Sharma–Tasso–Olver–Burgers equation

Different resonance constraints enrich the behavior of soliton solutions. The soliton molecules, which are the bound states of solitons, can be set off by the velocity resonance. The lump waves, which are localized in all directions in space, are theoretically regarded as a limit form of soliton in some ways. In this paper, a (2+1)-dimensional Sharma–Tasso–Olver–Burgers (STOB) equation is investigated. Soliton (kink) molecule, half periodic kink(HPK) and HPK molecule are studied. Then the lump solution is obtained and the interactions between lump and kink molecule are discussed. The kink molecule-lump solutions exhibit a fusion phenomenon and a rogue (instanton) phenomenon, respectively.     

The window Josephson junction: a coupled linear nonlinear system

We investigate the interface coupling between the two-dimensional sine-Gordon equation and the two-dimensional wave equation in the context of a Josephson window junction using a finite volume numerical method and soliton perturbation theory. The geometry of the domain as well as the electrical coupling parameters are considered. When the linear region is located at each end of the nonlinear domain, we derive an effective one-dimensional model, and using soliton perturbation theory, compute the fixed points that can trap either a kink or antikink at an interface between two sine-Gordon media. This approximate analysis is validated by comparing with the solution of the partial differential equation and describes kink motion in the one-dimensional window junction. Using this, we analyze steady-state kink motion and derive values for the average speed in the one- and two-dimensional systems. Finally, we show how geometry and the coupling parameters can destabilize kink motion.     

有限宽度背景中的啁啾灰孤子

基于在正常色散区的变系数非线性薛定谔方程,考虑一个带有微扰的参数渐减光纤系统,并利用数值模拟方法,对超高斯型有限宽度背景波和有限宽度背景中啁啾灰孤子的传输进行详细地研究.结果表明,超高斯背景波可以在带有微扰的参数渐减光纤系统中不受负载啁啾灰孤子的影响而稳定传输.当取背景波脉宽与啁啾孤子的初始脉宽比例大于或等于50时,有限宽度背景中啁啾灰孤子的数值结果基本与精确解相吻合.即使选取的背景波脉宽不宽,有限宽度背景中的啁啾灰脉冲仍可以很好的保持其孤子性质.     

Soliton solutions of some nonlinear evolution equations with time-dependent coefficients

In this paper, we obtain exact soliton solutions of the modified KdV equation, inhomogeneous nonlinear Schrödinger equation and G(m, n) equation with variable-coefficients using solitary wave ansatz. The constraint conditions among the time-dependent coefficients turn out as necessary conditions for the solitons to exist. Numerical simulations for dark and bright soliton solutions for the mKdV equation are also given.     

The extended auxiliary equation method for the KdV equation with variable coefficients

This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.     

A general method for the solution of nonlinear soliton and kink Schrödinger equations

We present a method by which one-dimensional nonlinear soliton and kink Schrödinger equations can be solved in closed form. The hermitean nonlinear soliton operator may contain up to second derivatives of the wave function and the vanishing condition must hold. The method is applied to solve known nonlinear Schrödinger equations for one-soliton and one-kink solutions and, by inverting the procedure, to derive new operators with wave packet solutions of algebraic and arbitrary shapes. One of them is equivalent to the Derivative Nonlinear Schrödinger equation.     

组合KdV方程的双扭结孤立波及其稳定性研究

利用扩展的双曲函数法得到了combined KdV-mKdV (cKdV)方程的几类精确解,其中一类为具有扭结—反扭结状结构的双扭结单孤子解.在不同的极限情况下,该解分别退化为cKdV方程的扭结状或钟状孤波解.理论分析表明,cKdV方程既有传播型孤立波解,也有非传播型孤立波解.文中对双扭结型孤立波解的稳定性进行了数值研究,结果表明,cKdV方程既存在稳定的双扭结型孤立波,也存在不稳定的双扭结型孤立波. 关键词： cKdV方程 双扭结单孤子 稳定性