A new method of new exact solutions and solitary wave-like solutions for the generalized variable coefficients Kadomtsev——Petviashvili equation

Using the solution of general Korteweg--de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev--Petviashvili (KP) equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.     

The exact solutions to (2 1)-dimensional nonlinear Schrdinger equation

1　IntroductionConsiderthe ( 2 1 ) dimensionalnonlinearSchr dingerequation (NLSE)iψt =ψxy γ2 vψ ( 1 )vx =2 ( ｜ψ｜2 ) y ( 2 )whereγ2 =± 1 .Eqs.( 1 )and ( 2 )playimportantroleinnonlinearopticalphysicalfield (see ,e .g .[1 ]) .InRef.[2 ],ZakharovappliedinversescatteringmethodtoderivethesolitonsolutionsforEqs.( 1 )and ( 2 ) ,andNsolitonsolutionscanbefoundinRef.[3].InRef.[4],RadhaandlakshmananhasinvestigatedthePainlev啨ｐｒｏｐｅｒｔｉｅｓｆｏｒＥｑｓ.( 1 )and ( 2 ) .Thehiddensymme…     

Lie group analysis,numerical and non-traveling wave solutions for the(2+1)-dimensional diffusion–advection equation with variable coefficients

In this paper, the variable-coefficient diffusion–advection(DA) equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended(G /G)-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.     

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrdinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.     

Chaotic behaviors of the （2＋1）-dimensional generalized Breor-Kaup system

With the help of the Maple symbolic computation system and the projective equation approach,a new family of variable separation solutions with arbitrary functions for the(2+1)-dimensional generalized Breor-Kaup(GBK) system is derived.Based on the derived solitary wave solution,some chaotic behaviors of the GBK system are investigated.     

Nonautonomous solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrdinger equation with time- and space-modulated coefficients

A class of analytical solitary-wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schrdinger equation with time- and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain (loss) coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.     

A series of new double periodic solutions to a (2+1)-dimensional asymmetric Nizhnik－Novikov－Veselov equation

By means of a new general ans?tz and with the aid of symbolic computation, a new algebraic method named Jacobi elliptic function rational expansion is devised to uniformly construct a series of new double periodic solutions to (2+1)-dimensional asymmetric Nizhnik－Novikov－Veselov (ANNV) equation in terms of rational Jacobi elliptic function.     

High-order rational solutions and resonance solutions for a (3+1)-dimensional Kudryashov–Sinelshchikov equation

We mainly investigate the rational solutions and N-wave resonance solutions for the(3+1)-dimensional Kudryashov–Sinelshchikov equation, which could be used to describe the liquid containing gas bubbles. With appropriate transformations, two kinds of bilinear forms are derived. Employing the two bilinear equations, dynamical behaviors of nine district solutions for this equation are discussed in detail, including bright rogue wave-type solution, dark rogue wave-type solution, bright W-shaped solution, dark W-shaped rational solution, generalized rational solution and bright-fusion, darkfusion, bright-fission, and dark-fission resonance solutions. In addition, the generalized rational solutions, which depending on two arbitrary parameters, have an interesting structure: splitting from two peaks into three peaks.     

New explicit exact solutions to a nonlinear dispersive－dissipative equation

Using the first-integral method, we obtain a series of new explicit exact solutions such as exponential function solutions, triangular function solutions, singular solitary wave solution and kink solitary wave solution of a nonlinear dispersive－dissipative equation, which describes weak nonlinear ion-acoustic waves in plasma consisting of cold ions and warm electrons.     

The periodic wave solutions for the generalized Nizhnik－Novikov－Veselov equation

The periodic wave solutions expressed by Jacobi elliptic functions for the generalized Nizhnik－Novikov－Veselov equation are obtained using the F-expansion method proposed recently. In the limiting cases, the solitary wave solutions and other types of travelling wave solutions for the system are obtained.     

变系数(2+1)维Broer-Kaup方程新的类孤子解

基于齐次平衡原则和分离变量法的思想,通过两个推广的Riccati方程组和Mathematica软件,求出了变系数(2+1)维Broer-kaup方程的一些精确解,包括各种类孤立波解、类周期解,其中许多解是新的.     

变系数(2+1)维Broer-Kaup方程的精确解

利用齐次平衡原则,导出了变系数(2+1)维Broer-Kaup方程的B?cklund变换(BT),并由该BT,求出了(2+1)维Broer-Kaup方程的各种形式的精确解.     

变系数（2＋1）维Broe-Kaup方程的新精确解

通过一个简单的变换，变系数(2 1)维Broer-Kaup方程被简化为人们熟知的变系数Burgers方程。利用近年来广泛使用的齐次平衡法和tanh-函数法，获得了变系数(2 1)维Broer-Kaup方程的一些新的精确解。     

变系数(2+1)维Broer-Kaup方程的新精确解P

通过一个简单的变换,变系数(2+1)维Broer-Kaup方程被简化为人们熟知的变系数Burgers方程.利用近年来广泛使用的齐次平衡法和tanh-函数法,获得了变系数(2+1)维Broer-Kaup方程的一些新的精确解.     

变系数(2+1)维Nizhnik-Novikov-Vesselov方程的三孤子新解

本文为获得非线性发展方程的相互作用解,研究了辅助方程法,并扩展应用辅助方程法和(G'/G)展开法, 获得了变系数非线性(2+1)维Nizhnik-Novikov-Vesselov方程的由椭圆函数、双曲函数、 三角函数和有理函数混合构成的新相互作用解. 关键词： G'/G)展开法')" href="#">(G'/G)展开法 辅助方程法 三孤子解     

Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.     

New doubly periodic and multiple soliton solutions of the generalized （3＋l）-dimensional KP equation with variable coefficients

A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.     

Travelling wave-like solutions of the Zakharov-Kuznetsov equation with variable coefficients

Travelling wave-like solutions of the Zakharov-Kuznetsov equation with variable coefficients are studied using the solutions of Raccati equation. The solitary wave-like solution, the trigonometric periodic wave solution and the rational wave solution are obtained with a constraint between coefficients. The property of the solutions is numerically investigated. It is shown that the coefficients of the equation do not change the wave amplitude, but may change the wave velocity.      

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.