A Generalized Variable-Coefficient Algebraic Method Exactly Solving （3＋1）-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.

Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear SchrSdinger Equations with Variable Coefficient

In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schrodinger equation with variable-coefficient. These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time. Six figures are given to illustrate some features of these solutions. The method can be applied to other nonlinear evolution equations in mathematical physics.

High-Order Solitons and Hybrid Behavior of (3 + 1)-Dimensional Potential Yu-Toda-Sasa-Fukuyama Equation with Variable Coefficients

In this paper, some exact solutions of the (3 + 1)-dimensional variable-coefficient Yu-Toda-Sasa-Fukuyama equation are investigated. By using Hirota’s direct method and symbolic computation, we obtained N-soliton solution. By using the long wave limit method, the N-order rational solution can be obtained from N-order soliton solution. Then, through the paired complexification of parameters, the lump solution is obtained from N-order rational solution. Meanwhile, we obtained a hybrid solution between 1-lump solution and N-soliton (N=1,2) by using the long wave limit method and parameter complex. Furthermore, four different sets of three-dimensional graphs of solitons, lump solutions and hybrid solutions are drawn by selecting four different sets of coefficient functions which include one set of constant coefficient function and three sets of variable coefficient functions.

The extended auxiliary the KdV equation with equation method for 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 new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.

Exact Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilmear transformation from its Lax pairs and End solutions with the help of the obtained bilinear transformation.

Approximate Solution of Lane-Emden Type Equations Using Variation of Parameters Method with an Auxiliary Parameter

Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated using a semi-analytical method called the Variation of parameters method with an auxiliary parameter. In the applied technique, an unknown auxiliary parameter is inserted in Variation of Parameters Method to solve some special cases of these equations. The used algorithm is easy to implement and very effective. The obtained solutions are also fairly accurate.

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Backlund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variable coefficients can affect the conserved density, associated flux, and appearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.

Auxiliary Equation Method and New Exact Solutions of BKP Equation

In this paper, auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation. We study the （2＋1）-dimensional BKP equation and get a series of new types of traveling wave solutions. The method used here can be also extended to other nonlinear partial differential equations.

Using reproducing kernel for solving a class of partial differential equation with variable-coefficients

How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.

Multi-Waves,Breathers,Periodic and Cross-Kink Solutions to the(2+1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

The present article deals with multi-waves and breathers solution of the(2+1)-dimensional variable-coefficient CaudreyDodd-Gibbon-Kotera-Sawada equation under the Hirota bilinear operator method.The obtained solutions for solving the current equation represent some localized waves including soliton,solitary wave solutions,periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method.Mainly,by choosing specific parameter constraints in the multi-waves and breathers,all cases the periodic and cross-kink solutions can be captured from the 1-and 2-soliton.The obtained solutions are extended with numerical simulation to analyze graphically,which results in 1-and 2-soliton solutions and also periodic and cross-kink solutions profiles.That will be extensively used to report many attractive physical phenomena in the fields of acoustics,heat transfer,fluid dynamics,classical mechanics,and so on.We have shown that the assigned method is further general,efficient,straightforward,and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering.We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

New Type of Variable-coefficient KP Equation with Self-consistent Sources and Its Grammian Solutions

New type of variable-coefficient KP equation with self-consistent sources and its Grammian solutions are obtained by using the source generation procedure.

A novel (G'/G)-expansion method and its application to the Boussinesq equation

In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.

Using a New Auxiliary Equation to Construct Abundant Solutions for Nonlinear Evolution Equations

In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As m → 0 and m → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.

New exact periodic solutions to (2+1)-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Symbolic Computations and Exact and Explicit Solutions of Some Nonlinear Evolution Equations in Mathematical Physics

With the aid of symbolic computation system Mathematica, several explicit solutions for Fisher＇s equation and CKdV equation are constructed by utilizing an auxiliary equation method, the so called G′/G-expansion method, where the new and more general forms of solutions are also constructed. When the parameters are taken as special values, the previously known solutions are recovered.

Optical soliton and elliptic functions solutions of Sasa-satsuma dynamical equation and its applications

The Sasa-satsuma(SS)dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers.This dynamical model has important physical significance.In this article,two mathematical techniques namely,improved F-expansion and improved aux-iliary methods are utilized to construct the several types of solitons such as dark soliton,bright soliton,periodic soliton,Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation.These results have imperative applications in sciences and other fields,and construc-tive to recognize the physical structure of this complex dynamical model.The computing work and obtained results show the infuence and effectiveness of current methods.