Redundant exact solutions of nonlinear differential equations

We analyze the paper by Wazwaz and Mehanna [Wazwaz AM, Mehanna MS. A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation. Appl Math Comput 2010;217:1484–90]. The authors claim that they have found exact solutions of the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation using the tanh–coth method and the Exp-function method. We demonstrate that two of their solutions are incorrect. All the others can be simplified and they are the partial cases of the well-known solution. Wazwaz and Mehanna made a number of typical mistakes in finding exact solutions of nonlinear differential equations. Taking the results of this paper we introduce the definition of redundant exact solutions for the nonlinear ordinary differential equations.     

New exact travelling wave solutions of two nonlinear physical models

In this paper, an improved tanh function method is used with a computerized symbolic computation for constructing new exact travelling wave solutions on two nonlinear physical models namely, the quantum Zakharov equations and the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions.The exact solutions are obtained which include new soliton-like solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and its applications are promising.     

Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE, we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.     

累次齐次平衡法及其应用

在求非线性偏微分方程精确解的过程中两次使用了齐次平衡法(称为累次齐次平衡法),解决了齐次平衡法求解少的不足,从而改进了齐次平衡法.以高阶(2+1)维Kadomtsev-Petviashvili方程和变异的Boussinesq方程为应用实例,说明使用累次齐次平衡法可以求得大量的精确解,其中许多解是新解或覆盖了其他方法所得的解.方法可应用于大量的非线性物理模型.     

EXACT SOLUTIONS OF (2+1)-DIMENSIONAL NONLINEAR SCHRöDINGER EQUATION BASED ON MODIFIED EXTENDED TANH METHOD

In this paper, the modified extended tanh method is used to construct more general exact solutions of a(2+1)-dimensional nonlinear Schr¨odinger equation.With the aid of Maple and Matlab software, we obtain exact explicit kink wave solutions, peakon wave solutions, periodic wave solutions and their 3D images.     

Symbolic computation and new families of exact non-travelling wave solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

The improved tanh function method [Chaos, Solitons & Fractals 2005;24:257] is further improved by constructing new ansatz solution of the considered equation. As its application, the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are considered and abundant new exact non-travelling wave solutions are obtained.     

2+1维广义浅水波方程的类孤子解与周期解

该文基于一个Riccati方程组，提出了一个新的广义投影Ric cati展开法，该方法直接简单并能构造非线性微分方程更多的新的解析解。利用该算法研究了(2+1)维广义浅水波方程，并求得了许多新的精确解，包括类孤子解和周期解。该算法也能应用到其它非线性微分方程中。     

任意方程法及BKP方程新的精确解

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.     

Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics

In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.     

New multisoliton solutions of the (2+1)-dimensional dispersive long wave equations

IntroductionSoliton is a complicated mathematical structure based on the nonlinear evolution equation.(1+ 1)-dimensional soliton and solitary wave solutions have been studied we1l and widely appliedto many physics fields like the condense matter physics, fluid mechanics, plasma physics, optics,etc. However, to find some exact physically significant soliton solutions in (2+l)-dimensions ismuch more difficult than in (1+1)-dimensions. Recently, by using some different approashes,one special type…     

New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup–Kupershmidt Equations

In this paper, using the exp-function method we obtain some new exact solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt (KK) equations. We show figures of some of the new solutions obtained here. We conclude that the exp-function method presents a wider applicability for handling nonlinear partial differential equations.     

A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation

In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons & Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained.     

非线性系统的精确解

Based on the computerized symbolic, a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations （PDES） in a unified way. The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions. At the same time, we present a more general transformation, which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations （NLEEs）. More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.     

On the non-differentiable exact solutions of the (2 + 1)-dimensional local fractional breaking soliton equation on Cantor sets

In this article, a new (2 + 1)-dimensional local fractional breaking soliton equation is derived with the local fractional derivative. Applying the traveling wave transform of the non-differentiable type, the (2 + 1)-dimensional local fractional breaking soliton equation is converted into a nonlinear local fractional ordinary differential equation. By defining a set of elementary functions on Cantor sets, a novel analytical technique namely the Mittag–Leffler function-based method is employed for the first time ever to construct the exact solutions. The solutions on the Cantor sets are presented via the 3-D contours. It reveals that the proposed method is effective and powerful and is expected to give some inspiration for the study of the local fractional PDEs.     

Explicit exact solutions for (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, we construct explicit exact solutions for the coupled Boiti–Leon–Pempinelli equation (BLP equation) by using a extended tanh method and symbolic computation system Mathematica. By means of the method, many new exact travelling wave solutions for the BLP system are successfully obtained. the extended tanh method can be applied to other higher-dimensional coupled nonlinear evolution equations in mathematical physics.     

A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation

In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

(3+1)-维广义随机KP方程的精确解

利用统一方式构造非线性偏微分方程行波解的广义Jacobi椭圆函数展开法和Hermite变换来研究(3+1)-维广义随机KP方程,给出了它的随机对偶周期和多孤子解.     

Symmetry group analysis and similarity solutions of variant nonlinear long-wave equations

Symmetry group properties and similarity solutions of the variant nonlinear long-wave equations in the form of system of nonlinear partial differential equations are analyzed. Lie symmetry group analysis of the variant nonlinear long-wave equations presents that the system has only two-parameter point symmetry group that corresponds to only traveling wave solutions. The symmetry groups yield the general reduced similarity form of the system, which is in the system of nonlinear ordinary differential equations. By using the improved tanh method the similarity solutions are obtained from the reduced system of equations. In addition, some graphical representations of the solitary and periodic solutions are presented.     

A new approach for solutions of the (2 + 1)-dimensional cubic nonlinear Schrödinger equation

A new method to solve the nonlinear evolution equations is presented, which combines the two kind methods – the tanh function method and symmetry group method. To demonstrate the method, we consider the (2 + 1)-dimensional cubic nonlinear Schrödinger (NLS) equation. As a result, some novel solitary solutions of the Schrödinger equation are obtained. And graphs of some solutions are displayed.