New exact solutions of the Broer–Kaup equations in (2 + 1)-dimensional spaces

With the aid of Maple, several new kinds of exact solutions for the Broer–Kaup equations in (2 + 1)-dimensional spaces are obtained by using a new ansätz. This approach can also be applied to other nonlinear evolution equations.     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Soliton and Periodic Wave Solutions of the Nonlinear Loaded (3+1)-Dimensional Version of the Benjamin-Ono Equation by Functional Variable Method

In this article, we establish new travelling wave solutions for the nonlinear loaded (3+1)-dimensional version of the Benjamin-Ono equation by the functional variable method. The performance of this method is reliable and effective and the method provides the exact solitary wave solutions and periodic wave solutions. The solution procedure is very simple and the traveling wave solutions are expressed by hyperbolic functions and trigonometric functions. After visualizing the graphs of the soliton solutions and the periodic wave solutions, the use of distinct values of random parameters is demonstrated to better understand their physical features. It has been shown that the method provides a very effective and powerful mathematical tool for solving nonlinear equations in mathematical physics.     

On a 2+1‐Dimensional Whitham–Broer–Kaup System: A Resonant NLS Connection

It is established that the Whitham–Broer–Kaup shallow water system and the “resonant” nonlinear Schrödinger equation are equivalent. A symmetric integrable 2+1‐dimensional version of the Whitham–Broer–Kaup system is constructed which, in turn, is equivalent to a recently introduced resonant Davey–Stewartson I system incorporating a Madelung–Bohm type quantum potential. A bilinear representation is adopted and resonant solitonic interaction in this new 2+1‐dimensional Kaup–Broer system is exhibited.     

Multiple soliton solutions for (2 + 1)‐dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations

Multiple soliton solutions for the (2 + 1)‐dimensional Sawada–Kotera and the Caudrey–Dodd–Gibbon equations are formally derived. Moreover, multiple singular soliton solutions are obtained for each equation. The simplified form of Hirota's bilinear method is employed to conduct this analysis. Copyright © 2011 John Wiley & Sons, Ltd.     

Solving the (2 + 1)-dimensional higher order Broer–Kaup system via a transformation and tanh-function method

In this paper, the (2 + 1)-dimensional higher order Broer–Kaup system is reduced to a simple nonlinear partial differential equation by a transformation, and utilizing tanh-function method we obtain many new exact solutions for the (2 + 1)-dimensional higher order Broer–Kaup system.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

Consistent Riccati expansion for finding interaction solutions of (2+1)‐dimensional modified dispersive water‐wave system

A consistent Riccati expansion (CRE) is proposed to solve the (2+1)‐dimensional modified dispersive water‐wave (MDWW) system. It is proved that the MDWW system is CRE solvable. Furthermore, new exact interaction solutions, namely, soliton‐trigonometric waves, trigonometric waves‐soliton, soliton‐cosine periodic waves, and soliton‐cnoidal waves are explicitly derived.     

(3+1)维短波方程的不变子空间和精确解

利用不变子空间方法研究了(3+1)维短波方程的不变子空间和精确解.在(2+1)维短波方程增加一维的情形下,构造了更加广泛的精确解,同时也得到了超曲面的爆破解.主要结果不仅推广了不变子空间理论在高维非线性偏微分方程中的应用,而且对研究高维方程的动力系统有重要意义.     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and (2+1) dimensions

The tanh method is proposed to find travelling wave solutions in (1+1) and (2+1) dimensional wave equations. It can be extended to solve a whole family of modified Korteweg–de Vries type of equations, higher dimensional wave equations and nonlinear evolution equations.     

Jacobi elliptic function solutions of the (1 + 1)‐dimensional dispersive long wave equation by Homotopy Perturbation Method

In this article, we try to obtain approximate Jacobi elliptic function solutions of the (1 + 1)‐dimensional long wave equation using Homotopy Perturbation Method. This method deforms a difficult problem into a simple problem which can be easily solved. In comparison with HPM, numerical methods leads to inaccurate results when the equation intensively depends on time, while He's method overcome the above shortcomings completely and can therefore be widely applicable in engineering. As a result, we obtain the approximate solution of the (1 + 1)‐dimensional long wave equation with initial conditions. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

New (3+1)‐dimensional nonlinear equations with KdV equation constituting its main part: multiple soliton solutions

In thiswork,we present two new(3+1)‐dimensional nonlinear equationswith Korteweg‐de Vries equation constituting its main part. We show that the dispersive relation is distinct for each model, whereas the phase shift remains the same. We determine multiple solitons solutions, with distinct physical structures, for each established equation. The architectures of the simplified Hirota's method is implemented in this paper. The constraint conditions that fall out which must remain valid in order for themultiple solitons to exist are derived.Copyright © 2015 John Wiley & Sons, Ltd.     

Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko-Dubrovsky equations

In this paper, new exact solutions with two arbitrary functions of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by means of the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. It is shown that the Exp-function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.     

New exact solutions and fractal patterns of generalized Broer–Kaup system via a mapping approach

With the help of an extended mapping approach, a new type of variable separation solution with two arbitrary functions of the (2 + 1)-dimensional generalized Broer–Kaup (GBK) system is derived. Based on the derived solitary wave excitation, we reveal some regular fractal and stochastic fractal patterns in the (2 + 1)-dimensional GBK system.     

Explicit N-fold Darboux transformation and multi-soliton solutions for the (1 + 1)-dimensional higher-order Broer–Kaup system

In this paper, we present a new N-fold Darboux transformations of the (1 + 1)-dimensional higher-order Broer–Kaup (HBK) system with the help of a gauge transformation of the spectral problem. As an application, new explicit (2N − 1)-soliton solutions of the (1 + 1)-dimensional HBK system are obtained. Both the N-fold Darboux transformation and (2N − 1)-soliton solutions can be written explicitly in terms of Vandermonde-like determinants which are remarkable compactness and transparency.     

Bifurcations and parametric representations of traveling wave solutions for the (2+1)‐dimensional Boiti–Leon–Pempinelle system

By using the method of bifurcation theory of planar dynamical systems to the traveling wave system of the (2+1)‐dimensional Boiti–Leon–Pempinelle system, exact explicit parametric representations of the traveling wave solutions are obtained in different parameter regions. Copyright © 2010 John Wiley & Sons, Ltd.     

Group classification and symmetry reduction of a (2+1) dimensional diffusion-advection equation

Based on classical Lie group method, we consider the continuum problem of the driven diffusive flow of particles past an impenetrable obstacle (rod) of length L. The infinitesimals of the diffusion-advection equation in (2+1) dimensions were found for an arbitrary nonlinear advection. The symmetries corresponding to different forms of the nonlinear advection are obtained. Three models are studied in details. The results show that the presence of an obstacle, whether stationary or moving, in a driven diffusive flow with nonlinear drift will distort the local concentration profile to a state which divided the (x, y)-plane into two regions. The concentration is relatively higher in one side than the other side, apart from the value of where D is the diffusion coefficient and υ is the drift velocity. This problem has relevance for the size segregation of particulate matter which results from the relative motion of different-size particles induced by shaking. Also, the obtained solutions include soliton, periodical, rational and singular solutions. Received: November 10, 2003; revised: February 10, 2004     

Stochastic exact solutions to (2 + 1)-dimensional stochastic Borer–Kaup equation

(2 + 1)-dimensional Wick-type stochastic Borer–Kaup equations are researched by homogeneous balance method and tanh-function method. And some stochastic exact solutions of (2 + 1)-dimensional Wick-type stochastic Borer–Kaup equations are obtained via Hermite transformation.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.