Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation

Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. Secondly, the nonisospectral BKP can be proved to be Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification, some solutions are obtained by using the standard truncated Painlevé expansion. Finally, based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed.     

SYMMETRIES AND GROUP-INVARIANT SOLUTIONS OF DIFFERENTIAL EQUATIONS

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热方程的非古典势对称群与不变解

主要研究了热方程与波方程的非古典势对称群生成元及相应的群不变解．研究表明对于守恒形式的偏微分方程,可通过其伴随系统求得的非古典势对称群生成元来构造其显式解．这些显式解不能由方程本身的Lie对称群生成元或Lie-B?cklund对称群生成元构造得到．     

Nonclassical Symmetries for Nonlinear Diffusion and Absorption

Nonclassical symmetry methods are used to study the nonlinear diffusion equation with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the source term which permit nonclassical symmetry reductions. In addition to the known source terms obtained by classical symmetry methods, we find new source terms which admit symmetry reductions. We also deduce a class of nonclassical symmetries which are valid for arbitrary diffusivity and deduce corresponding new solution types. This is an important result since previously only traveling wave solutions were known to exist for arbitrary diffusivity. A number of examples are considered and new exact solutions are constructed.     

正交各向异性弹性力学正交关系的研究

新的正交关系被推广到正交各向异性三维弹性力学．将弹性力学新正交关系中构造对偶向量的思路推广到正交各向异性问题．将弹性力学求解辛体系的对偶向量重新排序后,提出了一种新的对偶向量．由混合变量求解法直接得到对偶微分方程．所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点．由于对偶微分矩阵的这一特点,对于正交各向异性三维弹性力学发现了2个独立的、对称的正交关系．采用分离变量法求解对偶微分方程．从正交各向异性弹性力学求解体系的积分形式出发,利用一些恒等式证明了新的正交关系．新的正交关系不但包含原有的辛正交关系,而且比原有的关系简洁．新正交关系的物理意义是对偶方程的解关于z坐标的对称性的体现．辛正交关系是一个广义关系,但辛正交关系可以在一定的条件下以狭义的强形式出现．新的研究成果将为研究正交各向异性三维弹性力学的解析解和有限元解提供新的有效工具．     

Symmetry reduction, exact solutions and conservation laws of the Sawada-Kotera-Kadomtsev-Petviashvili equation

In this paper, by applying a direct symmetry method, we obtain the symmetry reduction, group invariant solution and many new exact solutions of SK-KP equation, which include Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and so on. At last, we also give the conservation laws of SK-KP equation.     

Special two-soliton solution of the generalized Sine–Gordon equation with a variable coefficient

A new special two-soliton solution to the generalized Sine–Gordon equation with a variable coefficient is constructed analytically, by using the self-similar method and Hirota bilinear method. To construct this special solution, we do not utilize the pairs of one-soliton solutions, as is customarily done when solving the Sine–Gordon equation, but introduce two auxiliary self-similar variables in Hirota’s procedure. We also study features of this solution by choosing different self-similar variables. The results obtained confirm that the behavior of such Sine–Gordon solitons can be easily controlled by the selection of the self-similar variables.     

Bäcklund transformations and abundant exact explicit solutions of the Sharma-Tasso-Olver equation

By using an extension of the homogeneous balance method and Maple, the Bäcklund transformations for the Sharma-Tasso-Olver equation are derived. The connections between the Sharma-Tasso-Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma-Tasso-Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Bäcklund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma-Tasso-Olver equation is illustrated.     

Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.     

Symmetry reductions and explicit solutions of a (3 + 1)-dimensional PDE

The symmetry of the (3 + 1)-dimensional partial differential equation has been derived via a direct symmetry method and proved to be infinite dimensional non-Virasoro type symmetry algebra. Many kinds of symmetry reductions have been obtained, including the (2 + 1)-dimensional ANNV equation and breaking soliton equation. And some new soliton solutions and complex solutions are obtained due to the Riccati equation method and symbolic computation.     

Symmetry analysis and exact explicit solutions for Kadomtsev-Petviashvili-Burgers equation

We apply the group theory to Kadomtsev-Petviashvili-Burgers (KPBII) equation which is a natural model for the propagation of the two-dimensional damped waves. In correspondence with the generators of the symmetry group allowed by the equation, new types of symmetry reductions are performed. Some new exact solutions are obtained, which can be in the form of solitary waves and periodic waves. Specially, our solutions indicate that the equation may have time-dependent nonlinear shears. Such exact explicit solutions and symmetry reductions are important in both applications and the theory of nonlinear science.     

Bretherton方程的对称约化和精确解

运用李群对称方法解决Bretherton方程问题,得到方程的对称约化和群不变解,比如幂级数解,最后得出该问题的守恒率.     

New solutions of the 2 + 1 dimensional BKP equation through symmetry analysis: source and sink solutions, creation and diffusion of breathers…

Making use of the theory of symmetry transformations in PDEs we construct new solutions of a 2 + 1 dimensional integrable model in the BKP hierarchy.

First, we analyze its reductions and we obtain a BKP equation independent on time. Starting with a solution of this equation we find a family of solutions of the 2 + 1 dimensional BKP equation. These solutions depend on three arbitrary functions on t.

On the other hand, new solutions can also be constructed by applying some elements of the symmetry group to known solutions of the model.

We observed that the solutions found by using both approaches describe interesting processes. Among these solutions we present source and sink solutions, solutions describing the creation or the diffusion (or both) of a breather, finite time blow-up processes, finite time source solutions, line solitons and coherent structures moving at arbitrary velocities.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Bcklund transformation, non-local symmetry and exact solutions for (2+1)-dimensional variable coefficient generalized KP equations

IntroductionDuring the study of water wave, many completely iategrable models were derived, such as(1+1)-dimensional KdV equation, MKdV equation, (2+1)-dimensional KdV equation, Boussinesq equation and WBK equations etc. Many properties of these models had been researched,such as BAcklund transformation (BT), converse rules, N-soliton solutions and Painleve property etc.II--8]. In this paper, we would like to consider (2+1)-dimensional variable coefficientgeneralized KP equation which …     

Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

Residual symmetry,nth Bäcklund transformation,and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.     

Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients

This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.     

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada-Kotera, Sawada-Kotera-Parker-Dye, Caudrey-Dodd-Gibbon, Kaup-Kupershmidt, Kaup-Kupershmidt-Parker-Dye, and the Ito forms.     

Nonlocal Symmetries and Explicit Solutions of the Boussinesq Equation

The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev′e equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.