Painlevé analysis and new analytic solutions for variable-coefficient Kadomtsev-Petviashvili equation with symbolic computation

The variable-coefficient Kadomtsev-Petviashvili (KP) equation is hereby under investigation. Painlevé analysis is given out, and an auto-Bäcklund transformation is presented via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, new analytic solutions are given, including the soliton-like and periodic solutions. It is also reduced to a (1+1)-dimensional partial differential equation via classical Lie group method and the Painlevé I equation by CK direct method.     

Lie point symmetries, partial Noether operators and first integrals of the Painlevé-Gambier equations

We utilize the Lie-Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé-Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé-Gambier equations. It is also shown that those Painlevé-Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé-Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

Exact travelling wave solutions of the generalized Bretherton equation

We search for exact travelling wave solutions of the generalized Bretherton equation for integer, greater than one, values of the exponent m of the nonlinear term by two methods: the truncated Painlevé expansion method and an algebraic method. We find periodic solutions for m=2 and m=5, to add to those already known for m=3; in all three cases these solutions exist for finite intervals of the wave velocity. We also find a “kink” shaped solitary wave for m=5 and a family of elementary unbounded solutions for arbitrary m.     

Painlevé analysis, Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer-Kaup equations

A (2+1) dimensional Broer-Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equation to one-dimensional partial differential equations including Burger’s equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

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.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

Exact solitary wave solutions for a generalized KdV–Burgers equation

Using the special truncated expansion method, the solitary wave solutions are constructed for the compound Korteweg–de Vries–Burgers (KdVB) equation. Exact and explicit solitary wave solutions for a generalized KdVB equation are obtained by introducing a suitable ansatz equation. The generalized two-dimensional KdVB equation is discussed. Some particular cases of the generalized KdVB equation are solved by using these methods.     

Explicit solutions of the Bogoyavlensky-Konoplechenko equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the Bogoyavlensky-Konoplechenko equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the BK equation. Then we get the reductions using the symmetry and give some exact solutions of the BK equation.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Truncated Painleve expansion and reduction of sine-Poisson equation to a quadrature in collisionless cold plasma

The relativistic nonlinear self-consistent equations for a collisionlesscold plasma with stationary ions is considered. The resultingsystem of equations for stationary and non-stationary equationsin two-dimensions can be reduced to the sine-Poisson equation.The truncated Painlevé expansion and reduction of thepartial differential equation to a quadrature problem (RQ method)are described and applied to obtain the travelling wave solutionsof the sine-Poisson equation describing the charge density equilibriumconfiguration model.     

Finite symmetry transformation groups and exact solutions of the cylindrical Korteweg-de Vries equation

Based on the new symmetry group method developed by Lou et al. and symbolic computation, both the Lie point groups and the non-Lie symmetry groups of the cylindrical Korteweg-de Vries (cKdV) equation are obtained. With the transformation groups, a type of group invariant solutions of cKdV equation can be derived from a simple one. Furthermore, some transformations from the cKdV equation to KP equation can also be discovered by this method.     

On tronquée solutions of the first Painlevé hierarchy

It is well known that, due to Boutroux, the first Painlevé equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painlevé equation (the first Painlevé hierarchy) as well.     

Analytical solutions to Fisher’s equation with time-variable coefficients

This paper provides analytical solutions to the generalized Fisher equation with a class of time varying diffusion coefficients. To accomplish this we use the Painlevé property for partial differential equations as defined by Weiss in 1983 in “The Painlevé property for partial-differential equations”. This was first done for the variable coefficient Fisher’s equation by Ö?ün and Kart in 2007; we build on this work, finding additional solutions with a weaker restriction on the trial solution. We also use the same technique to find solutions to Fisher’s equation with time-dependent coefficients for both diffusion and nonlinear terms. Lastly we compute specific solutions to illustrate their behaviors.     

From Klein to Painleve Via Fourier, Laplace and Jimbo

We describe a method for constructing explicit algebraic solutionsto the sixth Painlevé equation, generalising that ofDubrovin and Mazzocco. There are basically two steps. Firstwe explain how to construct finite braid group orbits of triplesof elements of SL₂(C) out of triples of generators of three-dimensionalcomplex reflection groups. (This involves the Fourier–Laplacetransform for certain irregular connections.) Then we adapta result of Jimbo to produce the Painlevé VI solutions.(In particular, this solves a Riemann–Hilbert problemexplicitly.) Each step is illustrated using the complex reflection groupassociated to Klein's simple group of order 168. This leadsto a new algebraic solution with seven branches. We also provethat, unlike the algebraic solutions of Dubrovin and Mazzoccoand Hitchin, this solution is not equivalent to any solutioncoming from a finite subgroup of SL₂(C). The results of this paper also yield a simple proof of a recenttheorem of Inaba, Iwasaki and Saito on the action of Okamoto'saffine D₄ symmetry group as well as the correct connection formulaefor generic Painlevé VI equations. 2000 Mathematics SubjectClassification 34M55, 34M40, 20F55.     

The master Painlevé VI heat equation

Given the second-order scalar Lax pair of the sixth Painlevé equation, we build a generalized heat equation with rational coefficients which does not depend any more on the Painlevé variable.     

Nonlocal symmetries and exact solutions of the (2+1)-dimensional generalized variable coefficient shallow water wave equation

In this paper, using the standard truncated Painlevé analysis, the Schwartzian equation of (2+1)-dimensional generalized variable coefficient shallow water wave (SWW) equation is obtained. With the help of Lax pairs, nonlocal symmetries of the SWW equation are constructed which be localized by a complicated calculation process. Furthermore, using the Lie point symmetries of the closed system and Schwartzian equation, some exact interaction solutions are obtained, such as soliton–cnoidal wave solutions. Corresponding 2D and 3D figures are placed to illustrate dynamic behavior of the generalized variable coefficient SWW equation.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Solutions of Discrete-Velocity Boltzmann Equations via Bateman and Riccati Equations

We propose several approaches for solving two discrete-velocity Boltzmann equations using the rescaling ansatz and the truncated Painlevé expansions. We use solutions of the two- and three-dimensional Bateman equations for the singularity manifold conditions to reduce the problem to Riccati equations. Both equations fail the Painlevé test.     

Dirichlet and Periodic-Type Boundary Value Problems for Painlevé II

It is established that, under certain conditions, the Dirichlet problem on a bounded interval for the Painlevé II equation is uniquely solvable and solutions are constructed in an iterative manner. Moreover, conditions for the existence of periodic solutions are set down.