Construction of Special Solutions for Nonintegrable Systems

The Painleve test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and a search for solutions of the obtained system. It has been demonstrated by the example of the generalized Henon-Heiles system that the use of the Laurent series solutions of the initial differential equation assists to solve the obtained algebraic system. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial differential equation in the form of the Laurent or Puiseux series we use the Painleve test. This test can also assist to solve the inverse problem: to find the form of a polynomial potential, which corresponds to the required type of solutions. We consider the five-dimensional gravitational model with a scalar field to demonstrate this.Comment: LaTeX, 14 pages, the paper has been published in the Journal of Nonlinear Mathematical Physics (http://www.sm.luth.se/math/JNMP/

Exact Solutions for Domain Walls in Coupled Complex Ginzburg - Landau Equations

The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. A front (shock) is a transient layer between a plane-wave state and a zero background. We report exact solutions for domain walls, i.e., pairs of fronts with opposite polarities, in a system of two coupled CGLEs, which describe transient layers between semi-infinite domains occupied by each component in the absence of the other one. For this purpose, a modified Hirota bilinear operator, first proposed by Bekki and Nozaki, is employed. A novel factorization procedure is applied to reduce the intermediate calculations considerably. The ensuing system of equations for the amplitudes and frequencies is solved by means of computer-assisted algebra. Exact solutions for mutually-locked front pairs of opposite polarities, with one or several free parameters, are thus generated. The signs of the cubic gain/loss, linear amplification/attenuation, and velocity of the coupled-front complex can be adjusted in a variety of configurations. Numerical simulations are performed to study the stability properties of such fronts.Comment: Journal of the Physical Society of Japan, in pres

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo-Miwa equation is treated, together with a B\"acklund transformation.Comment: 13 page