Solitary and periodic waves in nonlinear nonintegrable systems.

摘要

The existence of solitons and periodic wave trains is an important question in the study of nonlinear evolution equations. The methods for finding such solutions for integrable equations are well-known: the inverse scattering theory, the Hirota bilinear method, and the singularity manifold method. The goal of this work is to develop some methods for finding the solitary and periodic solutions for some nonintegrable equations.; We consider the equations that are the least integrable systems in the Hietarinta classification. For such equations the generalization of the Painleve expansion is used to find the traveling wave solutions in the form of solitary waves and traveling wave fronts. Furthermore, if the soliton solution is found, the periodic wave train represented by the superposition of the solitons approximates the exact periodic solution as the spacing between pulses gets large. To characterize these solutions the partial fraction decomposition in exponentials is used. The superposition is shown to satisfy the original equation plus some small correction term. Then, the exact solution is obtained by taking into account the correction term and by scaling the traveling wave coordinate. Also the periodic solution is obtained even when the solitary wave solution is not known. The described algorithm gives an explicit expressions for the velocity and amplitude of the periodic pulse train in terms of period and reveals the dynamics of interacting localized structures. It is demonstrated that the soliton interaction is controlled by the singularity structure of the system.