General Exact Solutions For Linear And Nonlinear Waves In A Thirring Model

Keywords

Exact solution; Nonlinear coupled PDE; Solitons; Thirring model; Traveling waves

Abstract

In the present paper, we construct exact solutions to a system of partial differential equations iux + v + u|v|2 = 0, ivt + u + v|u|2 = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinearwave in terms of the amplitudes of bothwaves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems.

Publication Date

3-15-2015

Publication Title

Mathematical Methods in the Applied Sciences

Volume

38

Issue

4

Number of Pages

636-645

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1002/mma.3095

Socpus ID

84921857190 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84921857190

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