General exact solutions for linear and nonlinear waves in a Thirring model

    Authors

    R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Math. Meth. Appl. Sci.

    Keywords

    nonlinear coupled PDE; Thirring model; traveling waves; exact solution; solitons; QUANTUM ELECTRODYNAMICS; EQUATIONS; Mathematics, Applied

    Abstract

    In the present paper, we construct exact solutions to a system of partial differential equations iu(x)+v+u|v|(2)=0, iv(t)+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 nonlinear wave in terms of the amplitudes of both waves. 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. Copyright (c) 2014 John Wiley & Sons, Ltd.

    Journal Title

    Mathematical Methods in the Applied Sciences

    Volume

    38

    Issue/Number

    4

    Publication Date

    1-1-2015

    Document Type

    Article

    Language

    English

    First Page

    636

    Last Page

    645

    WOS Identifier

    WOS:000348847200007

    ISSN

    0170-4214

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