On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables

DOI

10.2991/jnmp.1995.2.3-4.1How to use a DOI?

Abstract

Bäcklund transformations, which are relations among solutions of partial differential equations­usually nonlinear­have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented. I Introduction The following discussion reports some completed work and some work in progress. In the study of various nonlinear partial differential equations (pde) that has taken place in the last 30 years, it has become recognized that certain pde's admit what are called "Bäcklund transformations" (BT's). These can be given a formal definition [3], but I shall speak of them here only as equations that allow one to find a new solution of the given partial differential equation from an old one, often by simple quadratures (integrations.) These can also give rise to BT "superposition" relations that enable one to find new solutions algebraically after one has performed the first integration steps. It should be noted in the following that, because of the variety of mathematical expressions, some latin and greek letters will be used more than once; generally the notation within any one section is unique. Copyright c 1995 by Mathematical Ukraina Publisher. All rights of reproduction in any form reserved. 202 B. KENT HARRISON II Sine-Gordon Equation A simple, standard example of a BT is that for the sine-Gordon equation, written as [4] uv = sin , (1) where subscripts indicate differentiation. The BT is u = u + 2k sin + 2 , v = -v + 2k-1 sin - 2 , (2) where is an old (seed) solution of Eq. (1), is a new solution, and k is a parameter. Eqs. (2) are to be integrated for . This is particularly easy if the seed solution is simply the zero solution; then integration gives = 4 arctan(exp(ku + k-1 v)), (3) the single soliton solution. If 0 is a beginning solution, and 1 and 2 are solutions obtained from applying the BT to 0, with parameters k1 and k2, respectively, then 3, defined by tan 3 - 0 4 = k1 + k2 k1 - k2 tan

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© 2006, the Authors. Published by Atlantis Press.

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This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - B. Kent Harrison
PY  - 1995
DA  - 1995/09/01
TI  - On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables
JO  - Journal of Nonlinear Mathematical Physics
SP  - 201
EP  - 215
VL  - 2
IS  - 3-4
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.1995.2.3-4.1
DO  - 10.2991/jnmp.1995.2.3-4.1
ID  - Harrison1995
ER  -