Linear integral equations and nonlinear difference-difference equations

G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. Van Der Linden

Physica A: Statistical Mechanics and its Applications, 1984, vol. 125, issue 2, 344-380

Abstract: In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Bäcklund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.

Date: 1984
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:125:y:1984:i:2:p:344-380

DOI: 10.1016/0378-4371(84)90059-1

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