FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2004, VOLUME 10, NUMBER 1, PAGES 57-165

Methods of geometry of differential equations in analysis of integrable models of field theory

A. V. Kiselev

Abstract

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In this paper, we investigate algebraic and geometric properties of hyperbolic Toda equations uxy = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogues of the potential modified Korteweg--de Vries equation ut = uxxx + ux3 is constructed and its relationship with the hierarchy for the Korteweg--de Vries equation Tt = Txxx + TTx is established. Group-theoretic structures for the dispersionless (2+1)-dimensional Toda equation uxy = exp(-uzz) are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems Yt = iYxx + if(|Y|)Y (multi-soliton complexes) are described.

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