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 $u$_xy = exp(Ku) associated with nondegenerate symmetrizable matrices $K$. A hierarchy of analogues of the potential modified Korteweg--de Vries equation $u$_t = u_xxx + u_x³ is constructed and its relationship with the hierarchy for the Korteweg--de Vries equation $T$_t = T_xxx + TT_x is established. Group-theoretic structures for the dispersionless $(2+1)$-dimensional Toda equation $u$_xy = exp(-u_zz) are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems $$Y_t = iY_xx + if(|Y|)Y (multi-soliton complexes) are described.

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Last modified: October 25, 2004