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

Abstract

In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations $u_{xy}=\exp (Ku)$ associated with nondegenerate symmetrizable matrices $K$. A hierarchy of analogs to the potential modified Korteweg-de Vries equation $u_t=u_{xxx}+u_x^3$ is constructed, and its relation with the hierarchy for the Korteweg-de Vries equation $T_t=T_{xxx}+T{T}_x$ 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 ${\Psi }_t=i{\Psi }_{xx}+if(|\Psi |)\Psi$ (multi-soliton complexes) are described.

LaTeX2e, 134 pages, no figures, uses diagrams.tex. Submitted to: Fundamental'naya i Prikladnaya Matematika/J. Math. Sci

References in corpus (1)

• The Extended Toda Hierarchy