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

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

2004-06-17

arxiv.org/abs/nlin/0406036v1

Fundamental'naya i Prikladnaya Matematika 10 (2004) n.1 "Geometry of Integrable Models", 57-165.

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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

Scientific paper

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 $T_t=T_{xxx}+TT_x$ 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 Schr\"odinger equation type systems $\Psi_t = i\Psi_{xx} + i f(|\Psi|) \Psi$ (multi-soliton complexes) are described.

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