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

Abstract

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 Ψt = iΨxx + i f(|Ψ|) Ψ (multi-soliton complexes) are described.

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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.

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Kiselev, A.V. Methods of geometry of differential equations in analysis of integrable models of field theory. J Math Sci 136, 4295–4377 (2006). https://doi.org/10.1007/s10958-006-0229-0

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Keywords

  • Liouville Equation
  • Vries Equation
  • Symmetry Algebra
  • Cartan Matrix
  • Recursion Operator