New reductions of integrable matrix PDEs: $Sp(m)$-invariant systems

Abstract

We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semi-discretizations of the obtained systems and present new soliton solutions to both continuous and semi-discrete systems. As a by-product, a new integrable semi-discretization of the Manakov model (self-focusing vector NLS equation) is obtained.

33 pages; (v4) to appear in JMP; This paper states clearly that the elementary function solutions of (a vector/matrix generalization of) the derivative NLS equation can be expressed as the partial $x$-derivatives of elementary functions. Explicit soliton solutions are given in the author's talks at http://poisson.ms.u-tokyo.ac.jp/~tsuchida/

References in corpus (4)

• N-soliton collision in the Manakov model
• Classification of integrable polynomial vector evolution equations
• Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
• From AKNS to derivative NLS hierarchies via deformations of associative products

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• Symplectically-invariant soliton equations from non-stretching geometric curve flows
• Direct and inverse scattering problems for a first-order system with energy-dependent potentials
• Two-component integrable generalizations of Burgers equations with nondiagonal linearity
• The Marchenko method to solve the general system of derivative nonlinear Schrödinger equations
• Direct and inverse scattering problems for the first-order discrete system associated with the derivative NLS system