The soliton correlation matrix and the reduction problem for integrable systems

DOI10.1007/BF01218638zbMath0549.58025OpenAlexW2027775091MaRDI QIDQ800002

J. Harnad, Yvan Saint-Aubin, Steven Shnider

Publication date: 1984

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf01218638

zbMATH Keywords

solitonintegrable systemsine-Gordon equationsmulti-Bäcklund transformation2-dimensional field equationsdressing matrixsoliton correlation matrix

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Applications of Lie groups to the sciences; explicit representations (22E70) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Constructive quantum field theory (81T08) Partial differential equations of mathematical physics and other areas of application (35Q99)

Related Items (24)

The supersymmetric soliton correlation matrix ⋮ Superposition formulas for pseudounitary matrix Riccati equations ⋮ Classification of systems of nonlinear ordinary differential equations with superposition principles ⋮ Hyperbolic complex numbers and nonlinear sigma models ⋮ Linear integral transformations and hierarchies of integrable nonlinear evolution equations ⋮ Isospectral Hamiltonian flows in finite and infinite dimensions. I: Generalized Moser systems and moment maps into loop algebras ⋮ Formal power series solutions of supersymmetric (N=3) Yang–Mills equations ⋮ Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles ⋮ Lie-theoretical generalization and discretization of the Pinney equation ⋮ Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation ⋮ Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system ⋮ Rational differential systems, loop equations, and application to the \(q\)th reductions of KP ⋮ Infinite-dimensional Lie algebras acting on the solution space of various σ models ⋮ Dressing Method for the Degasperis-Procesi Equation ⋮ Classical nonlinear σ models on Grassmann manifolds of compact or noncompact type ⋮ Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian σ models ⋮ General Zakharov-Shabat equations, multi-time Hamiltonian formalism, and constants of motion ⋮ Generalized Drinfeld-Sokolov reductions and KdV type hierarchies ⋮ Nonlinear superposition formulas based on imprimitive group action ⋮ An infinite number of hidden variables in hyper-Kähler metrics ⋮ Inverse scattering and solitons in \(A_{n-1}\) affine Toda field theories ⋮ Multi-soliton solutions to the Thirring model through the reduction method ⋮ The generalised Zakharov–Shabat system and the gauge group action ⋮ The unified approach to integrable relativistic equations: Soliton solutions over nonvanishing backgrounds. I

Cites Work

• Bäcklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces
• The reduction problem and the inverse scattering method
• Bäcklund transformations and soliton-type solutions for sigma models with values in real Grassmannian spaces
• Quadratic pseudopotentials for Gl(N,\({\mathbb{C}})\) principal sigma models
• Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II
• Superposition principles for matrix Riccati equations
• The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems

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