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Orbital stability of localized structures via Bäcklund transformations. (English) Zbl 1289.35225

Summary: The Bäcklund transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite-dimensional dynamical systems. It has recently been used to study the stability of these special solutions. We offer here a dynamical perspective on the Bäcklund transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.

MSC:

35L71 Second-order semilinear hyperbolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B35 Stability in context of PDEs
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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