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Obstacles to asymptotic integrability. (English) Zbl 0867.35091

Fokas, A. S. (ed.) et al., Algebraic aspects of integrable systems: in memory of Irene Dorfman. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 26, 173-204 (1997).
Summary: We study nonintegrable effects appearing in the higher order corrections of an asymptotic perturbation expansion for a given nonlinear wave equation, and show that the analysis of the higher order terms provides a sufficient condition for asymptotic integrability of the original equation. The nonintegrable effects, which we call “obstacles” to the integrability, are shown to result in an inelasticity in soliton interaction.
The main technique used in this paper is an extension of the normal form theory developed by Kodama and the approximate symmetry approach proposed by Mikhailov. We also discuss the case of the KP equation with the higher order corrections, a quasi-two-dimensional extension of weakly dispersive nonlinear waves.
For the entire collection see [Zbl 0851.00060].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58J70 Invariance and symmetry properties for PDEs on manifolds
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