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Breaking solitons in 2+1 dimensional integrable equations. (English. Russian original) Zbl 0754.35127

Russ. Math. Surv. 45, No. 4, 1-86 (1990); translation from Usp. Mat. Nauk 45, No. 4 (274), 17-77 (1990).
A new construction of nonlinear integrable equations and dynamical systems, which extends the Lax isospectral deformation method, and is based on the equation \(L_ t=P(L)+[L,A]\) is developed (here \(P(L)\) is a certain analytic function with constant coefficients).
The eigenvalues vary and satisfy an equation having the breaking wave equation as a special case. Such a behaviour of the eigenvalues leads to the so-called “breaking soliton” solutions. The breaking effect results in multiplicity of the solutions. A notable feature of the derived nonlinear equations is the presence of attractors (not strange attractors) in their phase space and an irregular behaviour of their solutions. Many of the methods used in the “classical” soliton equations (e.g. conservation laws, Miura transformation, Painlevé analysis) are described in this new case. In the last chapter breaking solitons for the equations which are continual limits of the Toda lattice and two-dimensional Fermi-Pasta-Ulam system are considered.
[In the translation the author has added some material not included in the Russian original].
Reviewer: Y.P.Mishev (Sofia)

MSC:

35Q51 Soliton equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q58 Other completely integrable PDE (MSC2000)
37A30 Ergodic theorems, spectral theory, Markov operators
34D45 Attractors of solutions to ordinary differential 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.)
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