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Asymptotics of some differential and pseudodifferential equations, and dynamical systems with small dispersion. (English. Russian original) Zbl 0567.35078

Math. USSR, Sb. 50, 191-212 (1985); translation from Mat. Sb., Nov. Ser. 122(164), No. 2, 197-219 (1983).
Asymptotic soliton-type solutions of the Whitham and Boussinesq equations and an asymptotic solution of shock-wave type of the Toda lattice with variable coefficients in the case of small dispersion are constructed. The solutions represent a ”distorted” solitary wave (a smoothed shock wave) with amplitude and speed of motion which depend on time, propagating on a smooth ”background”.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35S05 Pseudodifferential operators as generalizations of partial differential operators
35L67 Shocks and singularities for hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D10 Statistical mechanics of plasmas
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