Maslov, V. P.; Omel’yanov, G. A.; Tsupin, V. A. 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”. Cited in 1 Document 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 Keywords:Asymptotic soliton-type solutions; Whitham and Boussinesq equations; asymptotic solution; shock-wave; Toda lattice; solitary wave PDFBibTeX XMLCite \textit{V. P. Maslov} et al., Math. USSR, Sb. 50, 191--212 (1985; Zbl 0567.35078); translation from Mat. Sb., Nov. Ser. 122(164), No. 2, 197--219 (1983) Full Text: DOI EuDML