The Gelfand-Levitan-Marchenko equation and the long-time asymptotics of the solutions of the nonlinear Schrödinger equation. (English) Zbl 0966.35121

Budylin, A. M.; Buslaev, V. S.

Semenov-Tian-Shansky, Michael (ed.), L. D. Faddeev’s seminar on mathematical physics. Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 201, 63-78 (2000).

From the introduction: We consider two different problems and some links between them. These problems are: 1) the long-time behavior of the solutions to completely integrable nonlinear equations; 2) the asymptotic inversion of semiclassical pseudodifferential operators with doubly discontinuous symbols, i.e., with symbols that are discontinuous with respect to both dual variables.
For the entire collection see [Zbl 0947.00010].

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B40	Asymptotic behavior of solutions to PDEs
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

long-time behavior of the solutions to completely integrable nonlinear equations; semiclassical pseudodifferential operators; discontinuous symbols