The Gel’fand-Levitan-Marchenko equation, and the long-time asymptotics of solutions of the nonlinear Schrödinger equation. (English. Russian original) Zbl 0998.35050

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

St. Petersbg. Math. J. 12, No. 5, 761-789 (2001); translation from Algebra Anal. 12, No. 5, 64-105 (2001).

This paper deals with the long time behaviour of the solutions of the Cauchy problem to the nonlinear Schrödinger equation \[ i\psi_t+ \psi_{ xx} -2|\psi |^2\psi =0,\quad \psi|_{t=0} =\psi_0(x) \] and with the asymptotic inversion of the quasiclassical pseudodifferential operators with symbols discontinuous with respect to both dual variables. The asymptotic behaviour of the solution \(\psi(x,t)\) as \(t\to\infty\) is studied in the scattering domain \(0<C_1 \leq{x\over t}\leq C_2 <\infty\). The corresponding representation formula for \(\psi(x,t)\), \(t\to\infty\), is given.

Reviewer: Petar Popivanov (Sofia)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B40	Asymptotic behavior of solutions to PDEs
47G30	Pseudodifferential operators

Keywords:

nonlinear Schrödinger equation; Cauchy problem; long-time asymptotics