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

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.

MSC:

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