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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
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