The Gel'fand-Levitan-Marchenko equation, and the long-time asymptotics of solutions of the nonlinear Schrödinger equation (Q2761455)

This paper deals with the long time behaviour of the solutions of the Cauchy problem to the nonlinear Schrödinger equation NEWLINE\[NEWLINEi\psi_t+ \psi_{ xx} -2|\psi |^2\psi =0,\quad \psi|_{t=0} =\psi_0(x)NEWLINE\]NEWLINE 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.