Asymptotics of solutions of the nonlinear Schrödinger equation and isomonodromic deformations of systems of linear differential equations. (English. Russian original) Zbl 0534.35028

Sov. Math., Dokl. 24, 452-456 (1981); translation from Dokl. Akad. Nauk SSSR 261, 14-18 (1981).
In this paper the problem of constructing asymptotics as \(t\to +\infty\) and for a fixed ratio \(x/4t=\lambda_ 0\) of the solution of the Cauchy problem for the nonlinear Schrödinger equation \(ip_ t+p_{xx}-8| p|^ 2p=0\) in the class of rapidly decreasing data is considered. In correspondence with the method of the inverse problem, the solution of the problem reduces to asymptotically solving the Riemann problem which determines a matrix-valued function \(\psi\) (x,t,\(\lambda)\) which is analytic in \(\lambda\) for Im \(\lambda\neq 0\), and with certain specified properties. This is achieved by converting the Riemann problem into one which is exactly solvable. The results are described in Theorem 1 and a sketch of the proof is given.
Reviewer: V.K.Kumar


35J10 Schrödinger operator, Schrödinger equation
35B40 Asymptotic behavior of solutions to PDEs
35Q15 Riemann-Hilbert problems in context of PDEs