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Quasi-classical asymptotics of solutions to the matrix factorization problem with quadratically oscillating off-diagonal elements. (English. Russian original) Zbl 1307.35179

Funct. Anal. Appl. 48, No. 1, 1-14 (2014); translation from Funkts. Anal. Prilozh. 48, No. 1, 1-18 (2014).
The paper investigates the asymptotic behavior of solutions of the \(2\times 2\) matrix Riemann-Hilbert problem \(H^+=H^- V\) on the real axis where the matrix \(V(x)\) has the off-diagonal elements \(b(x)e^{-itx^2/2}\) and \(c(x)e^{itx^2/2}\). The matrix \(V(x)\) becomes the identity matrix as \(x \to \infty\). The authors prove that the factorization problem has a solution at sufficiently large \(t\). The asymptotics of this solution is described by an expansion in the power-logarithmic series in \(t\).

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
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