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Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data. (English) Zbl 1451.35194

Summary: We study the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ i q_t(x, t) + q_{xx}(x, t) + 2q^2(x, t) \overline{q}(-x, t) = 0 \] with a step-like initial data: \(q(x, 0) = q_0(x)\), where \(q_0(x) = o(1)\) as \(x \to -\infty\) and \(q_0(x) = A + o(1)\) as \(x \to \infty\), with an arbitrary positive constant \(A > 0\). The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane \(-\infty < x < \infty\), \(t > 0\): (i) for \(x < 0\), the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for \(x > 0\), the solution approaches the “modulated constant”. The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q15 Riemann-Hilbert problems in context of PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B40 Asymptotic behavior of solutions to PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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