Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data

By Yan Rybalko, Dmitry Shepelsky. Journal of Differential Equations, 2021

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Abstract We study the Cauchy problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation i q t ( x , t ) + q x x ( x , t ) + 2 q 2 ( x , t ) q ¯ ( − x , t ) = 0 with a step-like initial data: q ( x , 0 ) = q 0 ( x ) , where q 0 ( x ) = o ( 1 ) as x → − ∞ and q 0 ( x ) = A + o ( 1 ) as x → ∞ , 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 − ∞ x ∞ , 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.

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