Focusing NLS Equation: Long-Time Dynamics of Step-Like Initial Data

We consider the initial value problem for the focusing nonlinear Schrodinger equation with “step-like” initial data: q(x, 0) = 0 for x ≤ 0 and q(x, 0) = Aexp(−2iBx) for x > 0, where A > 0 and B ∈ ℝ are constants. The paper aims at studying the long-time asymptotics of the solution to this problem. We show that there are three regions in the half-plane −∞ 0, where the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov–Manakov type for x < −4Bt, a modulated elliptic wave for , and a plane wave for . The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem.

[1]  Stephanos Venakides,et al.  UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY , 1999 .

[2]  P. Deift,et al.  The collisionless shock region for the long-time behavior of solutions of the KdV equation , 1994 .

[3]  Percy Deift,et al.  Long-Time Asymptotics for Integrable Nonlinear Wave Equations , 1993 .

[4]  A. B. D. Monvel,et al.  Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition , 2008 .

[5]  S. Venakides.  LONG TIME ASYMPTOTICS OF THE KORTEWEG-DE VRIES EQUATION , 1986 .

[6]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.

[7]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[8]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[9]  A. Gurevich,et al.  Nonstationary structure of a collisionless shock wave , 1973 .

[10]  Complex whitham deformations in problems with “integrable instability” , 1995 .

[11]  A. B. D. Monvel,et al.  THE FOCUSING NONLINEAR SCHRÖDINGER EQUATION ON THE QUARTER PLANE WITH TIME-PERIODIC BOUNDARY CONDITION: A RIEMANN–HILBERT APPROACH , 2007, Journal of the Institute of Mathematics of Jussieu.

[12]  Stephanos Venakides,et al.  Long‐time asymptotics of the nonlinear Schrödinger equation shock problem , 2007 .

[13]  Saturation of modulational instability via complex whitham deformations: Nonlinear Schrödinger equation , 1997 .

[14]  Stephanos Venakides,et al.  New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems , 1997 .

[15]  S. V. Manakov,et al.  Nonlinear Fraunhofer diffraction , 1974 .

[16]  A. B. D. Monvel,et al.  Long-Time Asymptotics for the Focusing NLS Equation with Time-Periodic Boundary Condition on the Half-Line , 2009 .

[17]  Alexander Its,et al.  A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics , 1997 .