Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $i{q}_t(x,t)+q_{xx}(x,t)+2q^2(x,t)\bar{q}(-x,t)=0,x\in \mathbb{R},t>0,$ with a step-like boundary values: $q(x,t)\to 0$ as $x\to -\infty$ and $q(x,t)\to A$ as $x\to \infty$ for all $t\ge 0$, where $A>0$ is a constant. The long-time asymptotics of the solution $q(x,t)$ of this problem along the rays $x/t=C\ne 0$ is presented in \cite{RS2}. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray $x=0$ than these rays with any nonzero constant $C$. We specify a one-parameter family of wedges in the $x,t$-plane, with curved boundaries, characterized by qualitatively different asymptotic behavior of $q(x,t)$, and present the main asymptotic terms for each wedge. Particularly, for wedges with $x<0$, we show that the solution decays as $t^p\sqrt{\ln t}$ with $p<0$ depending on the wedge. For wedges with $x>0$, we show that the asymptotics has an oscillating nature, with the phase functions specific for each wedge and depending on a slow variable parametrizing the wedges. The main tool used in this work is an adaptation of the nonlinear steepest decent method to the case when the stationary phase point of the phase function in the jump of the associated Riemann-Hilbert problem merges with a point which is singular for the corresponding spectral functions.

arXiv comment25 pages, 4 figures

References in corpus (18)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions
• Continuous and discrete Schrodinger systems with PT-symmetric nonlinearities
• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• Alice-Bob systems, $P_s$-$T_d$-$C$ principles and multi-soliton solutions
• General rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation
• Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schrödinger Model
• Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
• Long-Time Asymptotics for the Korteweg-de Vries Equation with Steplike Initial Data
• Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data
• Integrable nonlocal asymptotic reductions of physically significant nonlinear equations
• Peregrine Rogue Wave dynamics in the continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity
• Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
• General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system
• Explicit solutions for nonlocal NLS: GBDT and algebro-geometric approaches
• Airy and Painlevé asymptotics for the mKdV equation
• Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations
• Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

Cited by in corpus (1)

• Large-time asymptotics to the focusing nonlocal modified Kortweg-de Vries equation with step-like boundary conditions