Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schr\"odinger equation

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation $iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \, x\in\mathbb{R},\,t>0,$ 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 $t\geq0$, 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.