Long-time asymptotics for the integrable nonlocal focusing nonlinear Schr\"odinger equation for a family of step-like initial data

We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr"odinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function’’ ${\chi }_R(x)=AH(x-R)$, where $H(x)$ is the Heaviside step function, and $A>0$ and $R>0$ are arbitrary constants. Our main aim is to study the large-$t$ behavior of the solution of this problem. We show that for $R\in (\frac{(2n-1)\pi }{2A},\frac{(2n+1)\pi }{2A})$, $n=1,2,\dots$, the $(x,t)$ plane splits into $4n+2$ sectors exhibiting different asymptotic behavior. Namely, there are $2n+1$ sectors where the solution decays to $0$, whereas in the other $2n+1$ sectors (alternating with the sectors with decay), the solution approaches (different) constants along each ray $x\mathrm{/}t=const$. Our main technical tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert problem and its subsequent asymptotic analysis following the ideas of nonlinear steepest descent method.