Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data

Abstract

We study the Cauchy problem for
the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation
\[
iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0
\]
with a step-like initial data: $q(x,0)=o(1)$ as $x\to-\infty$ and $q(x,0)=A+o(1)$ as $x\to\infty$, where $A>0$ is an arbitrary constant. We develop the inverse scattering transform method for this problem in the
form of the Riemann-Hilbert approach and obtain the representation of the solution of the Cauchy
problem in terms of the solution of an associated Riemann-Hilbert-type analytic factorization problem,
which can be efficiently used for further studying the properties of the solution, including the large time
asymptotic behavior.