V.N. Karazin Kharkiv National University Publishing
Abstract
We study the Cauchy problem for
the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation
iqt(x,t)+qxx(x,t)+2q2(x,t)qˉ(−x,t)=0
with a step-like initial data: q(x,0)=o(1) as x→−∞ and q(x,0)=A+o(1) as x→∞, 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