Curved wedges in the long‐time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
Yan Rybalko, Dmitry Shepelsky
Abstract
Abstract We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation , with a step‐like boundary values: as and as for all , where is a constant. In a recent paper, we presented the long‐time asymptotics of the solution of this problem along the rays , where is a constant. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray than any of these rays. We specify a one‐parameter family of wedges in the ‐plane, with curved boundaries, characterized by qualitatively different asymptotic behavior of , and present the main asymptotic terms for each wedge. Particularly, for wedges within , we show that the solution decays as with depending on the wedge. For wedges within , we show that the asymptotics has an oscillatory nature, with the phase functions specific for each wedge and depending on a slow variable parameterizing the wedges.