Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition

摘要

We consider the initial boundary value problem for the focusing nonlinear Schrödinger equation in the quarter plane , in the case of decaying initial data (for , as ) and Dirichlet boundary data (for ) approaching a periodic (single-frequency) background as . We first provide admissibility conditions for the normal derivative of the solution on the boundary, under the assumption that it behaves asymptotically in a similar (single-frequency) manner. We then show that for the range , the long-time asymptotics of the solution inside the quarter plane exhibits decaying oscillations of Zakharov–Manakov type.