Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrцdinger equation and to its linearized version in the domain $\{x\geq l(t),t\geq0\}$. We show that there exist two cases: a) if $l''(t)<0$, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann–Hilbert problem, which is defined on a time-dependent contour; b) if $l''(t)>0$, then the Riemann–Hilbert problem is replaced by a respective scalar or matrix $\overline\partial$ problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.