Integrable evolution equations in time-dependent domains

We discuss the implementation of a method of solving initial boundary value problems in the case of integrable evolution equations in a time-dependent domain. This method is applied to a dispersive linear evolution equation with spatial derivatives of arbitrary order and to the defocusing nonlinear Schrödinger equation, in the domain l(t)<x<∞, 0<t<T, where l(t) is a given real sufficiently smooth function whose first derivative is monotonic, and T is a fixed positive constant.