  We generalise a previously published approach based on a multi-domain
spectral method on the whole real line in two ways: firstly, a fully explicit
4th order method for the time integration, based on a splitting scheme and an
implicit Runge--Kutta method for the linear part, is presented. Secondly, the
1D code is combined with a Fourier spectral method in the transverse variable
both for elliptic and hyperbolic NLS equations. As an example we study the
transverse stability of the Peregrine solution, an exact solution to the one
dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent
solution to the 2D NLS.
  It is shown that the Peregrine solution is unstable against all standard
perturbations, and that some perturbations can even lead to a blow-up for the
elliptic NLS equation.
