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We generalise a previously published approach based on a multi-domainspectral method on the whole real line in two ways: firstly, a fully explicit4th order method for the time integration, based on a splitting scheme and animplicit Runge--Kutta method for the linear part, is presented. Secondly, the1D code is combined with a Fourier spectral method in the transverse variableboth for elliptic and hyperbolic NLS equations. As an example we study thetransverse stability of the Peregrine solution, an exact solution to the onedimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independentsolution to the 2D NLS.  It is shown that the Peregrine solution is unstable against all standardperturbations, and that some perturbations can even lead to a blow-up for theelliptic NLS equation.

Numerical study of the transverse stability of the Peregrine solution