Numerical study of the transverse stability of the Peregrine solution
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We numerically study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schrödinger (NLS) equation and thus a y-independent solution to the 2D NLS. To this end we generalise a previously published approach based on a multi-domain spectral method on the whole real line. We do this 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. It is shown, with several examples, that the Peregrine solution is unstable against all standard perturbations, and that some perturbations can even lead to a blow up.
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