The Peregrine solution to the nonlinear Schrodinger equations is widely discussed as a model for rogue waves in deep water. We present here a detailed fully nonlinear numerical study of high accuracy of perturbations of the Peregrine solution as a solution to the nonlinear Schrodinger (NLS) equations.We study localized and nonlocalized perturbations of the Peregrine solution in the linear and fully nonlinear setting. It is shown that the solution is unstable against all considered perturbations.

https://www.intlpress.com/site/pub/files/_fulltext/journals/amsa/2017/0002/0002/AMSA-2017-0002-0002-a001.pdf

Numerical study of the stability of the Peregrine solution

We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.

Spectral approach to D-bar problems

We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the "energy" parameter $ E $. We show that as $ |E| \to \infty $, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when $ | E | $ is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.

Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation

Simple high-order boundary conditions for computing rogue waves in the nonlinear Schrödinger equation

We discuss numerical approaches for nonlinear Schrödinger and Korteweg–de Vries equations with slowly decaying and piecewise smooth initial data. An efficient time integration for these equations is based on exponential integrators in the Fourier domain. The important step in such approaches is the computation of rapidly oscillating integrals on suitably chosen contours in the complex plane using spectral methods. Several examples mainly in the linear case are discussed. 

On numerical approaches to nonlinear Schrödinger and Korteweg–de Vries equations for piecewise smooth and slowly decaying initial data

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schr\"odinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schr\"odinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.

Multidomain spectral method for Schr\"odinger equations

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Multidomain spectral method for Schr\"odinger equations