6533b836fe1ef96bd12a1620

RESEARCH PRODUCT

Numerical study of the transverse stability of the Peregrine solution

Christian KleinNikola Stoilov

subject

Mathematics::Analysis of PDEsFOS: Physical sciences010103 numerical & computational mathematics01 natural sciencesStability (probability)spectral approachdispersive blow-upperfectly matched layersymbols.namesakeMathematics - Analysis of PDEsnonlinear Schrodinger equations0103 physical sciencesFOS: MathematicsMathematics - Numerical Analysis0101 mathematics[MATH]Mathematics [math]010306 general physicsNonlinear Sciences::Pattern Formation and SolitonsReal lineVariable (mathematics)Physicsschrodinger-equationsNonlinear Sciences - Exactly Solvable and Integrable SystemsApplied MathematicsMathematical analysisNumerical Analysis (math.NA)Nonlinear systemTransverse planeExact solutions in general relativityFourier transformPeregrine solutionsymbolsExactly Solvable and Integrable Systems (nlin.SI)Spectral methodAnalysis of PDEs (math.AP)

description

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.

yearjournalcountryeditionlanguage
2020-04-28
10.1111/sapm.12306https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02570854