0000000000763992

AUTHOR

Kristelle Roidot

showing 3 related works from this author

Parallel Computing for the study of the focusing Davey-Stewartson II equation in semiclassical limit

2012

The asymptotic description of the semiclassical limit of nonlinear Schrödinger equations is a major challenge with so far only scattered results in 1 + 1 dimensions. In this limit, solutions to the NLS equations can have zones of rapid modulated oscillations or blow up. We numerically study in this work the Davey-Stewartson system, a 2 + 1 dimensional nonlinear Schrödinger equation with a nonlocal term, by using parallel computing. This leads to the first results on the semiclassical limit for the Davey-Stewartson equations.

T57-57.97Work (thermodynamics)Applied mathematics. Quantitative methods010102 general mathematicsOne-dimensional spaceMathematics::Analysis of PDEsSemiclassical physics010103 numerical & computational mathematicsParallel computing01 natural sciencesSchrödinger equationsymbols.namesakeNonlinear systemNonlinear Sciences::Exactly Solvable and Integrable SystemsQA1-939symbolsLimit (mathematics)0101 mathematicsNonlinear Sciences::Pattern Formation and SolitonsNonlinear Schrödinger equationMathematicsMathematicsESAIM: Proceedings
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Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations

2013

The formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers' equation and for dispersive regularizations as the Korteweg-de Vries equation, the Fourier coefficients indicate as …

Mathematics::Analysis of PDEsFOS: Physical sciencesKadomtsev–Petviashvili equation01 natural sciences010305 fluids & plasmasDispersionless equationMathematics - Analysis of PDEsSingularity0103 physical sciencesFOS: MathematicsMathematics - Numerical Analysis0101 mathematicsKorteweg–de Vries equationFourier seriesMathematicsMathematical physicsNonlinear Sciences - Exactly Solvable and Integrable Systems010102 general mathematicsMathematical analysisStatistical and Nonlinear PhysicsNumerical Analysis (math.NA)Condensed Matter PhysicsBurgers' equationNonlinear Sciences::Exactly Solvable and Integrable SystemsDissipative systemGravitational singularityExactly Solvable and Integrable Systems (nlin.SI)Analysis of PDEs (math.AP)Physica D
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Etude numérique d'équations aux dérivées partielles non linéaires et dispersives

2011

Numerical analysis becomes a powerful resource in the study of partial differential equations (PDEs), allowing to illustrate existing theorems and find conjectures. By using sophisticated methods, questions which seem inaccessible before, like rapid oscillations or blow-up of solutions can be addressed in an approached way. Rapid oscillations in solutions are observed in dispersive PDEs without dissipation where solutions of the corresponding PDEs without dispersion present shocks. To solve numerically these oscillations, the use of efficient methods without using artificial numerical dissipation is necessary, in particular in the study of PDEs in some dimensions, done in this work. As stud…

Davey-Stewartson systems[ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM]equations dispersivesdispersive shocksexponential time-differencing[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM][MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]spectral methodschocs dispersifsnumerical methodsdispersive equationsNo english keywordssplit stepschemas de decomposition d'operateursmethodes spectrales[MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]Kadomtsev-Petviashvili equationintegrating factor methodparallel computing[ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph]Pas de mot clé en français[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM]methodes numeriquesblow upequation de Kadomtsev-PetviashviliIntegrateurs exponentielssystemes de Davey-Stewartsoncalcul parallele
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