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RESEARCH PRODUCT
On critical behaviour in generalized Kadomtsev-Petviashvili equations
Christian KleinBoris DubrovinTamara Gravasubject
Differential equationsShock waveSpecial solutionBlow-upKadomtsev–Petviashvili equations[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]Mathematics::Analysis of PDEsFOS: Physical sciencesPainlevé equationsKadomtsev-Petviashvili equationsKadomtsev–Petviashvili equation01 natural sciences010305 fluids & plasmasShock wavesDispersive partial differential equationMathematics - Analysis of PDEs0103 physical sciencesFOS: MathematicsCritical behaviourLong-time behaviourSupercriticalDispersion (waves)0101 mathematicsKP equationSettore MAT/07 - Fisica MatematicaMathematical PhysicsMathematicsMathematical physicsKadomtsev-Petviashvili equationPainleve equationsConjectureNonlinear Sciences - Exactly Solvable and Integrable Systems010102 general mathematicsMathematical analysisDispersive shocks Kadomtsev–Petviashvili equations Painlevé equations Differential equations Dispersion (waves) Ordinary differential equations Shock waves Blow-up Critical behaviour Dispersive shocks Kadomtsev-Petviashvili equation KP equation Long-time behaviour Special solutions Supercritical Partial differential equationsStatistical and Nonlinear PhysicsMathematical Physics (math-ph)Condensed Matter PhysicsDispersive shocksPartial differential equationsNonlinear Sciences::Exactly Solvable and Integrable SystemsOrdinary differential equationSpecial solutions[ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph]Exactly Solvable and Integrable Systems (nlin.SI)Ordinary differential equationsAnalysis of PDEs (math.AP)description
International audience; An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2016-10-15 |