6533b85dfe1ef96bd12be94e
RESEARCH PRODUCT
On critical behaviour in systems of Hamiltonian partial differential equations
Christian KleinAntônio Renato Pereira MoroTamara GravaTamara GravaBoris DubrovinBoris Dubrovinsubject
Hamiltonian PDEsFOS: Physical sciencesSemiclassical physicsPainlevé equationsArticleSchrödinger equationHamiltonian systemsymbols.namesakeMathematics - Analysis of PDEs37K05Modelling and SimulationGradient catastrophe and elliptic umbilic catastrophe34M55FOS: MathematicsInitial value problemSettore MAT/07 - Fisica MatematicaEngineering(all)Mathematical PhysicsMathematicsG100Partial differential equationConjectureNonlinear Sciences - Exactly Solvable and Integrable SystemsHyperbolic and Elliptic systemsApplied MathematicsMathematical analysisQuasi-integrable systemsGeneral EngineeringMathematical Physics (math-ph)35Q55Nonlinear systemModeling and SimulationsymbolsExactly Solvable and Integrable Systems (nlin.SI)Hamiltonian (quantum mechanics)Gradient catastrophe and elliptic umbilic catastrophe; Hamiltonian PDEs; Hyperbolic and Elliptic systems; Painlevé equations; Quasi-integrable systemsAnalysis of PDEs (math.AP)description
Abstract We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P $$_I$$ I ) equation or its fourth-order analogue P $$_I^2$$ I 2 . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2013-11-27 |