Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori
Dario Bambusi, Roberto Feola, Riccardo Montalto
Abstract
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
Topics & Concepts
TorusMathematicsMathematical analysisCauchy distributionInitial value problemSemigroupNonlinear systemHamiltonian (control theory)Nonlinear Schrödinger equationOperator (biology)Laplace transformLemma (botany)Cauchy problemHamiltonian systemWave equationMathematical physicsSchrödinger equationPhysicsGeometryQuantum mechanicsPoaceaeChemistryRepressorGeneBiochemistryMathematical optimizationTranscription factorBiologyEcologyAdvanced Mathematical Physics ProblemsQuantum chaos and dynamical systemsNonlinear Waves and Solitons