Multimodal Standing Gravity Waves: a Completely Resonant System
Copyright policy )Multimodal Standing Gravity Waves: a Completely Resonant System
The standing gravity wave problem on an infinitely deep fluid layer is considered under the form of a nonlinear non local scalar PDE of second order as in [6]. Nonreso-nance at quadratic order of the infinite dimensional bifurcation equation, allows to give the explicit form of the quadratic change of variables able to suppress quadratic terms in the nonlinear equation. We state precisely the equivalence between formulations in showing that the above unbounded change of variable is invertible. The infinite set of solutions which can be expanded in powers of amplitude ε is then given up to order ε 2 .
- Centre national de la recherche scientifique France
- French National Centre for Scientific Research France
- Nice Sophia Antipolis University France
- Russian Academy of Sciences Russian Federation
- Université Côte d'Azur France
Bifurcations in context of PDEs, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], Free-surface potential flows for incompressible inviscid fluids, nonlocal PDE, complete resonance AMS classification: 35B32, gravity waves, PDEs in connection with fluid mechanics, 35B34, Asymptotic expansions of solutions to PDEs, standing gravity waves, [NLIN.NLIN-PS] Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Resonance in context of PDEs, [PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], doubly 2\(\pi\)-periodic solutions, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], nonlinear water waves, Periodic solutions to PDEs, Water waves, gravity waves; dispersion and scattering, nonlinear interaction, Abstract inverse mapping and implicit function theorems involving nonlinear operators, formal solutions, standing waves, 76B07, 76B15, bifurcation theory, infinite dimensional bifurcation equation
Bifurcations in context of PDEs, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], Free-surface potential flows for incompressible inviscid fluids, nonlocal PDE, complete resonance AMS classification: 35B32, gravity waves, PDEs in connection with fluid mechanics, 35B34, Asymptotic expansions of solutions to PDEs, standing gravity waves, [NLIN.NLIN-PS] Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Resonance in context of PDEs, [PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], doubly 2\(\pi\)-periodic solutions, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], nonlinear water waves, Periodic solutions to PDEs, Water waves, gravity waves; dispersion and scattering, nonlinear interaction, Abstract inverse mapping and implicit function theorems involving nonlinear operators, formal solutions, standing waves, 76B07, 76B15, bifurcation theory, infinite dimensional bifurcation equation
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