Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system
Jaime Angulo Carlos Matheus Didier Pilod
Communications on Pure & Applied Analysis 2009, 8(3): 815-844 doi: 10.3934/cpaa.2009.8.815
The objective of this paper is two-fold: firstly, we develop a local and global (in time) well-posedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrödinger-Benjamin-Ono system) for low-regularity initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schrödinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called dnoidal, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.
keywords: Nonlinear PDE traveling wave solutions. initial value problem

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