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May  2009, 8(3): 815-844. doi: 10.3934/cpaa.2009.8.815

Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system

Jaime Angulo 1, , Carlos Matheus 2, and  Didier Pilod 3,

1. 

IME-USP, Department of Mathematics, Rua do Matão 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil
2. 

IMPA, Estrada Dona Castorina 110, CEP 22460-320 Rio de Janeiro, RJ, Brazil
3. 

UFRJ, Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, CEP: 21945-970, Rio de Janeiro, RJ

Received  May 2008 Revised  October 2008 Published  February 2009

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.
Mathematics Subject Classification: Primary: 35Q55, 35Q51; Secondary: 76B25, 76B1.
Citation: Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815
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