Stability of solitary-wave solutions to the Hirota-Satsuma equation
Jerry L. Bona Didier Pilod
Discrete & Continuous Dynamical Systems - A 2010, 27(4): 1391-1413 doi: 10.3934/dcds.2010.27.1391
The evolution equation

$ u_t- $ uxxt$ +u_x-$uut$ +u_x\int_x^{+\infty}u_tdx'=0, $ (1)

was developed by Hirota and Satsuma as an approximate model for unidirectional propagation of long-crested water waves. It possesses solitary-wave solutions just as do the related Korteweg-de Vries and Benjamin-Bona-Mahony equations. Using the recently developed theory for the initial-value problem for (1) and an analysis of an associated Liapunov functional, nonlinear stability of these solitary waves is established.

keywords: Solitary waves nonlinear dispersive wave equations stability Korteweg-de Vries-type equations.
Sharp well-posedness results for the Kuramoto-Velarde equation
Didier Pilod
Communications on Pure & Applied Analysis 2008, 7(4): 867-881 doi: 10.3934/cpaa.2008.7.867
We study the dispersive Kuramoto-Sivashinsky and Kuramoto-Velarde equations. We show that the associated initial value problem is locally (and globally in some cases) well-posed in Sobolev spaces $H^s(\mathbb R)$ for $s > -1$. We also prove that these results are sharp in the sense that the flow map of these equations fails to be $C^2$ in $H^s(\mathbb R)$ for $s < -1$. In addition, we determine the limiting behavior of the solutions when the dispersive parameter tends to zero.
keywords: initial value problem. Nonlinear PDE
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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