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### Open Access Journals

DCDS

The evolution equation

$ u_t- $ u_{xxt}$ +u_x-$uu_{t}$ +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.

CPAA

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.

CPAA

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.## Year of publication

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