Related papers: Nonlinear smoothing for the periodic dispersion ge…
We show that uniqueness results of the kind those obtained for KdV and Schr\"odinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spaces $$H^s(\R)\cap L^2(x^{2r}dx),$$ for…
We consider the periodic non-linear Schr\"odinger equation with non-linearity given by $|u|^{p-1}u$ for odd $p > 1$ in dimension $1$. We first establish that the difference between the non-linear evolution and a phase rotation of the the…
We consider semilinear equations of the form p(D)u=F(u), with a locally bounded nonlinearity F(u), and a linear part p(D) given by a Fourier multiplier. The multiplier p(\xi) is the sum of positively homogeneous terms, with at least one of…
For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) +…
We study the dispersion-generalized Benjamin-Ono equation in the periodic setting. This equation interpolates between the Benjamin-Ono equation ($\alpha=1$) and the viscous Burgers' equation ($\alpha=0$). We obtain local well-posedness in…
We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation,…
We provide a roadmap to establish improved lower bounds on the decay rate of the uniform radius of analyticity $\sigma(T)$ for a given nonlinear dispersive equation, reducing the problem to the derivation of nonlinear smoothing estimates…
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the…
We study the well-posedness of the initial value problem on periodic intervals for linear and quasilinear evolution equations for which the leading-order terms have three spatial derivatives. In such equations, there is a competition…
We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well posedness results in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties…
New local well-posedness results for dispersion generalized Benjamin-Ono equations on the torus are proved. The family of equations under consideration links the Benjamin-Ono and Korteweg-de Vries equation. For sufficiently high dispersion…
We consider the $L^2$ well-posedness of third order Benjamin-Ono equation. We show that by means of a normal form and a gauge transformation, the equation can be changed into an Airy-type equation. A second goal of this work is to establish…
This article represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to…
Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For…
We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter $\varepsilon$. The proposed splitting schemes are furthermore asymptotic convergent…
We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDE of KdV-type, including generalized KdV and Benjamin-Ono equations. In this investigation, we consider the…
Using new generalized Landen transformations, we prove that the solutions of the KdV and other nonlinear equations obtained recently by using a kind of superposition principle for periodic solutions are in fact novel re-expressions of well…
We study persistence properties of solutions to some canonical dispersive models, namely the semi-linear Schr\"odinger equation, the $k$-generalized Korteweg-de Vries equation and the Benjamin-Ono equation, in weighted Sobolev spaces…
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized…
In this paper, the local well-posedness of periodic fifth order dispersive equation with nonlinear term $P_1(u)\p_xu + P_2(u)\p_x u\p_xu $. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. We also get some new Strichartz estimates.