8 research outputs found

### On the cauchy problem for a coupled system of kdv equations : critical case

We investigate some well-posedness issues for the initial value problem associated to the system
\begin{equation*}
\begin{cases}
u_{t}+\partial_x^3u+\partial_x(u^2v^3) =0,\\
v_{t}+\partial_x^3v+\partial_x(u^3v^2)=0,
\end{cases}
\end{equation*}
for given data in low order Sobolev spaces $H^s(\mathbb{R})\times H^s(\mathbb{R})$.
We prove local and global well-posedness results utilizing the sharp smoothing estimates associated to the linear problem combined with the contraction mapping principle. For data with small Sobolev norm we obtain global solution whenever $s\geq 0$ by using global smoothing estimates.
In particular, for data satisfying
$\delta<\|(u_0, v_0)\|_{L^2\times L^2} < \|(S, S)\|_{L^2\times L^2}$,
where $S$ is solitary wave solution, we get global solution whenever $s>3/4$.
To prove this last result, we apply the splitting argument introduced by Bourgain [5] and further simplified by Fonseca, Linares and Ponce [6, 7].FundaÃ§Ã£o para a CiÃªncia e a Tecnologia (FCT) - POCI 2010/FEDER, bolsa SFRH/BPD/22018/2005FundaÃ§Ã£o de Amparo Ã Pesquisa do Estado de SÃ£o Paulo (FAPESP

### Well-posedness for some perturbations of the kdv equation with low regularity data

We study some well-posedness issues of the initial value problem associated with the equation
$u_t+u_{xxx}+\eta Lu+uu_x=0, \quad x \in \mathbb{R}, \; t\geq 0,$
where $\eta>0$, $\widehat{Lu}(\xi)=-\Phi(\xi)\hat{u}(\xi)$ and $\Phi \in \mathbb{R}$ is bounded above. Using the theory developed by Bourgain and Kenig, Ponce and Vega, we prove that the initial value problem is locally well-posed for given data in Sobolev spaces $H^s(\mathbb{R})$ with regularity below $L^2$. Examples of this model are the Ostrovsky-Stepanyams-Tsimring equation for $\Phi(\xi)=|\xi|-|\xi|^3$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $\Phi(\xi)=\xi^2-\xi^4$, and the Korteweg-de Vries-Burguers equation for $\Phi(\xi)=-\xi^2$.FundaÃ§Ã£o para a CiÃªncia e a Tecnologia (FCT

### Well-posedness of KdV type equations

In this work, we study the initial value problems associated to some linear perturbations of the KdV equations. Our focus is in the well-posedness issues for the initial data given in the $L^2$-based Sobolev spaces. We develop a method that allows us to treat the problem in the Bourgain's space associated to the KdV equation. With this method, we can use the multilinear estimates developed in the KdV context thereby getting analogous well-posedness results for the linearly perturbed equations.FundaÃ§Ã£o para a CiÃªncia e a Tecnologia (FCT

### Nonlinear dispersive models, well-posedness and unique continuation property

The local and global well-posedness issues of the Cauchy problem associated to the coupled system of dispersive equations with low regularity data are addressed. Fourier transform restriction norm space techniques and the recently introduced high-low frequency splitting and the I-method are the main tools to achieve the objectives. Also, a complex analysis technique is used to prove unique continuation property for some bi-dimensional versions.FundaÃ§Ã£o para a CiÃªncia e a Tecnologia (FCT

### Exponential energy decay for the kadomtsev-petviashvili (KP-II) equation

In this paper we study the exponential decay of the energy of the externally damped Kadomtsev-Petviashvili (KP-II) equation. Our main tool is the classical dissipation-observability method. We use multiplier techniques to establish the main estimates, and obtain exponential decay result.FundaÃ§Ã£o para a CiÃªncia e a Tecnologia (FCT

### On the critical KdV equation with time-oscillating nonlinearity

We investigate the initial value problem (IVP) associated to the equation
\begin{equation*}
u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^5) =0,
\end{equation*}
where $g$ is a periodic function. We prove that, for given initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$ converges to the solution $U$ of the initial value problem associated to
\begin{equation*}
U_{t}+\partial_x^3U+m(g)\partial_x(U^5) =0,
\end{equation*}
with the same initial data, where $m(g)$ is the average of the periodic function $g$. Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is sufficiently large.FundaÃ§Ã£o para a CiÃªncia e a Tecnologia (FCT

### A system of coupled SchrÃ¶dinger equations with time-oscillating nonlinearity

This paper is concerned with the initial value problem (IVP) associated to the coupled system of supercritical nonlinear SchrÃ¶dinger equations
\begin{equation}
\begin{cases}
iu_{t}+\Delta u+\theta_1(\omega t)(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u = 0, \\
iv_{t}+\Delta v+\theta_2(\omega t)(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v = 0,
\end{cases}
\end{equation}
where $\theta_1$ and $\theta_2$ are periodic functions, which has applications in many physical problems, especially in nonlinear optics. We prove that, for given initial data $\varphi,\psi\in H^{1}(\mathbb{R}^{n})$, as $|\omega|\;\rightarrow\;\infty$, the solution $(u_{\omega},v_{\omega})$ converges to the solution $(U,V)$ of the IVP associated to
\begin{equation}\label{eq-0.2}
\begin{cases}
iU_{t}+\Delta U+I(\theta_1)(|U|^{2p}+\beta|U|^{p-1}|V|^{p+1})U = 0, \\
iV_{t}+\Delta V+I(\theta_2)(|V|^{2p}+\beta|V|^{p-1}|U|^{p+1})V = 0,
\end{cases}
\end{equation}
with the same initial data, where $I(g)$ is the average of the periodic function $g$. Moreover, if the solution $(U,V)$ is global and bounded, then we prove that the solution $(u_{\omega},v_{\omega})$ is also global provided $|\omega|\gg 1$

### On uniqueness and decay of solution to the Hirota equation

We address the question of the uniqueness of solution to the initial value problem associated to the equation
\begin{equation*}
\partial_{t}u+i\alpha
\partial^{2}_{x}u+\beta \partial^{3}_{x}u+i\gamma|u|^{2}u+\delta
|u|^{2}\partial_{x}u+\epsilon u^{2}\partial_{x}\overline{u} = 0,
\quad x,t \in \mathbb{R},
\end{equation*}
and prove that a certain decay property of the difference $u_1-u_2$ of two solutions $u_1$ and $u_2$ at two different instants of times $t=0$ and $t=1$, is sufficient to ensure that $u_1=u_2$ for all the time.FC