Related papers: The Cauchy problem for a Schroedinger - Korteweg -…

We study the global Cauchy problem associated to the Davey-Stewartson system in $\re^n,\ n=2,3$. Existence and uniqueness of solution are stablished for small data in some weak $L^p$ space. We apply an interpolation theorem and the…

In this paper, we are concerned with the Cauchy problem for the generalized KdV equation with random data and rough data. Firstly, when $s\in\mathbf{R}$, by using the initial value randomization technique introduced by Shen et al.…

We consider a general Euler-Korteweg-Poisson system in $R^3$, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as…

We show that the Cauchy problem for the KdV equation can be solved by the inverse scattering transform (IST) for any initial data bounded from below, decaying sufficiently rapidly at plus infinity, but unrestricted otherwise. Thus our…

We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem…

We consider the Cauchy problem for an equation of Korteweg-de Vries-Kawahara type with initial data in the analytic Gevrey spaces. By using linear, bilinear and trilinear estimates in analytic Bourgain spaces, we establish the local…

Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x…

The two-dimensional Zakharov system is shown to have a unique global solution for data without finite energy if the L^2 - norm of the Schr\"odinger part is small enough. The proof uses a refined I-method originally initiated by Colliander,…

The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

The Cauchy problem is studied for systems of quasi-linear wave equations with multiple speeds in two space dimensions. Using the method of Klainerman and Sideris together with the localized energy estimate, we give an alternative proof of a…

The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof…

In this article, we address the Cauchy problem for the KP-I equation \[\partial_t u + \partial_x^3 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0\] for functions periodic in $y$. We prove global well-posedness of this problem for any…

We prove that the Cauchy problem for the 2D quintic defocusing biharmonic Schr\"odinger equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $\frac{8}{7}<s<2$. Our main ingredient to establish the result is the…

We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$…

Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation \[ u_t +uu_x +u_{xxx}=0, \quad \left. u \right |_{t=0}=u_{0} \] in the space $H^{s}…

In this paper we consider some dissipative versions of the modified Korteweg de Vries equation $u_t+u_{xxx}+|D_x|^{\alpha}u+u^2u_x=0$ with $0<\alpha\leq 3$. We prove some well-posedness results on the associated Cauchy problem in the…

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality…

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality…