Related papers: A Coupled Generalized Korteweg-de Vries System Dri…

We study stochastic Korteweg - de Vries equation driven by L\'evy noise consisting of the compensated time homogeneous Poisson random measure and a cylindrical Wiener process. We prove the existence of a martingale solution to the equation…

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 study the Cauchy problem to the KP-I equation posed on $\R^2$. We prove that it is $C^0$ locally well-posed in $H^{s,0}(\R\times \R)$ for $s>1/2$, which improves the previous results in \cite{GPW,GMo}.

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…

We prove that the Cauchy problem of the Schr\"odinger - Korteweg - deVries (NLS-KdV) system on $\mathbb{T}$ is globally well-posed for initial data $(u_0,v_0)$ below the energy space $H^1\times H^1$. More precisely, we show that the…

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_x=0, \ u(0)=\phi, \end{eqnarray*} where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued…

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u=2(u^{3})_x, u(0)=\phi$, where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued function, we show that it is…

Our goal in this paper is to investigate ergodicity of the randomly forced Korteweg-de Vries-Burgers(KdVB) equation driven by non-additive white noise. Under reasonable conditions, we show that exponential ergodicity for KdVB equation…

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and…

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving…

We prove that the quartic Korteweg-de Vries equation is globally well-posed for real-valued initial data in $H^s(\mathbb{R})$, $s>-1/24$.

We consider the initial value problem (IVP) associated to a Boussinesq type system. After rewriting the system in an equivalent form of coupled KdV-type equations, we prove that this is locally well-posed in $(H^s(\R^2))^4$, $s>3/2$, using…

We prove the local well-posedness for the generalized Korteweg-de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty_{t,x}$-function $\Psi(t,x)$, with…

In this paper, we study the Cauchy problem for the Chern-Simons gauged $O(3)$ sigma model under the Lorenz gauge condition. We prove the local well-posedness of solutions if the initial matter field and gauge field satisfy $(\bm{\phi}_0,…

The Cauchy problem for the Chern-Simons-Higgs system in the (2+1)-dimensional Minkowski space in temporal gauge is locally well-posed for low regularity initial data improving a result of Huh. The proof uses the bilinear space-time…

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The…

In this work we study the Cauchy problem in Gevrey spaces for a generalized class of equations that contains the case $b=0$ of the $b$-equation. For the generalized equation, we prove that it is locally well-posed for initial data in Gevrey…

The Cauchy problem for the modified KdV equation is shown to be locally well posed for data u_0 in the space \hat(H^r_s) defined by the norm ||u_0||:=||<\xi>^s \hat(u_0)||_L^r', provided 4/3 < r \le 2, s \ge 1/2 - 1/(2r). For r=2 this…

This work is concerned with the Cauchy problem for a coupled Schr\"odinger-Benjamin-Ono system $$\left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv,\qquad t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+\nu\mathcal…

In this paper we consider the supercritical generalized Korteweg-de Vries equation $\partial_t\psi + \partial_{xxx}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5\leq p\in\R$. We prove a local well-posedness result in the homogeneous…