Related papers: Sharp well-posedness for a coupled system of mKdV …

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.

We study the Cauchy problem of the Schr\"odinger-Korteweg-de Vries system. First, we establish the local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show…

We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{{R}})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map…

We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $…

We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local…

We study the local well-posedness in the Sobolev space H^s for the modified Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega \cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map fails to be…

We prove local-in-time a-priori estimates in $H^{-1}(\mathbb{R})$ for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of…

In this work we prove that the initial-boundary value problem (IBVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad x\in\mathbb R^+,\;…

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of…

In this paper we study the generalized Korteweg de Vries (KdV) equation with the nonlinear term of order three: $(u^{3+1})_x$. We prove sharp local well--posedness for the initial and boundary value problem posed on the right half line. We…

We prove local well-posedness of partially periodic and periodic modified KP-I equations, namely for $\partial_t u+(-1)^{\frac{l+1}{2}}\partial^l_x u-\partial_x^{-1}\partial_y^2 u+u^2\partial_x u=0$ in the anisotropic Sobolev space…

In this article we consider the Cauchy problem with large initial data for an equation of the form (\partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms. Local well-posedness was established in…

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low regularity…

We prove the local well-posedness of the periodic stochastic Korteweg-de Vries equation with the additive space-time white noise. In order to treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type…

We investigate some well-posedness issues for the initial value problem (IVP) associated to the system \begin{equation} \{ \begin{array} [c]{l} 2i\partial_{t}u+q\partial_{x}^{2}u+i\gamma\partial_{x}^{3}u=F_{1}(u,w)\\…

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(R)$, with $s>-1/2$, are sharp in the…

We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(\R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(\R)$. The new ingredient is that we use directly…

We prove the local well-posedness for the Cauchy problem of the Korteweg-de Vries equation in a quasi periodic function space. The function space contains functions such that f=f_1+f_2+...+f_N where f_j is in the Sobolev space of order…

In this work we derive a point-wise formula that will allows us to study the well-posedness of initial value problem associated to nonlinear dispersive equations in fractional weighted Sobolev spaces $H^s(\R)\cap L^2(|x|^{2r}dx)$, $s, r \in…

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s…