相关论文: Well-posedness for the fifth-order KdV equation in…

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 $…

In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s> 5/2. Also, we…

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^+,\;…

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, \{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x \Delta u+u^ku_x =…

In this work we study the initial value problem (IVP) for the fifth order KdV equations, \begin{align*} \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0,\text{} & \quad x,t\in \mathbb R, \quad k=1,2, \end{align*} in weighted Sobolev…

In this work, we consider the initial value problem (IVP) for a system of modified Korteweg-de Vries (mKdV) equations \begin{equation} \begin{cases} \partial_t v + \partial_x^3 v+ \partial_x (v w^2) = 0, \hspace{0.98 cm} v(x,0)=\psi(x),\\…

We prove that the initial value problem (IVP) for the critical generalized KdV equation $u_{t}+u_{xxx}+(u^5)_{x}=0$ on the real line is globally well-posed in $H^{s}(\R)$ provided $s>3/5$.

This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We…

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as $\partial_tu+\alpha\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$ while $\alpha\in \mathbb{R}$. We introduce an interpolated energy space $E_s$ to…

This paper is a continuation of the paper \emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as…

We consider the initial value problem (IVP) associated to the cubic nonlinear Schr\"odinger equation with third-order dispersion \begin{equation*} \partial_{t}u+i\alpha \partial^{2}_{x}u- \partial^{3}_{x}u+i\beta|u|^{2}u = 0, \quad x,t \in…

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and…

The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation \[\partial_t u - \partial_x^5 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0,~(t,x,y)\in\mathbb{R}\times\mathbb{T}^2\] We prove…

The periodic KP-I initial value problem $\partial_t u+\partial_x^3 u-\partial_x^{-1}\partial_y^2 u+\partial_x (u^2/2)=0$ on $T_{x,y}^2\times R_t, $u(0)=\phi$ is globally well-posed in the energy space $E^1 = E^1 (T^2)=\phi: T^2\to…

The initial value problem (IVP) for the non-isotropic Schr\"odinger equation posed on the two-dimensional cylinders and $\mathbb{T}^2$ is considered. The IVP is shown to be locally well-posed for small initial data in…

We prove that the KP-I initial value problem is globally well-posed in the natural energy space of the equation.

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)\\…

In the current paper, we investigate the fifth order modified KP-I eqaution, namely \begin{equation*} \partial_t u-\partial_{x}^{5}u-\partial_{x}^{-1}\partial_{y}u+\partial_{x}(u^3)=0. \end{equation*} This equation is $L^2$ critical and we…

We present ill-posedness results for the initial value problem (IVP) of the 5th-order Gardner equation. We use new breather solutions discovered for this higher order Gardner equation to measure the regularity of the Cauchy problem in…

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based…