We prove that the Cauchy problem of the Schrödinger - 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 non-resonant NLS-KdV is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T})$ with $s>11/13$ and the resonant NLS-KdV is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T})$ with $s>8/9$. The idea of the proof of this theorem is to apply the I-method of Colliander, Keel, Staffilani, Takaoka and Tao in order to improve the results of Arbieto, Corcho and Matheus concerning the global well-posedness of the NLS-KdV on $\mathbb{T}$ in the energy space $H^1\times H^1$.
We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schrödinger-Korteweg-de Vries'' non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $ν$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.
In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $2<p\leq 8$, while a single mode in…
Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system
我们证明了一个非线性薛定谔型方程耦合到Korteweg-de Vries型方程的系统的两参数族孤波解的存在性和稳定性结果。这种系统模拟了短色散波和长色散波之间的相互作用。这些结果推广了Angulo，Albert and Angulo和Chen的早期结果.我们的证明涉及到孤立波的解决方案的特征作为极小的能量泛函受到两个约束。为了通过集中紧性建立极小化序列的预紧性，我们建立了问题关于两个约束变量的次可加性。
We consider the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension. We prove that for each $s>-\frac12$ there exists a conserved energy which is equivalent to the $H^s$ norm of the solution. For the Korteweg-de Vries (KdV) equation there is a similar conserved energy for every $s\ge -1$.
We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.
Well-posedness and ill-posedness of the fifth order modifed KdV equation
在Sobolev空间上考虑五阶修正KdV方程的初值问题. \partial_t u = \partial_x^5 u + c_1 \partial_x^3（u^3）+\partial_x u + c_3uuPDMATHABMO u =0，u（x，0）= u_0（x）其中$ u:R\timesR \to R $和$c_j$是实数.利用$X^{s,b}$空间上的压缩原理证明了sPDMATHABMT 3/4在H^s（R）中的局部适定性.此外，我们表明，从数据到解决方案的解决方案映射未能一致连续低于$H^{3/4}(R)$。反例是通过用三次NLS方程近似五阶mKdV方程得到的。
In this article, we study an initial-boundary-value problem of a coupled KdV-KdV system on the half line $ \mathbb{R}^+ $ with non-homogeneous boundary conditions: \begin{equation*} \left\{ \begin{array}{l} u_t+v_x+u u_x+v_{xxx}=0, \quad v_t+u_x+(vu)_x+u_{xxx}=0, \quad u(x,0)=ϕ(x),\quad v(x,0)=ψ(x), \quad u(0,t)=h_1(t),\quad v(0,t)=h_2(t),\quad v_x(0,t)=h_3(t), \end{array} \right. \qquad x,\,t>0. \end{equation*} It is shown that the problem is locally unconditionally well-posed in $H^s(\mathbb{R}^+)\times H^s(\mathbb{R}^+)$ for $s> -\frac34 $ with initial data $(ϕ,ψ)$ in $H^s(\mathbb{R}^+)\tim…
We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(\mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(\mathbb{R}/\mathbb{Z})$. In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schrödinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a nonlocal behavior both in space and time which accounts for long-range interactions and a so-called memory effect. Because of a loss of derivatives produced by the latter and the lack of semigroup structure of the solution operator, we employ a strategy of proof based on exploiting some smoothing effect similar to that used by Kenig, Ponce and Vega for the KdV equa…
We show symplectic non-squeezing for the KdV equation on the line $\mathbb R$. This is achieved via finite-dimensional approximation. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough of Killip and Visan in the well-posedness theory of KdV in low regularity spaces, relying on its completely integrable structure. We also prove and exploit a weak well-posedness result for KdV in $H^{-1}(\mathbb R)$. Furthermore, the employment of our methods provides a new concise proof for the known result of symplectic non-sque…
Local well-posedness for the fifth-order KdV equations on $\mathbb{T}$
本文是论文\emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}的继续。本文考虑KdV族中的五阶方程：\begin{equation*} \begin{cases} \partial_t u - \partial_x^5 u + 30u^2\partial_x u + 20 u\partial_x u \partial_x^3u + 10u \partial_x^3 u = 0, \hspace{1em} (t,x) \in \mathbb{R} \times \mathbb{T}, u(0,x) = u_0(x) \in H^s(\mathbb{T}) \end{cases}. \end{equation*}我们利用能量方法证明了五阶KdV方程在低正则性Sobolev初值下的局部适定性。本文的思想和论点与文献\emph{Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$}基本相同。准确地说，我们利用KdV哈密顿量的某些守恒律来观察非线性解的演化方向。另外，在非共振非线性项中，由于\emph{high $\times$ lo…
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 $H^{s} (\mathbb{R})$ (or $H^{s} (\mathbb{T})$), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems \[\left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} &=& c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x},\\ v_t+a_{2}v_{xxx}&=& c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x},\\ \left. (u,v)\right |_{t=0} &=& (u_{0},v_{0}) \end{array}\…
Given a suitable solution $V(t,x)$ to the Korteweg--de Vries equation on the real line, we prove global well-posedness for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $V(0,x) \in H^5(\mathbb{R}/\mathbb{Z})$ satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In our companion paper we show that smooth step-like initial data also satisfy our hypotheses. …
In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in $H^s(\mathbb T)$, $s > 2$, via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173 (2) (2008) 265--304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estim…
We study the focusing energy-critical NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^{\frac{4}{d-1}} u\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}_x^d\times\mathbb{T}_y$ with $d\geq 2$. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of \eqref{nls_abstract} are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem $m_c$ defined on the semivirial-vanishing manifo…
In this paper, we prove the global well-posedness for the focusing, cubic nonlinear Schrödinger equation on the product space $\mathbb{R} \times \mathbb{T}^3$ with initial data below the threshold that arises from the the ground state in the Euclidean setting. The defocusing analogue was discussed and proved in Ionescu-Pausader \cite{IPRT3} (Comm. Math. Phys. 312 (2012), no. 3, 781-831).
Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2π\mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.
Generalized KdV equation subject to a stochastic perturbation
我们证明了次临界广义Korteweg-de Vries方程（mKdV和gKdV与四次幂的非线性）的全局适定性。更精确地说，我们证明了如果驱动噪声是$L^2(\mathbb{R})$上的圆柱Wiener过程，协方差算子是适当的索伯列夫空间中的Hilbert-Schmidt，那么具有$H^1(\mathbb{R})$数据的解在$H^1(\mathbb{R})$中是全局适定性的.这推广了A. de Bouard和A.德彪西的随机KdV方程。
Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \quad v(x,0)=v_0(x)\] posed on the torus $\mathbb{T}$ in the spaces \[ {\cal H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_2:=H^s_0 (\mathbb{T})\times H^s(\mathbb{T}), \quad {\cal H}^s_3:=H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_4:=H^s (\mathbb{T})\times H^s (\mathbb{T}).\] For $k=1,2,3,4$, it is…
We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^αu\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime $α\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the \eqref{nls_abstract} is independent of $y$, it reduces to the focusing intercritical NLS on $\mathbb{R}^d$, which is known to have standing wave and finite time blow-up solutions. Naturally, we ask whether these special solutions with non-trivial $y$-dependence exist. In this paper we give an affirmative answer to this question. To that end, we…
We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$. For $s < \frac 14$, we show that the solution map for mKdV is not locally uniformly continuous in $M^{2,p}_{s}(\mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $…
Given smooth step-like initial data $V(0,x)$ on the real line, we show that the Korteweg--de Vries equation is globally well-posed for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. The proof uses our general well-posedness result for exotic spatial asymptotics. As a prerequisite, we show that KdV is globally well-posed for $H^3(\mathbb{R})$ perturbations of step-like initial data. In the case $V \equiv 0$, we obtain a new proof of the Bona--Smith theorem using the low-regularity methods that established the sharp well-posedness of KdV in $H^{-1}$.
We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $|u|^{α- 1} u$ for $α\in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($α= 2$) we obtain global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb R) + H^1(\mathbb T))$ via Gronwall's inequality.
We consider the periodic cubic-quintic nonlinear Schrödinger equation \begin{align}\label{cqnls_abstract} (i\partial_t +Δ)u=μ_1 |u|^2 u+μ_2 |u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus $\mathbb{T}^3$ with $μ_1,μ_2\in \mathbb{R} \setminus\{0\}$. As a first result, we establish the small data well-posedness of \eqref{cqnls_abstract} for arbitrarily given $μ_1$ and $μ_2$. By adapting the crucial perturbation arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that \eqref{cqnls_abstract} is always globally well-posed in $H^1(\mathbb{T}^3)$ in the case $μ_2>0…
We consider the following variational problem: minimize the $(n+1)$st polynomial conserved quantity of KdV over $H^n(\mathbb{R})$ with the first $n$ conserved quantities constrained. Maddocks and Sachs used that $n$-solitons are local minimizers for this problem in order to prove that $n$-solitons are orbitally stable in $H^n(\mathbb{R})$. Given $n$ constraints that are attainable by an $n$-soliton, we show that there is a unique set of $n$ amplitude parameters so that the corresponding multisolitons satisfy the constraints. Moreover, we prove that these multisolitons are the unique global con…
We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+γδ(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] at the mass-energy threshold, namely, when $E_γ(u_{0})[M(u_{0})]^σ=E_{0}(Q)[M(Q)]^σ$ where $u_{0}\in H^{1}({\mathbb R})$ is the initial data, $Q$ is the ground state of the free NLS on the real line ${\mathbb R}$, $E_γ$ is the energy, $M$ is the mass and $σ=(p+3)/(p-5)$. We also prove failure of the uniform space-time bounds at the mass-energy thr…
