We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the $L^2$-norm if their initial data belong to a finite dimensional subspace $\M$. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for $\dim \M = 1$ or for the smallest critical length $L$ with $\dim \M = 2$ by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than $\ln (t + 2) / t$ for all critical lengths.
This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all non-critical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Crépeau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1, and later Cerpa, and then Cerpa and Crépeau established t…
Third order nonlinear evolution equations, that is the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered: they all are connected via Baecklund transformations. These links can be depicted in a wide Baecklund Chart} which further extends the previous one constructed in [22]. In particular, the Baecklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Baecklund Chart. That …
We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.
This paper is concerned with the local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity. The equation arises as a non-integrable and lower nonlinearity version of the higher-order KdV equation. As for the lower nonlinearity model of the KdV equation, Linares, the author and Ponce [11] prove the local well-posedness under a non-degenerate condition introduced by Cazenave and Naumkin [1]. In this paper, it turns out that the well-posedness result can be extended into the higher-order equation. We also give a lower bound for the lifespan of the sol…
We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm.
Variable depth KDV equations and generalizations to more nonlinear regimes
在这里，我们研究的水波问题不均匀的底部在一个高度非线性制度的小振幅假设的Korteweg-de Vries（KdV）方程是强制执行。众所周知，对于这样的制度，KdV方程的推广（以某种方式与Camassa-Holm方程相关联）可以由A. Constantin，D. Lannes“The hydrodynamic relevance of the Camassa-Holm and Degasperis-Processi equations”when the bottom is flat.我们在这里推广这个结果与一类新的方程，考虑到可变的底部地形。当然，文献中存在的许多变深度KdV方程是作为特殊情况恢复的。各种制度的地形制度进行了研究，我们证明这些模型的一致性，以及充分的理由，其中一些。我们还研究了我们的新的可变深度和高度非线性推广的KDV方程的波浪破碎的问题。
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.
This article concerns the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV e…
We study the well-posedness of the initial value problem for fully nonlinear evolution equations, $u_{t}=f[u],$ where $f$ may depend on up to the first three spatial derivatives of $u.$ We make three primary assumptions about the form of $f:$ a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of $u$ is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the init…
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 establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half…
Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrödingger equation (DNLS) with small mass in \cite{KlausS:DNLS}, we show the corresponding microscopic conservation laws for the Schwartz solutions of DNLS with small mass. The new ingredient is to make use of the logarithmic perturbation determinant introduced in \cite{Rybkin:KdV:Cons Law, Simon:Trace} to show one-parameter family of microscopic conservation laws of the $A(κ)$ flow and the DNLS flow, which is motivated by \cite{HKV:NLS,KV:KdV:AnnMath,KVZ:KdV:GAFA}.
In most of the studies concerning nonlinear wave equations of Korteweg-de Vries type, the authors focus on waves of elevation. Such waves have general form ~$u_{\text{u}}(x,t)=A f(x-vt)$, where ~$A>0$. In this communication we show that if ~$u_{\text{up}}(x,t)=A f(x-vt)$ is the solution of a given nonlinear equation, then $u_{\text{down}}(x,t)=-A f(x-vt)$, that is, an inverted wave is the solution of the same equation, but with changed sign of the parameter ~$α$. This property is common for KdV, extended KdV, fifth-order KdV, Gardner equations, and generalizations for cases with an uneven bott…
The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-ord…
We consider two types of the generalized Korteweg - de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of $H^1$ that includes functions with polynomial decay, extending the result of Linares et al [39] to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. W…
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 thus close the gap in the well--posedness theory of the generalized KdV which remained open after the seminal work of Colliander and Kenig in \cite{CK}.
In this article, we prove that small localized data yield solutions to Higher order Korteweg-de Vries type equation with scattering-supercritical nonlinearity have linear dispersive decay in only a finite length of time. The proof is done by using space-time resonance method and analyzing the oscillatory integrals on the Fourier side.
This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified …
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…
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.
