We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$ where $A$ is a linear operator and $B$ is quadratic. A particular example is the Korteweg-de Vries (KdV) equation $u_t-u u_x+u_{xxx}=0$. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.
The linearized Korteweg-De Vries equation can be written as a Hamilton-like system. However, the Hamilton energy depends on the time, and is a nonsymmetric operator on $L^2({\bf R})$. By performing suitable unitary transforms on the Hamilton energy, we can reduce this operator into one that is not independent on the time but nonsymmetric. In this study, we consider the $L^2$-stability issues and smoothing estimates for this operator, and prove that it has no eigenvalues.
Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the …
In this paper, we establish the optimal convergence result of a second order exponential-type integrator from (136, Numer. Math., 2017) for solving the KdV equation under rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides the second order accuracy in $H^γ$ for initial data in $H^{γ+4}$ for any $γ\geq0$, where the regularity requirement is lower than the classical methods. The result is confirmed by numerical experiments and comparisons are made with the Strang splitting scheme.
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。
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right. \end{equation*} where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I\!\!R)$ and sharp ill-posedness in $H^s(I\!\!R)$ when $s<-p/2$ with $p \geq 2$.
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr ad of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intri…
A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems
Hamiltonian-Krein（不稳定性）指标涉及确定具有正实部的Hamiltonian本征值问题$ J L u=λu$的特征值的数目，其中$J$是反对称的，并且$L$是自伴的。如果$J$有一个有界的逆的指数是良好建立的，它是由运营商$L$限制到一些有限余维子空间上的负特征值的数量。有一类重要的问题--即KdV型问题--对于这些问题，$J$没有有界逆。本文克服了这一困难，推导出了KdV型本征值问题的指标。利用该指数讨论了KdV类问题和BBM类问题的同宿行波的谱稳定性。
This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$. New harmonic analysis tools, including new averaging approximations to the exponential phase functions, new frequency decomposition techniques, and new trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new techniques are introduced to establish stability estimates of time discretizations under …
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…
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 …
Almost Periodicity in Time of Solutions of the KdV Equation
本文研究了KdV方程$\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$的Cauchy问题，其中初值为几乎周期初值$u(x,0)=V(x)$ .我们考虑初始数据$V$，其中相关的薛定谔算子是绝对连续的，并有一个频谱，是不是太薄，在某种意义上，我们指定，并显示的存在性，唯一性，几乎周期性的时间的解决方案。这建立了Percy Deift对这类初始数据的猜想。结果表明，适用于所有小的解析准周期初始数据与丢番图频率向量。
The linear operator $c + (-Δ)^{α/2}$, where $c > 0$ and $(-Δ)^{α/2}$ is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg--de Vries equation. We establish a relation of the Green's function of this linear operator with the Mittag--Leffler function, which was previously used in the context of Riemann--Liouville's and Caputo's fractional derivatives. By using this relation, we prove that Green's function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every $c > 0$ and e…
Stochastic representation of tau functions of Korteweg-de Vries equation
本文将Korteweg de Vries（KdV）方程的τ函数表示为迭代Skorohod积分的拉普拉斯变换。我们的主要工具是积分算子的弗雷德霍姆行列式的概念。我们的结果扩展了Ikeda和Taniguchi的论文，他们得到了KdV的$N$ -孤子解的τ函数的随机表示为$N$独立Ornstein-Uhlenbeck过程的二次泛函的拉普拉斯变换。我们的一般结果超越了$N$ -孤子的情况下，使我们能够考虑一个非孤子解的KdV与高斯过程与柯西协方差函数。
We consider the Cauchy problem for the KdV hierarchy -- a family of integrable PDEs with a Lax pair representation involving one-dimensional Schrödinger operators -- under a local in time boundedness assumption on the solution. For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (f…
