Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation
Université Paris Dauphine（巴黎第九大学）；Humboldt-Universitaet zu Berlin（柏林洪堡大学）；Institut für Mathematik（数学研究所）；The University of Edinburgh（爱丁堡大学）；Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）；Institut Universitaire de France（法国高等研究院）；Hausdorff Center for Mathematics（豪斯多夫数学中心）；Institut für Angewandte Mathematik Universität Bonn（波恩大学应用数学研究所）；University of Oxford（牛津大学）；Beijing Institute of Technology（北京理工大学）
94 pages, minor updates. We also added Remark 4.5, showing sharpness (modulo the endpoint) of local well-posedness of the modulated KdV on the real line
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We study dispersive equations with a time non-homogeneous modulation acting on the linear dispersion term. As primary models, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) and the intermediate long wave equation (ILW), imposing certain irregularity conditions on the time non-homogeneous modulation. In this work, we establish phenomena called regularization by noise in three-folds: (i) When the modulation is sufficiently irregular, we show that the modulated KdV on both the circle and the real line is locally well-posed in the regime where the (unmodulated) KdV equation is known to be ill-posed. In particular, given any $s \in \mathbb R$, we show that the modulated KdV on the circle with a sufficiently irregular modulation is locally well-posed in $H^s(\mathbb T)$. Moreover, by adapting the $I$-method to the current modulated setting, we prove global well-posedness of the modulated KdV in negative Sobolev spaces. (ii) It is known that certain (semilinear) dispersive equations such as BO and ILW exhibit quasilinear nature. We show that sufficiently irregular modulations make the modulated versions of these equations semilinear by establishing their local well-posedness by a contraction argument, providing local Lipschitz continuity of the solution map. (iii) We also prove nonlinear smoothing for these modulated equations, where we show that a gain of regularity of the nonlinear part becomes (arbitrarily) larger for more irregular modulations.
As applications of our approach, we also include further examples.
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。
This paper is dedicated to finding the solutions of the equation of the loaded modified Korteweg-de Vries. By the way, it is shown to find the solutions via $(G'/G)$-expansion method that is one of the most effective ways of finding solutions.
We are concerned with the decay of long time solutions of the initial value problem associated with the Schrödinger-Korteweg-de Vries system. We use recent techniques in order to show that solutions of this system decay to zero in the energy space. The result is independent of the integrability of the equations involved and it does not require any size assumptions.
For more than 20 years, the Korteweg-de Vries equation has been intensively explored from the mathematical point of view. Regarding control theory, when adding an internal force term in this equation, it is well known that the Korteweg-de Vries equation is exponentially stable in a bounded domain. In this work, we propose a weak forcing mechanism, with a lower cost than that already existing in the literature, to achieve the result of the global exponential stability to the Korteweg-de Vries equation.
In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.
The initial-boundary value problem for Schrödinger-Korteweg-de Vries system on the half-line
本文证明了Schrödinger-Korteweg de Vries系统在左右半直线上的初边值问题的局部适定性。在低正则性条件下，利用Holmer为研究Korteweg-de Vries方程的IBVP而发展的两个边界强迫算子族，得到了这些结果（偏微分方程中的通信，31（2006））和另一个家庭最近推出的卡瓦尔坎特（微分和积分方程（2017））在非线性薛定谔与二次非线性的背景下。
In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a threshold value which may provides us a rupture in the spectral stability. Concerning the second equation, we establish the existence of periodic waves using a Galilean transformation on the periodic cnoidal solution for the modified Korteweg-de Vries equation and for both equations, the threshold values are the same. The main advantage presented in our pape…
We consider the Cauchy problem for an equation of Korteweg-de Vries-Kawahara type with initial data in the analytic Gevrey spaces. By using linear, bilinear and trilinear estimates in analytic Bourgain spaces, we establish the local well-posedness for this problem. By using an approximate conservation law, we extend this to a global result in such a way that the radius of analyticity of solutions is uniformly bounded below by a fixed positive number for all time.
This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and nonlinear Schrödinger (modified fNLS) equations, with the dispersions |D|^α\partial_x and |D|^{α+1}, respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified scattering which has a logarithmic phase correction. Our results cover the full range -1<α<1, α\neq 0 for both the modified fKdV and fNLS equations.
