Solitary wave interactions with a periodic forcing: the extended Korteweg-de Vries framework

Soliton interactions with an external forcing: the modified Korteweg-de Vries framework

The aim of this work is to study asymptotically and numerically the interaction of solitons with an external forcing with variable speed using the forced modified Korteweg-de Vries equation (mKdV). We show that the asymptotic predictions agree well with numerical solutions for forcing with constant speed and linear variable speed. Regarding forcing with linear variable speed, we find regimes in which the solitons are trapped at the external forcing and its amplitude increases or decreases in time depending on whether the forcing accelerates or decelerates.

Comment on the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" by M. Fokou, T.C. Kofané, A. Mohamadou and E. Yomba, Eur. Phys. J. Plus, 132, 410 (2017)

The authors of the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" [1] claim that they have derived the full third order perturbed KdV equation for the case of uneven bottom. We show that the authors' derivation is not consistent due to the fact that they took into account only some of the third order corrections but not all of them. Moreover, we show that a consistent third order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom cannot be derived for a general form of bottom function.

科特韦格与德弗里斯的合作--对人格的探讨

The collaboration between Korteweg and de Vries -- An enquiry into personalities

多年来，Korteweg和de Vries的名字已经紧密联系在一起。以它们命名的方程在非线性偏微分方程理论中起着基础性的作用。什么是起源的博士论文德弗里斯和Korteweg德弗里斯文件？这两位数学家的远亲巴斯蒂安·威林克（Bastiaan Willink）试图回答这些问题。这篇文章是根据作者在2003年9月阿姆斯特丹大学专门讨论Korteweg和de Vries的研讨会上发表的演讲撰写的。

Long-time evolution of pulses in the Korteweg-de Vries equation in the absence of solitons revisited: Whitham method

We consider the long-time evolution of pulses in the Korteweg-de Vries equation theory for initial distributions which produce no soliton, but instead lead to the formation of a dispersive shock wave and of a rarefaction wave. An approach based on Whitham modulation theory makes it possible to obtain an analytic description of the structure and to describe its self-similar behavior near the soliton edge of the shock. The results are compared with numerical simulations.

Whitham Shocks and Resonant Dispersive Shock Waves Governed by the Higher Order Korteweg-de Vries Equation

The addition of higher order asymptotic corrections to the Korteweg-de Vries equation results in the extended Korteweg-de Vries equation. These higher order terms destabilise the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theo…

广义Riemann型和Korteweg-de Vries流体动力学方程的微分代数可积性分析

Differential-Algebraic Integrability Analysis of the Generalized Riemann Type and Korteweg-de Vries Hydrodynamical Equations

利用微分代数方法研究了N = 3; 4时广义Riemann型流体力学方程的Lax型可积性。该方法也被应用到著名的Korteweg-de Vries动力系统的Lax型可积性的研究。