色散水波和修正Boussinesq方程的特殊多项式和精确解

Boussinesq's equation for water waves: the soliton resolution conjecture for Sector IV

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by $\frac{x}{t}\in (\frac{1}{\sqrt{3}},1)$, in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction…

Dynamics of Higher-order Bright and Dark Rogue Waves in a New (2+1)-Dimensional Integrable Boussinesq Model

This work deals with the dynamics of higher-order rogue waves in a new integrable (2+1)-dimensional Boussinesq equation governing the evolution of high and steep gravity water waves. To achieve this objective, we construct rogue wave solutions by employing Bell polynomial and Hirota's bilinearization method, along with the generalized polynomial function. Through the obtained rogue wave solutions, we explore the impact of various system and solution parameters in their dynamics. Primarily, these parameters determine the characteristics of rogue waves, including the identification of their type…

耗散Boussinesq方程

Dissipative Boussinesq equations

水波的经典理论是建立在无粘流理论的基础上的。然而，在某些应用中包括粘性效应是重要的。提出了两种模型，以增加耗散效应的Boussinesq方程的上下文中，其中包括弱色散和非线性在浅水框架的影响。耗散Boussinesq方程的数值积分。

Localized Nonlinear Waves on Spatio-Temporally Controllable Backgrounds for a (3+1)-Dimensional Kadomtsev-Petviashvili-Boussinesq Model in Water Waves

Physics of nonlinear waves on variable backgrounds and the relevant mathematical analysis continues to be the challenging aspect of the study. In this work, we consider a (3+1)-dimensional nonlinear model describing the dynamics of {water waves} and construct nonlinear wave solutions on spatio-temporally controllable backgrounds for the first time by using a simple mathematical tool auto-Bäcklund transformation. Mainly, we unravel physically interesting features to control and manipulate the dynamics of nonlinear waves through the background. Adapting an exponential function and general polyno…

Surface gravity waves of the Boussinesq equation

This article is devoted to exact solutions of the Boussinesq equation that models nonlinear shallow water waves. For this we use the Hirota bilinear method and differential constrains. Out solutions describe in particular the motion of the wave packets, waves on soliton and "dancing" waves. We present a simple method for multiplication of solutions of the Hirota equation which allow us to generate more complex structures from the waves.

关于水波的 Boussinesq 方程

On Boussinesq's equation for water waves

本文针对不适定的水波 Boussinesq 方程，通过求解三阶谱问题发展了逆散射变换（IST）方法，建立了初值问题的解的表达式，并得出了全局解的存在性、唯一性、精确爆破时间及十个渐近扇区的首项与误差估计。