Breather solutions of the modified Benjamin-Bona-Mahony equation

Solitary and Jacobi elliptic wave solutions of the generalized Benjamin-Bona-Mahony equation

Exact bright, dark, antikink solitary waves and Jacobi elliptic function solutions of the generalized Benjamin-Bona-Mahony equation with arbitrary power-law nonlinearity will be constructed in this work. The method used to carry out the integration is the F-expansion method. Solutions obtained have fractional and integer negative or positive power-law nonlinearities. These solutions have many free parameters such that they may be used to simulate many experimental situations, and to precisely control the dynamics of the system.

Solitons, compactons and undular bores in Benjamin-Bona-Mahony-like systems

We examine the effect of dissipation on traveling waves in nonlinear dispersive systems modeled by Benjamin- Bona- Mahony (BBM)-like equations. In the absence of dissipation the BBM-like equations are found to support soliton and compacton/anticompacton solutions depending on whether the dispersive term is linear or nonlinear. We study the influence of increasing nonlinearity of the medium on the soliton- and compacton dynamics. The dissipative effect is found to convert the solitons either to undular bores or to shock-like waves depending on the degree of nonlinearity of the equations. The an…

Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x = u_{xxt}$ are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg-de Vries equation $u_t + uu_x + u_{xxx} =0.$ The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompa…

浅水波的新包络方程与调制不稳定性

New envelope equations for shallow water waves and modulational instability

本文基于BBM方程框架研究长波波群动力学，推导出新的包络方程，发现聚焦与散焦两种非线性薛定谔约化情形，并给出调制不稳定性条件，解释了畸形波的产生机制。

一类与BBM型流体动力流相关的新动力系统的可积性研究

On integrability of a new dynamical system associated with the BBM-type hydrodynamic flow

本文针对一类高阶BBM型非线性色散方程的可积性问题，采用梯度-完整可积性方案，通过构造无穷守恒律层级与相容Poisson结构，证明了相关两个动力系统的完全可积性，揭示了其与Riemann型流体动力方程层级的关联。