Soliton dynamics in damped and forced Boussinesq equations

Abstract

We investigate the dynamics of a lattice soliton on a monatomic chain in the presence of damping and external forces. We consider Stokes and hydrodynamical damping. In the quasi-continuum limit the discrete system leads to a damped and forced Boussinesq equation. By using a multiple-scale perturbation expansion up to second order in the framework of the quasi-continuum approach we derive a general expression for the first-order velocity correction which improves previous results. We compare the soliton position and shape predicted by the theory with simulations carried out on the level of the monatomic chain system as well as on the level of the quasi-continuum limit system. For this purpose we restrict ourselves to specific examples, namely potentials with cubic and quartic anharmonicities as well as the truncated Morse potential, without taking into account external forces. For both types of damping we find a good agreement with the numerical simulations both for the soliton position and for the tail which appears at the rear of the soliton. Moreover we clarify why the quasi-continuum approximation is better in the hydrodynamical damping case than in the Stokes damping case.

Evolution equation for bidirectional surface waves in a convecting fluid

“…28,29 In the present case, r Ͼ 0 and it represents the destabilizing term. The equation we find has been proposed as one of two possible heuristic generalizations of the KdV-KS equation to bidirectional waves.…”

Blow-up for the sixth-order multidimensional generalized Boussinesq equation with arbitrarily high initial energy

“…Note that as far as we know, in the known works (e.g., [27][28][29][30][31]), as seen from the above explanations, there is no obvious result on the blow-up of problem (1.1)-(1.2) in finite time with arbitrarily high initial energy. Therefore, the main purpose of the present paper is to solve the problem.…”

A Compact Fourth-Order Finite Difference Scheme for the Improved Boussinesq Equation with Damping Terms

“…It can be seen that the wave becomes a dwarf and breaks down drastically, and the wave profile is destroyed at about t = 20. When τ s = 0.4 or even bigger value is chosen, we find that the wave stands still, but the wave is destroyed instantly, as perhaps the condition of oscillation is not satisfied [30], the long-wave components of a wave packet with a wave number that does not satisfy the condition do not propagate. The variations of solitary wave height (left) and the relative error err(t) (right) for τ s = 0.2 and τ s = 0.02 are shown in Fig.…”