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We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.

The system of Coupled Nonlinear Schrodinger's Equations (CNLSE) is solved numerically by means of a conservative difference scheme. A new kind of repelling collision is discovered for negative values of the cross-modulation coupling parameter, $\alpha_2$. The results show that as the latter becomes increasingly negative, the behavior of the solitons during interaction change drastically. While for $\alpha_2 >0$, the solitons pass through each other, a negative threshold value $\alpha^*_2 < 0$ is found below which the solitons repell each other. This is a novel result for this kind of models and the conservation of momentum for the system of quasi-particles (QPs) is thoroughly investigated.

We study the dynamics of the sine-Gordon equation's kink soliton solutions under the coarse-grain description via two "collective variables": the position of the "center" of a soliton and its characteristic width ("size"). Integral expressions for the interaction potential and the quasi-particles' cross-masses are derived. However, these cannot be evaluated in closed form when the solitons have varying widths, so we develop a perturbation approach with the velocity of the faster soliton as the small parameter. This enables us to derive a system of four coupled second-order ODEs, one for each collective variable. The resulting initial-value problem is very stiff and numerical instabilities make it difficult to solve accurately, so a semi-empirical iterative approach to its solution is proposed. Then, we demonstrate that, even though it appears the solitons pass through each other, the quasi-particles actually "exchange" their pseudomasses during a collision.