| Abstract: |
| Boussinesq`s equation was the first model for the propagation of surface waves over shallow inviscid fluid layer. Boussinesq found an analytical solution of his equation and thus proved that the balance between the steepening effect of the nonlinearity and the flattening effect of the dispersion maintains the shape of the wave. This discovery can be properly termed `Boussinesq Paradigm.` Apart from the significance for the shallow water flows, this paradigm is very important for understanding the particle-like behavior of nonlinear localized waves. The localized solutions (with finite energy in infinite region) which can retain their identity during the interaction are called quasi-particles if some mechanical properties (such as mass, energy, momentum) are conserved by the governing equations. Special interest are the generalized wave equations containing both a nonlinearity and a dispersion as well as the nonlinear evolution equations.
The original Boussinesq equation is fully integrable but incorrect in sense of Hadamard. 2D Boussinesq paradigm is correct but fully nonintegrable. The Manakov system is fully integrable but models only an elastic interaction. As it should have been expected, most of the physical systems are not fully integrable (even in one spatial dimension) and only a numerical approach
can lead to unearthing the pertinent physical mechanisms of the interactions. In this paper we study numerically the soliton dynamics of Manakov system with gain/loss, cross modulation, and external potentials (vector Schrodinger equation). The system is not integrable and admits only 3 conservation laws. The results obtained shed the mechanism of soliton interactions. |
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