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A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg–de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth-order Boussinesq equation is considered for illustrations.

Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

Pattern formation in a finite layer of fluid induced either by buoyancy or by a surface-tension gradient is considered. The fluid is confined between poor conducting horizontal boundaries, leading to patterns with a characteristic horizontal scale much larger than the fluid depth. The evolution of the system is studied by numerical integration of the (1+2)D equation introduced by Knobloch [1990]: [Formula: see text] Here µ is the scaled bifurcation parameter, κ=1, and a represents the effects of a heat transfer finite Biot number. The coefficients β, δ and γ do not vanish when the boundary conditions at top and bottom are not identical (β≠0, δ≠0) or when non-Boussinesq effects are taken into account (γ≠0). When the conductive state becomes unstable due to surface-tension inhomogeneities, it is shown that the system evolves towards a stationary pattern of hexagons with up or down flow depending on the relative value of the coefficients β and δ. In the case of buoyancy-driven convection (β=δ≠0), the system displays a tesselation of steady squares. Knobloch’s equation also describes time-dependent patterns at high thermal gradients, including spatio-temporal chaos, due to the non-variational character of the equation.

Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.