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The influence of internal hydrogen in metals is modeled using a bi-continual model. The internal medium is introduced in the bi-continuum for modeling the diffusion of mobile hydrogen. Study of the model equations was carried out under the assumption of a small fluctuation of the concentration of mobile hydrogen under the action of external mechanical load. The case of appearance of the nonlinear solitary waves exceeding the average concentration is considered. The conditions are obtained under which the appearance of a decaying solitary wave is possible.

It is shown that travelling wave asymptotic solutions of the perturbed Burgers equation may describe dispersive/dissipative features of the difference schemes used for simulations of the non-linear advection equation. Also these solutions may be employed to improve numerical modelling by choosing suitable artificial non-linear additions to the scheme.

Two-dimensional long wave nonlinear models are developed to describe the propagation of the rogue waves. It is shown that the simplest model corresponds to the Kadomtsev-Petviashvili equation, while the influence of the atmosphere movement and the current in the ocean on the generation of the rogue waves is resulted in derivation of new nonlinear integro-differential equation. Two mechanisms of the rogue wave formation are proposed on. the basis of the model equations. First is based on a resonant interaction of inclined plane solitary waves. According to the second one the rogue wave is single two-dimensionally localized travelling wave having the pit shape.