The interaction of short single-component vector solitons in the frame of the coupled third–order nonlinear Schrodinger equations taking into account third–order linear dispersion, self–stepping, self–stimulated Ramanscattering, cross–stepping and cross–stimulated Raman-scattering terms is considered. Conditions of reflection and propagation of the solitons through each other and also the condition of oscillator interaction (vector breather) are obtained.
Chemical reactions in a porous medium are found in many natural phenomena and technological processes. Reactive substances dissolved in groundwater can significantly change the soil strength. The precipitate formed as a result of the reaction changes the porous medium structure and affects the porosity and permeability. A one-dimensional model of the reaction of two reagents in a homogeneous porous medium with a linear reaction function is considered. The model includes the mass balance equations of each reagent and precipitate, and the kinetic equation of precipitate growth. It is assumed that the precipitate is stationary and the growth rate of the precipitate is proportional to the reagents’ concentration. A carrier fluid with constant concentration reagents is injected at the empty porous medium entrance. The reaction front moves in a porous medium at a constant speed. The exact solution to the problem is constructed by eliminating the unknown functions and lowering the equations’ order. A Riemann invariant that relates the concentration of sediment and reagents to the system’s characteristics was found. The reaction’s numerical simulation is performed. It is shown that, for a long time, the reagents’ concentrations and the precipitate tend to final limit values. Sediment profiles always decrease monotonously, and the type of the profiles’ convexity changes.
Standing surface waves in a viscous infinite-depth fluid are studied. The solution of the problem is obtained in the linear and quadratic approximations. The case of long, as compared with the boundary layer thickness, waves is analyzed in detail. The trajectories of fluid particles are determined and an expression for the vorticity is derived.
Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the third power of the steepness. The conditions, under which the packet envelope evolution is described by the nonlinear Schrödinger equation with a dissipative term linear in the amplitude, are determined. The rule, in accordance with which the term of this type can be correctly added in the evolutionary equation of an arbitrary order is formulated.