Konstantin A. Gorshkov

Perturbation Theories for Nonlinear Waves

Some ideas and theories developed since 1960s to describe nonlinear waves with slowly varying parameters (modulated waves)are outlined. These theories are associated with different versions of the asymptotic perturbation method. In this framework, both quasi-periodic and solitary waves (solitons)can be treated. A scheme for reduction of a quasihyperbolic system to one or more evolution equations is also presented. Some challenges for the theory are briefly discussed.

Asymptotic theory of plane soliton self-focusing in two-dimensional wave media

Abstract An asymptotic method is developed to describe a long-term evolution of unstable quasi-plane solitary waves in the Kadomtsev-Petviashvili model for two-dimensional wave media with positive dispersion. An approximate equation is derived for the parameters of soliton transversal modulation and a general solution of this equation is found in an explicit form. It is shown that the development of periodic soliton modulation, in an unstable region, leads to saturation and formation of a two-dimensional stationary wave. This process is accompanied by the radiation of a small-amplitude plane soliton. In a stable region, an amplitude of the modulation is permanently decreasing due to radiation of quasi-harmonic wave packets. The multiperiodic regime of plane soliton self-focusing is also investigated.

Perturbation theory for Rankine vortices

A perturbation scheme is constructed to describe the evolution of stable, localized Rankine-type hydrodynamic vortices under the action of disturbances such as density stratification. It is based on the elimination of singularities in perturbations by using the necessary orthogonality conditions which determine the vortex motion. Along with the discrete-spectrum modes of the linearized problem which can be kept finite by imposing the orthogonality conditions, the continuous-spectrum perturbations play a crucial role. It is shown that in a stratified fluid, a single (monopole) vortex can be destroyed due to the latter modes before it drifts very far, whereas a vortex pair preserves its stability for a longer time. The motion of the latter is studied in two cases: smooth stratification and a density jump. For the motion of a pair under a small angle to the interface, a complete description is given in the framework of our theory, including the effect of reflection of the pair from a region with slightly larger density.

Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions

A brief review of soliton dynamics constituting one-dimensional periodic chains is presented. It is shown that depending on the governing equation, solitons may have either exponential or oscillatory-exponential decaying tails. Under certain conditions, solitons interaction can be considered within the framework of Newtonian equations describing the dynamics of classical particles. Collective behaviour of such particles forming a one-dimensional chain may be simple or complex and even chaotic. Specific features of soliton motions are presented for some popular models of nonlinear waves (Korteweg-de Vries, Toda, Benjamin-Ono, Kadomtsev-Petviashvili, and others).

Decay of Kadomtsev–Petviashvili lumps in dissipative media

Abstract The decay of Kadomtsev–Petviashvili lumps is considered for a few typical dissipations—Rayleigh dissipation, Reynolds dissipation, Landau damping, Chezy bottom friction, viscous dissipation in the laminar boundary layer, and radiative losses caused by large-scale dispersion. It is shown that the straight-line motion of lumps is unstable under the influence of dissipation. The lump trajectories are calculated for two most typical models of dissipation—the Rayleigh and Reynolds dissipations. A comparison of analytical results obtained within the framework of asymptotic theory with the direct numerical calculations of the Kadomtsev–Petviashvili equation is presented. Good agreement between the theoretical and numerical results is obtained.

Perturbation Theory for the Compound Soliton of the Gardner’s Equation; Their Interaction and Evolution in a Media with Variable Parameters

This paper is a brief review of the results of an approximate description of the evolution and interaction of composite solitons, obtained by the authors in 2001–2016. As one of the applications of the theory, the features of the evolution of intense internal waves in the shelf zone of the ocean are analyzed.

Evolution of internal soliton groups

As known, acoustic wave propagation through a group of internal solitary waves (a solibore) in a coastal zone has a number of peculiarities; among them is a possible damping of sound upon passing a periodic group of solitons when the group period resonates with the interference distance of acoustic modes. However, solibores are typically not periodic, and the distance between solitons and possibly their order in the group can vary upon the onshore propagation. In this presentation, an evolution of a multisoliton group is considered in the framework of an evolution Gardner equation that takes both quadratic and cubic nonlinearity into account. For that, a perturbation method is used which allows the description of solitons as compounds of interacting fronts‐kinks, and reduces the problem to a set of ordinary differential equations. The results are applied to strong solitons observed near the Oregon coast in 1995 where the same wave group was registered at two sites separated by 20 km. Although nonperiodic,...