E. A. Kuznetsov

Soliton stability in plasmas and hydrodynamics

Abstract The stability of solitons is reviewed for nonlinear conservative media. The main attention is paid to the description of the methods: perturbation theory, inverse scattering transform, Lyapunov method. Its applications are demonstrated in detail for the nonlinear Schrodinger equation, the KdV equation, and their generalizations. Applications to problems in plasma physics and hydrodynamics are considered.

Multi-scale expansions in the theory of systems integrable by the inverse scattering transform

Abstract It is shown that using multi-scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the IST method into other systems of this type.

On the topological meaning of canonical Clebsch variables

Abstract A class of flows of an ideal incompressible liquid with nontrivial topology is considered. Parametrization of these flows by the n -field is shown to result in hamiltonian equations.

Optical solitons and quasisolitons

Optical solitons and quasisolitons are investigated in reference to Cherenkov radiation. It is shown that both solitons and quasisolitons can exist, if the linear operator specifying their asymptotic behavior at infinity is sign-definite. In particular, the application of this criterion to stationary optical solitons shifts the soliton carrier frequency at which the first derivative of the dielectric constant with respect to the frequency vanishes. At that point the phase and group velocities coincide. Solitons and quasisolitons are absent, if the third-order dispersion is taken into account. The stability of a soliton is proved for fourth order dispersion using the sign-definiteness of the operator and integral estimates of the Sobolev type. This proof is based on the boundedness of the Hamiltonian for a fixed value of the pulse energy.

Nonlinear interaction of solitons and radiation

Abstract In the framework of the one-dimensional nonlinear Schrodinger equation a nonlinear interaction between solitons and radiation is studied both analytically and numerically. The results are applied for analysis of the relaxation of amplified (perturbed) optical solitons in fiber communications. It is shown that as a result of the nonlinear interference between solitons and radiation the relaxation of pulses to a new soliton has an oscillatory behavior. The oscillations are damping in a power-law fashion. A new effect is found: a mutual attraction of solitons appearing due to their scattering on a nonsoliton part.

Nonlinear mirror mode dynamics: Simulations and modeling

With the aim to understand the origin of the pressure-balanc ed magnetic struc- tures in the form of holes and humps commonly observed in the solar wind and planetary mag- netosheaths, high-resolution hybrid numerical simulatio ns of the Vlasov-Maxwell (VM) equa- tions using both Lagrangian (particle in cells) and Euleria n integration schemes are presented and compared with asymptotic and phenomenological models for the nonlinear mirror mode dynamics. It turns out that magnetic holes do not result from direct nonlinear saturation of the mirror instability that rather leads to magnetic humps. Nevertheless, both above and be- low threshold, there exist stable solutions of the VM equati ons in the form of large-amplitude magnetic holes. Special attention is paid to the skewness of the magnetic fluctuations (that is negative for holes and positive for humps) and its dependency on the distance to threshold and the beta of the plasma. Furthermore, the long-time evolution of magnetic humps result- ing from the mirror instability in an extended domain far eno ugh from threshold may, when the beta of the plasma is not too large, eventually lead to the formation of magnetic holes.

Sharper criteria for the wave collapse

Abstract Sharper criteria for three-dimensional wave collapse described by the Nonlinear Schrodinger Equation (NLSE) are derived. The collapse threshold corresponds to the ground state soliton which is known to be unstable. Thus, for nonprefocusing distributions this represents the separatrix between collapsing and noncollapsing sectors. Numerical results support the theoretical results. Generalizations of the criteria for the NLSE with arbitrary power nonlinearity are also presented.

On the stability of nonlinear waves in integrable models

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.

Turbulence spectra generated by singularities

The problem of turbulence spectra generated by the singularities located on lines and planes is considered. It is shown that the frequency spectrum of fluid-surface displacements due to whitecaps (linear singularities) is scaled like a weakly turbulent Zakharov-Filonenko spectrum. The corresponding wave-vector spectrum may be highly anisotropic with a decrease in maximum, as in the Phillips spectrum. However, in the isotropic situation, the spectrum differs markedly from the Phillips form. For a highly anisotropic two-dimensional turbulence, the vorticity jumps can generate the Kraichnan power-law distribution in the region of maximal angular peak. For the isotropic distribution, the turbulence spectrum coincides with the Saffman spectrum. For the shock-generated acoustic turbulence, the spectrum has the form of the Kadomtsev-Petviashvili spectrum Eω∼ ω−2 for all spatial dimensionalities.

Talanov transformations in self-focusing problems and instability of stationary waveguides

Abstract With the help of the Talanov transformation the stationary two-dimensional waveguides in homogeneous media with the quadratic Kerr effect are shown to be unstable. The exact collapsing solution is found for the one-dimensional model.