Nonlinear scattering: The states which are close to a soliton

We assume that the nonlinear Schroedinger equation with sufficiently general nonlinearity admits solutions of the soliton type. The Cauchy problem with initial data close to a soliton is considered. We also assume that the linearization of the equation in the vicinity of the soliton possesses only a real spectrum. The main result claims that the asymptotic behavior of the solution as t→+∞ is given by the sum of a soliton with deformed parameters and a dispersive tail, i.e., a solution of the linear Schroedinger equation. The case of the minimal spectrum has been considered in the previous paper. Bibliography: 1 title. Dedicated to O. A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 38–50. Translated by G. S. Perelman     

Soliton solutions of an integrable boundary problem on the half-line for the discrete toda chain

We write formulas for soliton solutions of the discrete Toda chain and pose the integrable boundary value problem for this chain. We find conditions for the parameters (discrete spectrum points, transmission coefficients, and the corresponding factors) whereby solutions of the integrable boundary value problem are selected from all soliton solutions. As a result, we construct two hierarchies of soliton solutions of the specified problem with even and odd soliton numbers and find an explicit form of the conditions for the parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 387–397, September, 2006.     

Perturbation of the two-soliton solution to the KdV equation

The Cauchy problem for the perturbed Korteveg-de Vries equation with the two-soliton initial data is considered. Differential equations for the slow deformation of the parameters—amplitudes and phase shifts—are derived. It is shown that the phase shift of the slow soliton depends on the deformation of the fast solition. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 92–102.     

Structural instability of a soliton for the Davey-Stewartson II equation

An algebraic soliton of the Davey-Stewartson II equation looses structural stability under a small perturbation of the initial data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 354–361, March, 1999.     

An elementary approach to the polynomial τ-functions of the KP Hierarchy

We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 23–36, January, 1999.     

Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance

We consider a simple model system supporting stable solitons in two dimensions. The system is the parametrically driven damped nonlinear Schrodinger equation, and the soliton stabilizes for sufficiently strong damping. We elucidate the stabilization mechanism by reducing the partial differential equation to a finite-dimensional dynamical system and conclude that the negative feedback loop occurs via enslaving the soliton phase, locked to the driver, to its amplitude and width. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 226–233, August, 2005.     

Nonlinear localized waves in a medium with nonlocal interaction

Soliton solutions are found for nonlinear integro-differential equations with a type λ/(τ-τ′) kernel used to describe particle tunneling and magnetic and superconducting vortices in a medium with nonlocal interaction. The Fourier transform method is applied to derive asymptotic formulas for even and odd localized solutions. Analytical solutions are found for particular parameter values. A complete pattern is constructed for the behavior of soliton solutions in an arbitrary range of the interaction parameter λ by means of numerical calculations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 3, pp. 366–379, March, 1998.     

Soliton in Gravitating Gas: Hoag's Object

We explore the possibility of creating of solitons in gravitating gas. We show that the virial arguments do not create an obstacle to the existence of localized static solutions. The simplest toroidal soliton of gravitating gas could explain the peculiar galaxy named Hoag's Object. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 103–114, January, 2006.     

Perturbation of a two-soliton solution of the Korteweg-de vries equation in the case of close amplitudes

We investigate the interaction process for two solitons with close amplitudes under a small perturbation. The leading term of the formal asymptotic solution is found as the sum of two solitons with slowly varying parameters. The equations of slow variations are derived for the soliton phase shifts. The effects related to the interaction between the perturbed solitons can compensate the velocity difference in some conditions, which can result in the formation of the so-called quasi-stationary soliton pair. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 434–440, March, 1999.     

Asymptotic behavior of singular solitons and the inverse scattering method for solving boundary value problems

We solve the boundary value problems for the sine-Gordon equation and the elliptic Toda lattice in the framework of the inverse scattering method. This approach allows incorporating the singular soliton solutions of the spiral vortex form. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 279–291, August, 2000.     

Simulation of soliton solutions for the propagation of a femtosecond pulse in a medium with a cubic nonlinearity

Soliton solutions are constructed numerically for the problem of propagation of a femtosecond pulse in a medium with a cubic nonlinearity. The problem is posed as an eigenvalue problem with an operator nonlinear in the eigenfunctions. For given values of the propagation parameter we find the real eigenvalue λ and the corresponding eigenvector. This eigenvector is a soliton, i.e., a solution that does not vary in the coordinate of propagation of the light pulse. An algorithm is proposed to find the minimum eigenvalue and the corresponding eigenfunctions that satisfy given conditions. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 63–68, 1999.     

A new conservative finite difference scheme for Boussinesq paradigm equation

A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the proposed family of schemes is about two times more accurate than the family of schemes studied in [Kolkovska N., Two families of finite difference schemes for multidimensional Boussinesq paradigm equation, In: Application of Mathematics in Technical and Natural Sciences, Sozopol, June 21–26, 2010, AIP Conf. Proc., 1301, American Institute of Physics, Melville, 2010, 395–403].     

Generation of soliton pairs in nonlinear media with weak dissipation

Weak interaction processes are studied for two KdV solitons at low viscosity. A model is suggested for describing the interaction in terms of slowly varying parameters of the exact two-soliton solution to the KdV equation under perturbation. It is shown that both the inverse scattering problem method and the Witham method lead to the same system of reduced equations. The resulting solutions are in good agreement with numerical calculation data. The main effect is the formation of a bound quasistationary soliton pair and its subsequent dissipation. It turns out that although both the process of soliton merger and the process of dissipation are due to low viscosity, the former is substantially faster. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 254–271, February, 1997.     

<Emphasis Type="Italic">N</Emphasis>-soliton train and generalized complex Toda chain for the Manakov system

We analyze the dynamical behavior of the N-soliton train of the Manakov system and of the vector NLS equation in the adiabatic approximation. We prove that the dynamics of the N-soliton train in both cases are described by a generalized version of the complex Toda chain model. This fact can be used to predict the asymptotic regimes of the N-soliton train provided the initial soliton parameters are given. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 391–404, June, 2007.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of [Novikov in Int. Math. Res. Papers 2005(6):287–349, [2005]] and implemented numerically in [Alekseenko et al. in Acoust. J. 54(3), [2008]]. This method is obtained as a development of the -approach to inverse scattering at fixed energy in dimension d≥3 of [Beals and Coifman in Proc. Symp. Pure Math. 43:45–70, [1985]] and [Henkin and Novikov in Usp. Mat. Nauk 42(3):93–152, [1987]] and involves, in particular, some results of [Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93–180, [1965], [1974]] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov in Usp. Mat. Nauk 31(5):245–246, [1976]] and [Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15–18, [1976]]). Also, our studies go back, in particular, to [Regge in Nuovo Cimento 14:951–976, [1959]]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.      

The space of solvsolitons in low dimensions

Up to now, the only known examples of homogeneous nontrivial Ricci soliton metrics are the so-called solvsolitons, i.e., certain left-invariant metrics on simple connected solvable Lie groups. In this article, we describe the moduli space of solvsolitons of dimension ≤ 6 up to isomorphism and scaling. We start with the already known classification of nilsolitons and, following the characterization given by Lauret in (J. Reine Angew. Math. 650:1–21, 2011), we describe the subspace of solvsolitons associated to a given nilsoliton, as the quotient of a Grassmanian by a finite group.     

On the Toda and Kac–van Moerbeke hierarchies

We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac–van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N-soliton solution of the Toda and Kac–van Moerbeke hierarchy. Received November 20, 1997; in final form July 7, 1998     

The dressing chain of discrete symmetries and proliferation of nonlinear equations

In the examples of sine-Gordon and Korteweg-de Vries (KdV) equations, we propose a direct method for using dressing chains (discrete symmetries) to proliferate integrable equations. We give a recurrent procedure (with a finite number of steps in general) that allows the step-by-step production of an integrable system and its L-A pair from the known L-A pair of an integrable equation. Using this algorithm, we reproduce a number of known results for integrable systems of the KdV type. We also find a new integrable equation of the sine-Gordon series and investigate its simplest soliton solution of the double π-kink type. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115. No. 2. pp. 199–214. May. 1998.     

Effect of polarization on the trajectory of dissipative solitons

We show that the optical Magnus effect for dissipative solitons is determined not only by the helicity but also by the topological index, i.e., by the magnetic quantum number or by the projection of the soliton orbital moment on its trajectory. In the case of inhomogeneous media, we find a relation between the optical Magnus effect and the nonholonomy of the field of unit vectors tangent to the trajectory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 126–134, January, 2009.