V.G. Makhankov

One more example of inelastic soliton interaction

Abstract Inelastic interaction of solitons has been obtained via computer within the framework of the “improved” Kortewegde Vries type equation, ut+(un)x+uxxx−uxxt = 0, n = 2, 3, which is important from the point of view of the common Fermi-Pasta-Ulam problem.

On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying boussinesq's equation

Abstract We derive an equation for a dense perturbation valid in a resonance region, M → 1, associated with the Schrodinger equation for a complex amplitude of the high-frequency Langmuir-oscillation electrical field. A solution of the above system in the linear approximation is obtained and discussed.

Soliton-like “bubbles” in a system of interacting bosons

Abstract We study the ψ3–ψ5 nonlinear Schrodinger equation describing the boson gas with 2- and 3-body interactions. This equation is shown to possess, in 1, 2, and 3 dimensions, a new type of solitons which may be interpreted as bubbles in Bose condensate. The static “bubbles” turn out to be essentially unstable solutions.

On the integrability and isotopic structure of the one-dimensional Hubbard model in the long wave approximation

Abstract The Lax representation for the continuum limit of the Hubbard model is found. The lagrangian of the system is shown to be invariant under four parametric U(1, 1) internal symmetry group transformations. The group properties and the set of solutions generated by this group are considered. A possible generalization of the model considered is presented for the case of the U(p, q) group.

Dynamics of Langmuir turbulence. Formation and interaction of solitons

Abstract We present a computer study of the formation and interaction of Langmuir solitons allowing for the generation of ion-sound waves.

On the existence of non-one-dimensional soliton-like solutions for some field theories

Abstract Some general conditions of the existence of non-dimensional spherically symmetric solitons are discussed. As a particular case, the condition of the existence of instanton-like solutions is obtained which coincides in an amasing way with the well-known formula p = 2D (D−2) the condition for theory to be renormalizable. (D is the space time dimension, p is the degree of nonlinearity.)

Stability of some one-field solitons

Abstract The problem of soliton solution stability for some types of nonlinear wave equation permitting Lagrangian formulation is discussed. The main attention is paid to the soliton stability in the direction transversal to their motion.

Continual classical Heisenberg models defined on graded su(2,1) and su(3) algebras

Continual integrable Heisenberg models are constructed on real subalgebras of the superalgebra spl(2/1). Two Heisenberg models are shown to exist on the compact subalgebra uspl(2/1)≊su(2/1). One of these, SU(2/1)/S(U(2)×U(1)), is gauge equivalent to SU(2) nonlinear vector Schrodinger equation (NLSE) expressed in odd Grassman variables, the other, SU(2/1)/S(L(1/1)×U(1)), to ‘‘super’’ NLSE which is invariant under global supersymmetry transformations of SL(1/1). Also constructed are a Heisenberg model on the noncompact subalgebra ospu(1,1/1), with higher nonlinearities, and its gauge equivalent analog. Hamiltonian structure and classical solutions are studied and the possible connection of the given models with a version of the Hubbard one is discussed.

On the gauge equivalence of the Landau-Lifshitz and the nonlinear Schrödinger equations on symmetric spaces

Abstract The gauge equivalence between a generalized Heisenberg spin chain (G/H) in the classical and continuum limit and the nonlinear Schrodinger equation (NLSE), with special attention to noncompact groups, is established. It has been demonstrated that noncompact groups allow a richer spectrum of possible reductions of the Heisenberg system to the NLSE. Some specialities of the model with nontrivial boundary conditions are discussed. The gauge equivalence between single-axis anisotropic Landau-Lifshitz equations (LLE) and isotropic LLE is briefly discussed.

On the stability of soliton solutions to the Schrödinger equation with nonlinear term of the form ψ|ψ|ν

The stability of soliton solutions ψ = A0 sech2νν2Aν202βν+2(x−υt)expiυ24+2βν+2Aν0t+υ2(x−υt) to the nonlinear Schrodinger equation iψt + ψxx + β|ψ|νψ = 0 is investigated for arbitrary positive ν.