A. V. Mikhailov

The reduction problem and the inverse scattering method

Abstract The problem of reduction for systems of nonlinear equations integrable by the inverse scattering method is discussed. It is shown that reduction can be constructed by predetermining the representation of the group (the reduction group) acting on a set of eigenfunctions of the linear problem associated with an integrable equation. An infinite set of conservation laws is constructed for the system of equations for a two-dimensional Toda chain, the inverse problem is solved and exact N-soliton solutions are found.

The Symmetry Approach to Classification of Integrable Equations

In this volume each of the contributors proposes his own test to recognize integrable PDEs. We believe that, independently from the basic definition of integrability, the test must satisfy some general requirements. Namely, it has to be effective (in other words, if an equation has passed through the test, then there are almost no doubts about its integrability); sufficiently algorithmical, yet able to admit a proper realization in a symbolic computer language (like Reduce, Formac, Macsyma, MuMath, AMP, etc.); applicable to a large class of PDEs.

Two-dimensional generalized Toda lattice

The zero curvature representation is obtained for the two-dimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.

On the integrability of classical spinor models in two-dimensional space-time

Well known classical spinor relativistic-invariant two-dimensional field theory models, including the Gross-Neveu, Vaks-Larkin-Nambu-Jona-Lasinio and some other models, are shown to be integrable by means of the inverse scattering problem method. These models are shown to be naturally connected with the principal chiral fields on the symplectic, unitary and orthogonal Lie groups. The respective technique for construction of the soliton-like solutions is developed.

Perturbative symmetry approach

The aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non-evolution equations. Our formalism is the development and incorporation of the perturbative approach of Zakharov and Schulman, the symbolic method of Sanders and Wang and the standard symmetry approach of Shabat et al. We apply our theory to describe integrable generalizations of the Benjamin-Ono type equations and to isolate integrable cases of the Camassa-Holm type equations.

Inverse scattering method with variable spectral parameter

In the traditional scheme of the inverse scattering method, the spectral parameter of the auxiliary linear problem is assumed to be a constant. It is here proposed to regard the parameter as a variable quantity that satisfies an overdetermined system of differential equations which is uniquely determined by the auxiliary linear problem. The nonlinear equations that arise in such an approach contain, as a rule, an explicit dependence on the coordinates. This makes it possible to construct not only the well-known equations (gravitation equation, Heisenberg equation in axial geometry, etc.) but also a number of new integrable equations that have applied significance.

Extension of the module of invertible transformations. Classification of integrable systems

We demonstrate that for the systems of equations, which are invariant under a point group or possess conservation laws of the zeroth or first order, a nontrivial extension of the module of invertible transformations is possible. That simplifies greatly a classification of the integrable systems of equations. Here we present an exhaustive list and a classification of the second order systems of the formut=uxx+f(u, v, uxvx), −vt=vxx+g(u, v, ux,vx), which possess the conservation laws of higher order. The reduction group approach allows us to define the Lax type representations for some new equations of our list.

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.

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.

The Landau-Lifschitz equation and the Riemann boundary problem on a torus

Abstract The Cauchy problem for the Landau-Lifschitz equation is studied by the inverse transform method. We reduce this problem to the matrix Riemann problem on a torus and then to a certain Fredholm integral equation. An exact N -soliton solution is constructed. The reduction group approach is a main tool of our study.