G. Soliani

Integrable dissipative structures in the gauge theory of gravity

The Jackiw - Teitelboim gauge formulation of (1 + 1)-dimensional gravity allows us to relate different gauge-fixing conditions to integrable hierarchies of evolution equations. We show that the equations for the zweibein fields can be written as a pair of time-reversed evolution equations of the reaction - diffusion type, admitting dissipative solutions. The spectral parameter for the related Lax pair appears as the constant-valued spin connection associated with the SO(1,1) gauge symmetry. Spontaneous breaking of the non-compact symmetry and irreversible evolution are discussed.

Group analysis of the three‐wave resonant system in (2+1) dimensions

The three‐wave resonant interaction equations (2D‐3WR) in two spatial and one temporal dimension within a group framework are analyzed. The symmetry algebra of this system, which turns out to be an infinite‐dimensional Lie algebra whose subalgebra is of the Kac–Moody type, is found. The one‐ and two‐dimensional symmetry subalgebras are classified and the corresponding reduction equations are obtained. From these the new invariant and the partially invariant solutions of the original 2D‐3WR equations are obtained.

Deformation of surfaces, integrable systems, and Chern–Simons theory

A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern–Simons action via the reduction of the gauge connection in Hermitian symmetric spaces. In this article we show that the methods developed in studying classical non-Abelian pure Chern–Simons actions can be naturally implemented by means of a geometrical interpretation of such systems. The Chern–Simons equation of motion turns out to be related to time evolving two-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss–Mainardi–Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.

On the isospectral-eigenvalue problem and the recursion operator of the Harry-Dym equation

SummaryThe isospectral-eigenvalue problem for the Herry-Dym equation is derived using a prolongation technique. The eigenvalue equation is then exploited to find a recursion operator.

An analysis of the Transverse Optical Klystron (TOK) and the Free Electron Laser (FEL) through the exact solution of their evolution equation

Abstract We give the exact analytical solution of the evolution equation of an electron beam interacting in a wiggler with an electromagnetic wave. The initial energy distribution is assumed both monochromatic and gaussian. Furthermore, the evolution in a drift space is considered. Some basic characteristics of the FEL and TOK are derived from the mathematical properties of the solution.

Gauge equivalence theory of the noncompact Ishimori model and the Davey-Stewartson equation

The gauge equivalence between a noncompact version of the Ishimori spin model and the Davey–Stewartson equation is established. Explicit relationships connecting the corresponding two sets of fields involved in these systems are obtained via any pair of complex functions satisfying an equation of the Schrodinger type for a free particle. Using these formulas, two examples of classes of nontrivial exact singular solutions to the Davey–Stewartson equation are given. One of them is of the closed stringlike type, while the other is doubly periodic and is expressed in terms of Riemann theta functions. The role played by the symmetry group associated with the gauge equivalent equations under consideration is also clarified.

Integrable nonlinear field equations and loop algebra structures

Abstract We apply the (direct and inverse) prolongation method to a couple of nonlinear Schrodinger equations. These are taken as a laboratory field model for analyzing the existence of a connection between the integrability property and loop algebras. Exploiting a realization of the Kac-Moody type of the incomplete prolongation algebra associated with the system under consideration, we develop a procedure which allows us to generate a new class of integrable nonlinear field equations containing the original ones as a special case.

On the Painlevé property of nonlinear field equations in 2+1 dimensions: The Davey–Stewartson system

With the purpose of clarifying some aspects of the complete integrability of nonlinear field equations, a singular‐point analysis is performed of the Davey–Stewartson system, which can be considered as an extension in 2+1 dimensions of the nonlinear Schrodinger equation. It is found that the system under consideration possesses the Painleve property and allows a set of Backlund transformations obtained by truncating the series expansions of the solutions about the singularity manifold.

Symmetry properties and exact patterns in birefringent optical fibers.

We carry out a group-theoretical study of the pair of nonlinear Schrödinger equations describing the propagation of waves in nonlinear birefringent optical fibers. We exploit the symmetry algebra associated with these equations to provide examples of specific exact solutions. Among them, we obtain the soliton profile, which is related to the coordinate translations and the constant change of phase.

Topological field theories and integrable models

We show that the classical non-Abelian pure Chern–Simons action is related to nonrelativistic models in (2+1) dimensions, via reductions of the gauge connection in Hermitian symmetric spaces. In such models the matter fields are coupled to gauge Chern–Simons fields, associated with the isotropy subgroup of the considered symmetric space. Thus a relation between the Chern–Simons theory and the Davey–Stewartson hierarchy is established in a natural way. The Backlund transformations are interpreted in terms of Chern–Simons constraints. Moreover, certain nonintegrable Heisenberg models can be embedded into the pure Chern–Simons theory. The main classical and quantum properties of these systems are discussed.