L. Martina

Scattering of localized solitons in the plane

Abstract Localized (exponentially decaying in all directions) soliton solutions of the evolution equations related to the Zakharov-Shabat spectral problem in the plane are explicitly given. They can move along any direction in the plane and the only effect of their interaction is a shift in their position both in the x- and y-directions, independently of their relative initial position in the plane.

Group foliation and non-invariant solutions of the heavenly equation

The main physical results of this paper are new exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining non-invariant solutions of partial differential equations with infinite-dimensional symmetry groups. The method involves the compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup, the latter acting transitively on the submanifold of the common solutions. By studying the integrability of the resulting conditions, one can provide an explicit foliation of the entire solution manifold of the considered equations.

Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem

Abstract The Lie algebra of the symmetry group of the three-wave resonant interaction system in three (or more) space dimensions is shown to be infinite-dimensional and to have the structure of the direct sum of three Kac-Moody-Virasoro u (1) algebras. The symmetry group is obtained and its one- and two-dimensional subgroups are used to perform symmetry reduction. A wide range of new solutions is obtained. The methods presented in this article are applicable to any system of partial differential equations with an infinite-dimensional symmetry group.

The Weierstrass–Enneper system for constant mean curvature surfaces and the completely integrable sigma model

The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints. A new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples, including a multi-soliton solution.

Galilean symmetry in noncommutative field theory

Abstract When the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended “exotic” Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an “exotic” two-parameter central extension of the Schrodinger group. The conformal symmetry is, however, broken by the interaction.

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.

Integrable hierarchies of nonlinear difference-difference equations and symmetries

We construct the hierarchy of nonlinear difference-difference equations associated with the discrete Schrodinger spectral problem. As examples of equations contained in this hierarchy we obtain the discrete-time Toda and Volterra lattice equations. In the case of the time-discrete Toda lattice, we construct its Lie point and generalized symmetries. Finally, we present its Backlund transformations and relate it to the already constructed symmetries.

Multidimensional localized solitons

Abstract Recently it has been discovered that some nonlinear evolution equations in 2 + 1 dimensions, which are integrable by the use of the Spectral Transform, admit localized (in the space) soliton solutions. This article briefly reviews some of the main results obtained in the last 5 years thanks to the renewed interest in soliton theory due to this discovery. The theoretical tools needed to understand the unexpected richness of behaviour of multidimensional localized solitons during their mutual scattering are furnished. Analogies and especially discrepancies with the unidimensional case are stressed.

Dynamics of multidimensional solitons

Abstract The N 2 -soliton solution of the Davey-Stewartson equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The solution can simulate quantum effects as ineleastic scattering, fusion and fission, creation and annihilation.

The scaling reduction of the three-wave resonant system and the Painlevé VI equation

Abstract The scaling invariant solutions of the three-wave resonant system in one spatial and one temporal dimension satisfy a system of three first-order nonlinear ordinary differential equations. These equations can be reduced to one second-order equation quadratic in the second derivative. This equation is outside the class of equations classified by Painleve and his school. However, it is a special case of an equation recently found to be related via a one-to-one transformation to the Painleve VI equation.