Helmholtz algebraic solitons

We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons

Soliton solution and bifurcation analysis of the KP–Benjamin–Bona–Mahoney equation with power law nonlinearity

This paper studies the Kadomtsev–Petviashvili–Benjamin–Bona–Mahoney equation with power law nonlinearity. The traveling wave solution reveals a non-topological soliton solution with a couple of constraint conditions. Subsequently, the dynamical system approach and the bifurcation analysis also reveals other types of solutions with their corresponding restrictions in place

Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies

We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on.Comment: 22 pages, LaTeX, v2: typos corrected, references added, version to appear in JM

Bicomplexes and Integrable Models

We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are presented, including the nonlinear Schrodinger and sine-Gordon equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st

Soliton dynamics in nonlinear planar systems

The work in this thesis is concerned with the study of stability and scattering of solitons in planar models ie where spacetime is (2+l)-dimensional. We consider both integrable models, where exact solutions can be written in closed form, and non-integrable models, where approximations and numerical methods must be employed. In chapter III we use a 'collective coordinate' approximation to study the scattering of solitons in a model motivated by elementary particle physics. In chapter IV we discuss a method to obtain approximate soliton configurations which can then be used to investigate soliton dynamics. In chapter V we perform a test of the 'collective coordinate' approximation by applying it to the study of classical and quantum soliton scattering in an integrable model, where exact results are known. Chapters VI and VII are concerned with an integrable chiral model. First we construct exact solutions using twistor methods and then we go on to study soliton stability using numerical techniques. Through computer simulations we find that there exist solitons which scatter in a way unlike any previously found in integrable models. Furthermore, this soliton scattering resembles very closely that found in certain non-integrable models, thereby providing a link between the two classes. Finally, chapter VIII is an outlook on current and future research