Dynamics of solitons in nearly integrable systems

Physica D: Nonlinear Phenomena, 1985

In the nonlinear Schrodinger equation with the dissipative perturbation the collision of a Cast soliton with a breather. i.e. a bound state of two solutions which is stable in the presence of the perturbation, is considered. It is demonstrated that the collision results in breaking the breather into two solitons. The relative velocity of the "splinters" and the change of the Cast soliton parameters are calculated. The same inelastic process is considered under the action of the conservative perturbation of a special Corm. The inelastic collision of two breathers is investigated too.

Bharathidasan University, India: Ph.D. Thesis, 2015

Solitons are very important nonlinear entities in the recent years which find multifaceted applications in different branches of science, engineering, and technology, due to their ability to propagate over extraordinary distances without any loss of energy and remarkable stability under collisions. These solitons give rise to several interesting features in the associated nonlinear dynamical systems. Especially, the multicomponent solitons show distinct propagation characteristics and posses fascinating energy sharing collisions which are not possible in scalar (single component) solitons. In this thesis, we consider a set of multicomponent nonlinear dynamical systems, such as multicomponent Yajima-Oikawa equations in (1+1)-dimension, long-wave– short-wave resonance interaction equations in (2+1)-dimensions, coherently coupled nonlinear Schro ̈dinger equations in the presence of four-wave mixing nonlinearities (with same type as well as opposite signs for nonlinearities), and three-coupled Gross-Pitaevskii equations with spin-mixing nonlinearities, arising in the context of nonlinear optics and Bose-Einstein condensates. After studying the integrability nature of these equations, we construct explicit multicomponent soliton solutions which supports a variety of profiles like single-hump, double-hump, flat-top, dark (hole), and gray solitons for different choices of parameters. Our studies on bright soliton collisions reveal different types of energy-sharing and energy-switching collisions in addition to their elastic collisions. Studies on special solutions like bound states and resonant solitons show periodic oscillations which can be controlled by altering the polarization parameters. But, the dark solitons admit only elastic collisions. Our results can find interesting applications in different context of nonlinear science.

Lettere Al Nuovo Cimento Series 2, 1980

In this thesis we investigate solitons in nonintegrable nonlinear evolution equations, with particular reference to their radiation. First we study the effects of damping and driving on the kink of φ 4 theory. Our investigation includes the effects of radiation for the first time, and as a result we discover a new feature, namely a resonance between the driving and the wobbling mode at its natural oscillating frequency. We show that this resonance is unique in that the driving transfers energy to the translational mode of the kink as well as to its oscillatory mode. We also show that the wobbling kink does not behave chaotically for small oscillating amplitudes. Continuing the theme of radiation from solitary waves, we study kinks in discrete φ 4 models with the exceptional property that stationary kinks may be centred at arbitrary points between the lattice sites. We prove that, while this does lead to a drastic increase in mobility of the kinks, they still lose energy to radiation waves for most nonzero velocities; however, we discover isolated values of velocity for which radiationless kink propagation becomes possible. We provide a unification of the known exceptional models by showing that their equations of motion are composed from an underlying one-dimensional map. A simple algorithm based on this observation generates three new families of exceptional discretisations. I would like first and foremost to thank my supervisor, Prof. Igor Barashenkov for being such an inspiration to me. None of this would have been possible without him. I am indebted to Prof. Dmitry Pelinovsky for useful comments on my work and for updating me on the latest research in the field of nonlinear travelling waves. Also, most of the results described in this thesis came about through collaborations with Profs Barashenkov and Pelinovsky, from which I have benefitted enormously. In addition I would like to thank Dr Nora Alexeeva for assistance with references and Prof. Alexander Tovbis for helpful discussions. I am grateful for the generous funding provided by the National Research Foundation and the University of Cape Town over the last four years to make this work possible. Opinions expressed and conclusions arrived at are my own and are not necessarily to be attributed to the NRF. Much appreciation goes to Igor and Nora for repeatedly writing reference letters for my funding applications, and to the staff of the Postgraduate Funding Office for their friendly and efficient help. Last but not least I would like to thank my family and friends for keeping me sane and happy. Special thanks go to Amanda for the holiday of a lifetime, Peter for making sure my weekends were not squandered on work, Chris for long and entertaining discussions about sport, my dad for his guiding hand in my life, and my mum for all the lovely food that over the course of the last 3 1 2 years has gone into producing this stack of pages! I'd like to say an extra-special thankyou to my 'little' brother Sven for being my companion down the years and a willing victim for technical and mathematical discussions. Finally, a grudging mention must go to Harry the cat for pinning me to my chair during the latter phases of writing this thesis.

Chaos (Woodbury, N.Y.), 2005

We introduce the concept of this special focus issue on solitons in nonintegrable systems. A brief overview of some recent developments is provided, and the various contributions are described. The topics covered in this focus issue are the modulation of solitons, bores, and shocks, the dynamical evolution of solitary waves, and existence and stability of solitary waves and embedded solitons.

The European Physical Journal Plus, 2021

Nonlinear waves have long been at the research focus of both physicists and mathematicians, in diverse settings ranging from electromagnetic waves in nonlinear optics to matter waves in Bose-Einstein condensates, from Langmuir waves in plasma to internal and rogue waves in hydrodynamics. The study of physical phenomena by means of mathematical models often leads to nonlinear evolution equations known as integrable systems. One of the distinguished features of integrable systems is that they admit soliton solutions, i.e., stable, localized traveling waves which preserve their shape and velocity in the interaction. Other fundamental properties of integrable systems are their universal nature, and the fact that they can be effectively linearized, e.g., via the Inverse Scattering Transform (IST), or reduced to appropriate Riemann-Hilbert problems. Moreover, explicit solutions can often be derived by the Zakharov-Shabat dressing method, by Bäcklund or Darboux transformations, or by Hirota's bilinear method. Prototypical examples of such integrable equations in 1+1 dimensions are the nonlinear Schrödinger (NLS) equation and its multicomponent generalizations, the sine-Gordon equation, the Korteweg-de Vries (KdV) and the modified KdV equations, the Kulish-Sklyanin model, etc. The most notable examples of integrable systems in 2+1-dimensions are the Kadomtsev-Petviashvili (KP) equations, and the Davey-Stewartson equations. The aim of this special issue is to present the latest developments in the theory of nonlinear waves and integrable systems, and some of their applications. Below, we briefly outline the contributions to the present Focus Point (FP) summarizing their achievements in nonlinear wave phenomena and integrable systems. Some open problems and questions are also identified.

Progress of Theoretical Physics, 1987

Journal of Physics A: Mathematical and Theoretical

We present a general scheme for constructing robust excitations (solitonlike) in non-integrable multicomponent systems. By robust, we mean localised excitations that propagate with almost constant velocity and which interact cleanly with little to no radiation. We achieve this via a reduction of these complex systems to more familiar effective chiral field-theories using perturbation techniques and the Fredholm alternative. As a specific platform, we consider the generalised multicomponent Nonlinear Schrödinger Equations (MNLS) with arbitrary interaction coefficients. This non-integrable system reduces to uncoupled Korteweg-de Vries (KdV) equations, one for each sound speed of the system. This reduction then enables us to exploit the multi-soliton solutions of the KdV equation which in turn leads to the construction of soliton-like profiles for the original non-integrable system. We demonstrate that this powerful technique leads to the coherent evolution of excitations with minimal radiative loss in arbitrary non-integrable systems. These constructed coherent objects for non-integrable systems bear remarkably close resemblance to true solitons of integrable models. Although we use the ubiquitous MNLS system as a platform, our findings are a major step forward towards constructing excitations in generic continuum non-integrable systems.

Physica D: Nonlinear Phenomena, 1998

Advances in nonlinear science have been plentiful in recent years. In particular, interest in nonlinear wave propagation continues to grow, stimulated by new applications, such as ber optic communication systems, as well as the many classical unresolved issues of uid dynamics. What is arguably the turning point for the modern perspective of nonlinear systems took place at Los Alamos over forty years ago with the pioneering numerical simulations of Fermi, Pasta, and Ulam. A decade later, this research initiated the next major advance of Zabusky and Kruskal that motivaded the revolution in completely integrable systems. With this in mind, the conference on Nonlinear Waves in Solitons in Physical Systems (NWSPS) was organized by the Center for Nonlinear Studies (CNLS) at Los Alamos National Laboratory in May of 1997, to assess the current state-of-the-art in this very active eld. Papers from the conference attendees as well as from researchers unable to attend the conference were collected in this special volume of Physica D. In this paper, the contributions to the conference and to this special issue are reviewed, with an emphasis on the many unifying principles that all these works share. be broken into pieces, which are then solved independently by, for example, the Fourier or Laplace transform, and then added back to form a solution to the original problem.

2011

The interactions of N solitons of the vector NLS eq. in adiabatic approximation is modeled by a generalized complex Toda chain (GCTC). A comparative analysis with the scalar N -soliton interactions is given. Additional constraints that have to be imposed on the polarization vectors which make the adiabatic approximation applicable are formulated. The effects of two types of external potentials are discussed.