Terms Of Solitons Research Articles

The symmetries of solitons

In this article we will retrace one of the great mathematical adventures of this century—the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment carried out by Fermi, Pasta, and Ulam on one of the first electronic computers, and with Zabusky and Kruskal’s insightful explanation of the surprising results of that experiment (and of a follow-up experiment of their own) in terms of a new concept they called “solitons”. Solitons however raised even more questions than they answered. In particular, the evolution equations that govern solitons were found to be Hamiltonian and have infinitely many conserved quantities, pointing to the existence of many non-obvious symmetries. We will cover next the elegant approach to solitons in terms of the Inverse Scattering Transform and Lax Pairs, and finally explain how those ideas led step-by-step to the discovery that Loop Groups, acting by “Dressing Transformations”, give a conceptually satisfying explanation of the secret soliton symmetries.

Surfaces of revolution in terms of solitons

The global Weierstrass representation and its spectral properties are studied in detail for surfaces of revolution. It is also shown that the modified Korteweg–de Vries equations generate global integrable deformations of tori of revolution. The geometric meanings of first integrals of these evolutions are discussed.

The Lax pair by dimensional reduction of Chern–Simons gauge theory

We show that the Nonlinear Schrödinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1-dimensional Chern–Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebra structure and the Inverse Spectral Transform, has to appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coordinate. In terms of solitons, a natural interpretation for the one-dimensional analog of Chern–Simons Gauss law is given.

An attempt at a soliton approach to plastic flow of crystals

In order to eliminate the discrepancy between theoretical and experimental data on the low-temperature dependence of the strain rate sensitivity in copper single crystals, a rough quantum-mechanical approach is proposed. Its general idea consists in that the dislocation kinks are discussed in terms of solitons and the steady state of plastic flow is considered as a soliton gas governed by Bose-Enstein statistics. Particularly, in order to apply this approach to the plastic flow of fcc crystals both a very simplified soliton mechanism for partial dislocation construction and a very simplified quantum-mechanical approach to the formation of the jog on the gliding dislocation are proposed. The calculations results obtained from the simple model are in quite good agreement with those obtained experimentally by Basinski.

Shot noise in some morpholinium (TCNQ)2salts: an explanation in terms of soliton transport

Shot noise was observed in the electric transport in uniform MEM(TCNQ)2, MEtM(TCNQ)2 and MBtM(TCNQ)2. The shot noise occurred in the ohmic regime. The mean distances that the charge carriers travel between trapping events are larger than the lengths of the samples ( approximately 3 mm). An explanation in terms of solitons with charges +or-1/2e is proposed. Although the soliton picture is, at this stage, rather speculative, it appears to be the only charge-transport mechanism currently to be found in the literature that is compatible with the experimental results. In MEtM(TCNQ)2 and MBtM(TCNQ)2 two conduction mechanisms were found, but the precise nature of these mechanisms and their interactions could not be revealed.

Fermion-induced spin of solitons: Vacuum and collective aspects

We study quantization of collective rotational motions of a soliton in the presence of vacuum fluctuations of fermions coupled to the soliton's background field. Under certain consistency conditions, we express the canonically conjugate momentum (“spin”) of the soliton in terms of spectral properties of the Dirac hamiltonian describing the fermion-soliton interaction. In particular, if fermions belong to a double-valued representation of the considered rotation group, a soliton of a unit fermion number has a half-odd-integer spin.

Soliton quantization in lattice field theories

Quantization of solitons in terms of Euclidean region functional integrals is developed, and Osterwalder-Schrader reconstruction is extended to theories with topological solitons. The quantization method is applied to several lattice field theories with solitons, and the particle structure in the soliton sectors of such theories is analyzed. A construction of magnetic monopoles in the four-dimensional, compactU(1)-model, in the QED phase, is indicated as well.

Dynamics of the order parameter of superfluid phases of helium-3

The spin dynamics and the orbital dynamics in the A and B phases of superfluid 3He are analyzed. Attention is focused on solitons and instantons: nonsingular configurations of the field of the order parameter of the 3He superfluid phases. A qualitative explanation in terms of solitons and instantons is offered for several experiments involving the A and B phases of 3He. The analysis of these questions is prefaced by some general information on the order parameter and the free energy in superfluid 3He.

Solitons and the electrical and mobility properties of dislocations in silicon

The experimental evidence for an explanation of the electrical and mobility properties of dislocations in silicon in terms of solitons (i.e. antiphase defocts) associated with strongly reconstructed partials is presented. This paper uses the results of calculations of the electronic properties of two types of defect (the soliton and its vacancy complex) given in an accompanying paper. We suggest that the absence of spin centres in high-temperature-deformed silicon can be interpreted through the soliton being a negative-U Anderson centre. We support this hypothesis with theoretical arguments. These solitons form obvious nucleation points for double kinks and a theory of dislocation velocity is given, starting from this premiss. The expression for the dislocation velocity is formally identical to that of Hirth and Lothe, its doping dependence is the same as that of Hirsch’s theory and the calculated effective donor and accoptor levels are in good agreement with observation. © 1983 Taylor & Francis Group, LLC.

Dynamics of the Toda lattice: A soliton-phonon phase-shift analysis

The dynamics of the Toda lattice can be described in terms of solitons and nonlinear phonons (ripples). The latter are shown to be obtainable from a general multisoliton solution by allowing the soliton parameters to become imaginary. This device yields phonon phase shifts due to a soliton via a linearization procedure and shows that the zone-edge phonon (standing wave) is removed in the presence of a soliton. An expansion of the soliton-ripple solution up to the second order in the phonon amplitude allows the calculation of typical nonlinear quantities such as momentum and mass of the ripple and the spatial shift of a soliton due to its collision with a ripple. Finally, the relationship is established between the form of a ripple, viewed as a phonon wave packet, and its associated action variable within the framework of inverse scattering theory.

Are Solitons Already Seen in Heavy-Ion Reactions?

An explanation in terms of solitons is given for the recent observation by Galin et al. that the linear momentum transfer in C12-induced heavy-ion reactions is limited.Received 30 August 1982DOI:https://doi.org/10.1103/PhysRevLett.50.407©1983 American Physical Society

Solitons in ferromagnetic spin-1 chains?

We report finite-ring calculations revealing that spin-1 ferromagnetic easy-plane chains in a symmetry-breaking field are well described by a weakly interacting Bose gas. The thermally-induced central peak (CP) previously observed by neutron scattering from CsNiF3, and interpreted in terms of solitons, is traced back to annihilation/creation processes of the excitations. Its temperature variation shows a pronounced interaction-mediated saturation. The essential features of the spin dynamics and of the field and temperature dependence of the specific heat can be explained without solitons. We predict corresponding CP phenomena in a large variety of spin systems.

An investigation of internal solitary waves in a two-fluid system

The results of an experimental investigation dealing with finite-amplitude internal solitary waves in a two-fluid system are presented. Particular attention is paid to characterizing solitons in terms of their shape and amplitude-wavelength scale relationship. Two cases are considered, viz., a shallow- and a deep-water configuration, in order to study the depth effect upon the propagational characteristics of the waves. Comparisons are made between the experimental results and existing internal-wave theories. In addition, discussion is presented describing how these existing theories may be extended to include higher-order nonlinear and viscous effects.

Soliton-like states for opening in polynucleotide double helices

Abstract The polar hydrogens of the nucleobases in DNA and synthetic duplexes exchange with solvent hydrogens under conditions in which these molecules are ordered and remote from any denaturation transitions. This behavior of both natural and synthetic polynucleotide duplexes has led to the proposal that ordered double helices contain small amounts of open states, in which bases are unpaired, and that these open states mediate the exchange of otherwise inaccessible hydrogen-bonded protons. In this paper we conclude from analysis of the experimental data that these open configurations can consist of mobile segments on the order of ten base pairs in length. We suggest that these extended open states may be described in terms of solitons, which are now being recognized as important conformational excitations in polymers and other chain compounds.

An analysis of the dielectric α-relaxation of crystaline polyethene in terms of solitons

A previously proposed crystal defect providing a mechanism for the dielectric α-relaxation of polyethylene can be interpreted as a soliton. The soliton dynamics are consistent with an earlier observation that absolute rate theory provides excellent agreement with the experimental loss frequencies

Solutions of classical gauge field theories with spin and internal symmetry

We examine finite-dimensional matrix solutions to the equations of classical gauge field theories and interpret them in terms of solitons with spin and internal symmetry. The equations are reduced to relations among scalar quantities which turn out to be the same as those obtained for spinless solitons with no internal degrees of freedom, both for spherically symmetric and cylindrically symmetric situations. General formulas are derived for the invariant magnetic field associated with the theory. In the spherical case with degrees of freedom corresponding to spin 1 2 and isospin 1 2 we find that the solitons are monopoles with magnetic charge satisfying the quantization condition eg = 1 2 . Finally, we discuss the difficulties associated with our interpretation.