Symmetric Soliton Solutions Research Articles

Asymptotic Stability in Geometric σ–models

Geometric σ–models have been defined as purely geometric theories of scalar fields in interaction with gravity. By construction, these theories possess soliton solutions with topologically nontrivial scalar sectors. We perform a detailed analysis of the stability of the effective scalar field theory far from the soliton core. It is shown that the requirement for the asymptotic stability is consistent with the existence of massive, static, spherically symmetric soliton solutions.

ON THE GAUGED NONCOMPACT SPIN SYSTEM

We examine classical and quantum aspects of the planar noncompact spin system coupled with Chern–Simons gauge field in the presence of background charge. We first define our classical spin system as nonrelativistic nonlinear sigma model in which the order parameter spin takes value in the noncompact manifold ℳ= SU(1, 1)/U(1) . Although the naive model does not allow any finite energy self-dual solitons, it is shown that the gauged system admits static Bogomol'nyi solitons with finite energy whose rotationally symmetric soliton solutions are analyzed in detail. We also discuss the large spin limit in which the self-dual equation reduces to the well-known gauged nonlinear Schrödinger model or Abelian Higgs model, depending on the choice of the background charge term. Then, we perform quantization of the model. We find that the spin algebra satisfies anomalous commutation relations, and the system is a field theoretic realization of the anyons.

Breakup of two-dimensional into three-dimensional Kadomtsev-Petviashvili solitons

This paper reports on three-dimensional simulations that follow exact, $z$-symmetric soliton solutions to an important model equation of plasma physics and superfluid helium (Bose condensate). This is the Kadomtsev-Petviashvili equation. Solitons are seen to break up when perturbed along $z$. Dependence of growth on the wave number of the perpendicular perturbation is found numerically. This leads to a wave number producing the maximum rate of breakup. Due to numerical instabilities, a somewhat smaller wave number must be used. Fully three-dimensional entities are produced. After a while they become virtually identical to known, azimuthally symmetric solutions. Based on this, implications for the reconnection hypothesis formulated by Feynman, used in superfluid helium II theory, are indicated.

Two‐dimensional Models for Nonlinear Vorticity Waves in Shear Flows

The evolution of weakly nonlinear wave perturbations in shear flows of stratified fluid is investigated for large Reynolds numbers. The study is focused on the vorticity waves, i.e., the wave‐like motions caused by the mean flow vorticity gradient. A situation typical of the upper ocean is considered.The shear flow is supposed to be localized near the surface and to have no inflection points. The vertical scale of stratification is much larger than that of the shear current. Description of the dynamics of essentially three‐dimensional wave perturbations is reduced by a systematic asymptotic procedure to a single nonlinear evolution integrodifferential equation for (2+1)‐dimensions. The small parameters are the ratio of the vertical scale of the shear to the typical wavelength of the perturbations and the amplitude parameter. The equation does not contain viscous terms, but the regime of evolution it describes occurs owing to small but finite viscosity. The viscosity inhibits generation of strongly nonlinear vortices in the critical layer. Possible existence of localized two‐dimensional stationary solutions of the equation is investigated. Axially symmetric soliton solutions are found for a fluid of arbitrary depth in the limit of vanishing stratification. In stratified flows a linear resonant interaction between shear flow perturbations and internal waves is found to play the major role. The radiation damping of vorticity waves due to these resonances makes the existence of similar lump solitary structures in stratified fluid impossible.

Self-Dual Chern–Simons Solitons in the Planar CP(1) Model

We consider a nonrelativistic CP (1) system coupled minimally to an Abelian Chern–Simons gauge field and study the self-dual solitons which saturate the Bogomol'nyi bound. We find a rich structure of rotationally symmetric static soliton solutions for various uniform background charge densities. Possible application to spin system is mentioned.

Simulations of Two-Dimensional Kadomtsev-Petviashvili Soliton Dynamics in Three-Dimensional Space.

This Letter reports on three-dimensional simulations that follow exact, {ital z} symmetric soliton solutions to an important equation of plasma physics and fluid dynamics. These solitons are seen to break up when perturbed slightly along {ital z}. Some fairly robust, three-dimensional entities are subsequently produced. However, they do not seem to resemble known, azimuthally symmetric solutions. An explanation of this interesting fact is offered. {copyright} {ital 1996 The American Physical Society.}

Pulsations of spherically symmetric systems in general relativity.

The pulsation equations for spherically symmetric black hole and soliton solutions are brought into a standard form. The formulas apply to a large class of field theoretical matter models and can easily be worked out for specific examples. The close relation to the energy principle in terms of the second variation of the Schwarzschild mass is also established. The use of the general expressions is illustrated for the Einstein-Yang-Mills and the Einstein-Skyrme system. \textcopyright{} 1996 The American Physical Society.

Symmetric and antisymmetric solutions for nonlinear wave modulations in plasmas

The existence of symmetric and antisymmetric stationary localized solutions for nonlinear amplitude-modulated Langmuir waves and circularly polarized electromagnetic waves in plasmas is considered. It is shown that the antisymmetric Langmuir solitons exist in the entire range of the Mach number, namely, 0<M<1, whereas antisymmetric electromagnetic solitons exist for all values of the Mach number. The parameter regions for the existence of different types of symmetric soliton solutions are obtained explicitly. It is found that both the density-hump and the density-dip electromagnetic solitons exist for subsonic as well as supersonic values of the Mach number.

Soliton solutions for self-dual SU(N) gauge fields on Euclidean space

Axially symmetric soliton solutions for self-dual SU(N) gauge fields on Euclidean four-dimensional flat space are found using an extension of the Belinsky–Zakharov solution generating technique. The new solutions depend at most on N−1 arbitrary solutions of the usual Laplace equations in cylindrical coordinates. The inverse scattering method using the chosen seen solution reduces to the computation of at most N−1 quadratures. The n-soliton solution is written in a determinantal form. Three particular cases of one-soliton solutions for the SU(5) group are exhibited.

Soliton solutions for self-dual SU(2) gauge fields on Euclidean space

Axially symmetric soliton solutions for self-dual SU (2) gauge fields on Euclidean four-dimensional flat space are found using an extension of the Belinsky–Zakharov solution generating technique.

On axially symmetric soliton solutions of the coupled scalar-vector-tensor equations in general relativity

The inverse scattering theory used by Belinsky and Zakharov to obtain soliton solutions of the of the Einstein equations is here applied to the case of a five-dimensional space and interpreted in the framework of the Jordan-Kaluza-Klein theory. For two solitons exact, stationary, axially symmetric and asymptotically flat solutions are obtained.