Soliton Limit Research Articles

Self-localized states for electron transfer in nonlocal continuum deformable media

We consider the problem of electron transport in a deformable continuum medium subjected to an external harmonic substrate potential. We then consider the quasi-stationary state of the full problem to find a Gross–Pitaevskii type equation with a nonlocal external potential, which is solved by variational and numerical means (considered as the exact solution) to find the parameter conditions for the existence of self-localized solutions. The variational approach predicts a threshold on the on-site or nonlocality parameter where localized solutions cease to exist from the Non-Linear Schrödinger soliton limit. A numerical continuation of stationary state solutions in the corresponding discrete system is used to confirm the prediction of the turning value in the on-site term. We finally study the full stationary state and make use of an approximation, proposed by Briedis et al. [17], for the nonlocal term, corresponding to strong nonlocalities, to find analytic expressions for self-localized states in terms of the series solutions of a nonlinear modified Bessel equation.

Generation of stable cnoidal waves in an erbium doped fiber laser system

In this work an all-fiber laser system able to produce stable cnoidal waves in both limits, i.e. in sinusoidal and soliton limits, is presented. The system is constructed using an erbium highly-doped fiber as an active media and an electrooptic modulator as the core of the control tool. This control tool is able to modulate the cavity losses. The direct modulation allows to control the pulses properties, shape, width and intensity. The proposed system is well described by a three level laser model based on the Statz-de Mars rate equations. Numerical and Experimental results are presented.

The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

The nonlinear Dirac equation for Bose–Einstein condensates (BECs) in honeycomb optical lattices gives rise to relativistic multi-component bright and dark soliton solutions. Using the relativistic linear stability equations, the relativistic generalization of the Boguliubov-de Gennes equations, we compute soliton lifetimes against quantum fluctuations and classify the different excitation types. For a BEC of atoms, we find that our soliton solutions are stable on time scales relevant to experiments. Excitations in the bulk region far from the core of a soliton and bound states in the core are classified as either spin waves or as a Nambu–Goldstone mode. Thus, solitons are topologically distinct pseudospin- domain walls between polarized regions of . Numerical analysis in the presence of a harmonic trap potential reveals a discrete spectrum reflecting the number of bright soliton peaks or dark soliton notches in the condensate background. For each quantized mode the chemical potential versus nonlinearity exhibits two distinct power law regimes corresponding to the free-particle (weakly nonlinear) and soliton (strongly nonlinear) limits.

Computational Approach to Hyperelliptic Riemann Surfaces

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the curve. A canonical basis of the homology is introduced algorithmically for this curve. The periods of the holomorphic differentials and the Abel map are computed with the Clenshaw–Curtis method to achieve spectral accuracy. The code can handle almost degenerate Riemann surfaces. This work generalizes previous work on real hyperelliptic surfaces with prescribed cuts to arbitrary hyperelliptic surfaces. As an example, solutions to the sine-Gordon equation in terms of multi-dimensional theta functions are studied, also in the solitonic limit of these solutions.

Soliton turbulence in shallow water ocean surface waves.

We analyze shallow water wind waves in Currituck Sound, North Carolina and experimentally confirm, for the first time, the presence of soliton turbulence in ocean waves. Soliton turbulence is an exotic form of nonlinear wave motion where low frequency energy may also be viewed as a dense soliton gas, described theoretically by the soliton limit of the Korteweg-deVries equation, a completely integrable soliton system: Hence the phrase "soliton turbulence" is synonymous with "integrable soliton turbulence." For periodic-quasiperiodic boundary conditions the ergodic solutions of Korteweg-deVries are exactly solvable by finite gap theory (FGT), the basis of our data analysis. We find that large amplitude measured wave trains near the energetic peak of a storm have low frequency power spectra that behave as ∼ω-1. We use the linear Fourier transform to estimate this power law from the power spectrum and to filter densely packed soliton wave trains from the data. We apply FGT to determine the soliton spectrum and find that the low frequency ∼ω-1 region is soliton dominated. The solitons have random FGT phases, a soliton random phase approximation, which supports our interpretation of the data as soliton turbulence. From the probability density of the solitons we are able to demonstrate that the solitons are dense in time and highly non-Gaussian.

Exact Temporal Evolution of Two-Species Bose—Einstein Condensates

We construct exact stationary solutions to the one-dimensional coupled Gross—Pitaevskii equations for the two-species Bose—Einstein condensates with equal intraspecies and interspecies interaction constants. Three types of complex solutions as well as their soliton limits are derived. By making use of the SU(2) unitary symmetry, we further obtain analytical time-evolving solutions. These solutions exhibit spatiotemporal periodicity.

Skyrmion crystals in pseudo-spin-1/2 Bose—Einstein condensates

Exact two-dimensional solutions are constructed for the pseudo-spin-1/2 Bose—Einstein condensates, which are described by the coupled nonlinear Gross—Pitaevskii equations where the intra- and inter-species coupling constants are assumed to be equal. The equations are decoupled by means of re-combinations of the nonlinear terms of the hyperfine states according to the spatial dimensions. The stationary solutions form various spin textures which are identified as skyrmion crystals. In a special case, a crystal of skyrmion—anti-skyrmion pairs is formed in the soliton limit.

On elliptic solutions of a coupled nonlinear Schrödinger system

An explicit formula is obtained for single-phase bounded elliptic solutions of the Manakov system of integrable coupled nonlinear Schrödinger equations in terms of the Weierstrass sigma function with a real quasiperiod. The parametrization is effective in the sense that the reality conditions are completely characterized for each of the three possible couplings: focusing–focusing, defocusing–defocusing and focusing–defocusing. The Manakov soliton is recovered in the soliton limit and the small-wave-modulation limit is shown to satisfy the linearized dispersion relation of planewave solutions.

Black brane solutions and their solitonic extremal limit in Einstein-scalar gravity

We investigate static, planar, solutions of Einstein-scalar gravity admitting an anti-de Sitter (AdS) vacuum. When the squared mass of the scalar field is positive and the scalar potential can be derived from a superpotential, minimum energy theorems indicate the existence of a scalar soliton. On the other hand, for these models, no-hair theorems forbid the existence of hairy black brane solutions with AdS asymptotics. By considering a specific example (an exact integrable model which has the form of a Toda molecule) and by deriving explicit exact solution, we show that these models allow for hairy black brane solutions with non-AdS domain wall asymptotics, whose extremal limit is a scalar soliton. The soliton smoothly interpolates between a non-AdS domain wall solution at $r=\infty$ and an AdS solution near $r=0$.

The Ricci flow for simply connected nilmanifolds

We prove that the Ricci flow g(t) starting at any metric on R that is invariant by a transitive nilpotent Lie group N can be obtained by solving an ODE for a curve of nilpotent Lie brackets on R. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))| g(t) converges to a Ricci soliton in C∞, uniformly on compact sets in R. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly nonisomorphic to N .

Constant mean curvature surfaces in AdS 3

We construct constant mean curvature surfaces of the general finite-gap type in AdS_3. The special case with zero mean curvature gives minimal surfaces relevant for the study of Wilson loops and gluon scattering amplitudes in N=4 super Yang-Mills. We also analyze properties of the finite-gap solutions including asymptotic behavior and the degenerate (soliton) limit, and discuss possible solutions with null boundaries.

Bright-dark soliton complexes in spinor Bose-Einstein condensates

We present bright-dark vector solitons in quasi-one-dimensional spinor (F=1) Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to a completely integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the $m_{F}=+1,-1,0$ spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasi-elastic collisions. It is also shown that, in the presence of a parabolic trap of strength $\omega $, the bright component(s) is (are) guided by the dark one(s), and, as a result, the small-amplitude vector soliton as a whole performs harmonic oscillations of frequency $\omega/ \sqrt{2}$ in the shallow soliton limit. We investigate numerically deviations from this prediction, as the depth of the solitons is increased, as well as when the strength of the spin-dependent interaction is modified.

Hyperelliptic Theta-Functions and Spectral Methods: KdV and KP Solutions

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristic quantities of the Riemann surface with high precision even in almost degenerate situations as in the solitonic limit where the branch points coincide pairwise. As an example we consider hyperelliptic solutions to the Kadomtsev–Petviashvili and the Korteweg–de Vries equations. Tests of the numerics using identities for periods on the Riemann surface and the differential equations are performed. It is shown that an accuracy of the order of machine precision can be achieved.

Local magnon modes and the dynamics of a small-radius two-dimensional magnetic soliton in an easy-axis ferromagnet

The internal dynamics of a small-radius precession magnetic soliton is considered. A variational formulation of the problem on the soliton-magnon interaction is proposed and used to calculate the frequency of a truly local mode. It is shown that this mode, as well as the conventional translational mode, remains localized in the small soliton radius limit. The presence of the local mode is confirmed by the numerical solution of the scatter- ing problem. © 2005 Pleiades Publishing, Inc.

Instabilities and oscillations of one- and two-dimensional Kadomtsev-Petviashvili waves and solitons II. Linear to nonlinear analysis

We further investigate the dynamics of nonlinear structures that arise from the Kadomtsev–Petviashvili equation. When the analysis is linear, assuming perturbations grow exponentially in time, we find the growth rates of two important instabilities numerically. These are the only purely growing modes. The wavelength–doubling instability is seen to dominate its rival, that of Benjamin and Feir, at least when the amplitude of the wave is not too large. Approximate formulae, found to higher order than in part I (referenced in 1), are checked against the numerically found values. The models are seen to be better than expected. For the dominant wavelength–doubling instability, our model extends beyond the assumed region of validity. It is surprisingly close, almost up to the soliton limit. When we depart from linear stability analysis and include terms nonlinear in the perturbation, a simple analysis shows that the linear instability eventually drives a doubly space–periodic hyperbolic secans pulse in time. After a long time, initial conditions are reproduced. A proof that the maximum amplitude achieved by the perturbation is approximately proportional to the linear growth rate is given within the limitations of the calculation. This fact was suspected from numerics. A second class of possible dynamic behaviour, not arising from initially linear growth of a perturbation, is found. This class involves fully two–dimensional stationary solutions and their possible oscillations.

On the nonlinear growth of two-dimensional Tollmien–Schlichting waves in a flat-plate boundary layer

We study the nonlinear development of two-dimensional Tollmien-Schlichting waves in an incompressible flat-plate boundary layer at asymptotically large values of the Reynolds number. Attention is restricted to the far-downstream lower-branch' regime where a multiple-scales analysis is possible. It is supposed that to leading-order the waves are inviscid and neutral, and governed by the [Davis-Acrivos-]Benjamin-Ono equation. This has a three-parameter family of periodic solutions, the large-amplitude (soliton) limit of which bears a qualitative resemblance to the 'spikes' observed in certain K-type'transition experiments. The variation of the parameters over slow length- and timescales is controlled by a viscous sublayer. For the case of a purely temporal evolution, it is shown that a solution for this sublayer ceases to exist when the amplitude reaches a certain finite value. For a purely spatial evolution, it appears that an initially linear disturbance does not evolve to a fully nonlinear stage of the envisaged form. The implications of these results for the soliton theory of spike formation are discussed

Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

The real, nonsingular elliptic solutions of the Korteweg-de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit.

Periodic processes and dispersion relations for Ablowitz–Ladik equations

There is considered the system of algebraic dispersion equations for the complexified Ablowitz–Ladik discrete equations and relevant reductions to “dark” and “white” versions, in an analogy to the NLS equation. An universal formalism presented here allows to determine effectively parameters of multiphase quasiperiodic solutions, multisoliton ones and of solutions in a form of solitons on a quasiperiodic background. This idea is connected with so-called soliton limit, which in the case of the white type solution, differs essentially from those introduced in the past and must be preceded by some modular transformation. There is a full correspondence with a similar approach for NLS equations. A few subtleties concerning solvability and examples are also discussed.

Quantization ambiguities of the SU(3) soliton

We reconsider canonical quantization of the rotating soliton in the SU(3) chiral quark-soliton model. We show that at the level of 1 N c , in contrast to the SU(2) version of the model, there appear terms which spoil the commutation rules of the flavor generators. Terms of similar origin are also present in the expressions for axial couplings and magnetic moments. We investigate the small soliton limit of the model, and require that the results for the physical observables reduce to the ones of the non-relativistic quark model. This procedure allows us to identify the troublesome terms. Next, we introduce a symmetry conserving approach which consists in subtracting the previously identified terms.

Soliton dynamics and nuclear spin-lattice relaxation in incommensurately modulated crystals

The nuclear spin-lattice relaxation rate in incommensurate systems is analysed for the the so-called soliton limit which often can be applied near the transition to a subsequent commensurate phase. Previous calculations are corrected by taking into account adequately the eigenfunctions of the relevant fluctuations. In contrast to previous conclusions, it is shown that for nuclei in the discommensurations the spin-lattice relaxation rate is expected to increase on approaching the phase transition. The theoretical predictions are confirmed by experimental data obtained from NMR measurements on the prototype incommensurate systems Rb2ZnCl4 and BCCD. b]References