Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.     

应变对二维蜂巢晶格光子晶体能带结构的影响

基于平面波法,本论文对应变引起的二维蜂巢晶格光子晶体的能带结构进行了数值计算。选取的两个方向分别是锯齿型边界(zigzag)方向和扶手椅型边界(armchair)方向,在这两个典型方向上对二维蜂巢晶格进行了正负各20%的单轴应变。由于应变导致的对称性破缺,能带结构会有显著的变化。在沿锯齿型边界方向上,TE模带隙随着晶格被拉伸逐渐减小,TM模带隙在应变量大于16%时消失。对于沿扶手椅型边界方向,TE模带隙在压缩15%以上时逐渐减小,在其他应变量的情况下几乎保持不变;TM模带隙在应变量大于18%时消失。这些结果对于完善应力工程和设计二维光子晶体器件有重要的指导意义。     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

Gap generation for Dirac fermions on Lobachevsky plane in a magnetic field

We study symmetry breaking and gap generation for fermions in the 2D space of constant negative curvature (the Lobachevsky plane) in an external covariantly constant magnetic field in a four-fermion model. It is shown that due to the magnetic and negative curvature catalyses phenomena the critical coupling constant is zero and there is a symmetry breaking condensate in the chiral limit even in free theory. We analyze solutions of the gap equation in the cases of zero, weak, and strong magnetic fields. As a byproduct, we calculate the density of states and the Hall conductivity for noninteracting fermions that may be relevant for studies of graphene.     

Soliton models for the nucleon and predictions for the nucleon spin structure

In these lectures the three flavor soliton approach for baryons is reviewed. Effects of flavor symmetry breaking in the baryon wave functions on axial current matrix elements are discussed. A bosonized chiral quark model is considered to outline the computation of spin dependent nucleon structure functions in the soliton picture.     

Chiral symmetry breaking at finite temperature in Coulomb gauge QCD

Chiral symmetry breaking at finite temperature is analysed in Coulomb gauge QCD, using a suitably renormalised gap equation. In Coulomb gauge the gap equation is derived using the Ward identities and the Dyson equations for the vector and axial-vector vertices. Making the ladder approximation to the Bethe-Salpeter kernel relates the chiral symmetry breaking parameters to the static quark potential. It is thus possible to use a confining potential in the analysis of chiral symmetry breaking. We extend this to finite temperature. For a confining potential there is no chiral symmetry restoration at any finite temperature.     

The explicit symmetry breaking solutions of the Kadomtsev-Petviashvili equation

To describe two correlated events, the Alice–Bob (AB) systems were constructed by Lou through the symmetry of the shifted parity, time reversal and charge conjugation. In this paper, the coupled AB system of the Kadomtsev–Petviashvili equation, which is a useful model in natural science, is established. By introducing an extended Bäcklund transformation and its bilinear formation, the symmetry breaking soliton, lump and breather solutions of this system are derived with the aid of some ansatze functions. Figures show these fascinating symmetry breaking structures of the explicit solutions.     

Topological Fermi liquids from Coulomb interactions in the doped honeycomb lattice

We propose a simple method for obtaining time reversal symmetry (T) broken phases in simple lattice models based on enlarging the unit cell. As an example we study the honeycomb lattice with nearest neighbor hopping and a local nearest neighbor Coulomb interaction V. We show that when the unit cell is enlarged to host six atoms that permits Kekulé distortions, self-consistent currents spontaneously form creating nontrivial magnetic configurations with total zero flux at high electron densities. A very rich phase diagram is obtained within a variational mean field approach that includes metallic phases with broken time reversal symmetry (T). The predominant (T) breaking configuration is an anomalous Hall phase, a realization of a topological Fermi liquid.     

SU(3) symmetry breaking for masses, magnetic moments and sizes of baryons

We use the SU(3) Skyrme model to investigate the effects of symmetry breaking imposed by different pseudoscalar meson masses on the structure of baryons. Specifically, we calculate their mass splittings, magnetic moments, charge radii, semileptonic decays as well as different strangeness contents of the proton. Since the Skyrme soliton is allowed to deform under the pressure of the symmetry breaking we find significant variations in baryon sizes with respect to strangeness.     

Symmetry breaking and enhanced condensate fraction in a matter-wave bright soliton

An exact diagonalization study reveals that a matter-wave bright soliton and the Goldstone mode are simultaneously created in a quasi-one-dimensional attractive Bose-Einstein condensate by superpositions of quasidegenerate low-lying many-body states. Upon formation of the soliton the maximum eigenvalue of the single-particle density matrix increases dramatically, indicating that a fragmented condensate converts into a single condensate as a consequence of the breaking of translation symmetry.     

THEORY OF SOLITON GENERATION AND LATTICE RELAXATION IN POLYACETYLENE (Ⅰ) GENERAL FORMALISM

This series of papers is devoted to the theory of soliton and polaron generation in polyacetylene by both radiative and non-radiative processes. In the first paper of this series a general formalism is presented which combines the Lattice relaxation approach for the multiphonon process developed by Huang and many others with the consistency condition in terms of the Bogoliubov-de Gennes equation and the gap equation. The exact adiabatic electron wave functions consistent with the different lattice syimnetry breaking in the initial and final states are used to calculate the transition probability between these multi-electron states. The formalism presented is illustrated by deriving a general expression for the non-radiative decay rate using the steepest descent method in the case of stronc coupling. The theory developed here canbe applied to treating the quantum. transitions between states of differnet symmetry breaking when the multi-electron background effect and the consistency condition are essential.The applications of the general formalism to polyacettyene and the comparison with numerical results and the recent experiments tvi11 be given in the succeeding papers.     

Origin of Symmetry Breaking and Confinement in Conducting Polymers with Ring Structures

A model to describe the main features of conjugated polymers with ring structures, such as polythiophene and polypyrrole, is constructed. It is shown that the origin of the symmetry breaking and confinement of a soliton and anti-soliton pair is branch hopping in the polymer rings.     

Lie Symmetry and Nonlinear Instability in Computation of KdV Solitons

The soliton calculation method put forward by Zabusky and Kruskal has played an important role in the development of soliton theory, however numerous numerical results show that even though the parameters satisfy the linear stability condition, nonlinear instability will also occur. We notice an exception in the numerical calculation of soliton, gain the linear stability condition of the second order Leap-frog scheme constructed by Zabusky and Kruskal, and then draw the perturbed equation with the finite difference method. Also, we solve the symmetry group of the KdV equation with the knowledge of the invariance of Lie symmetry group and then discuss whether the perturbed equation and the conservation law keep the corresponding symmetry. The conservation law of KdV equation satisfies the scaling transformation, while the perturbed equation does not satisfy the Galilean invariance condition and the scaling invariance condition. It is demonstrated that the numerical simulation destroy some physical characteristics of the original KdV equation. The nonlinear instability in the calculation of solitons is related to the breaking of symmetry.     

Origin of Symmetry Breaking and Confinement in Conducting Polymers with Ring Structures

A model to describe the main features of conjugated polymers with ring structures, such as polythiophene and polypyrrole, is constructed. It is shown that the origin of the symmetry breaking and confinement of a soliton and anti-soliton pair is branch hopping in the polymer rings.     

Soliton ratchets in homogeneous nonlinear Klein-Gordon systems

We study in detail the ratchetlike dynamics of topological solitons in homogeneous nonlinear Klein-Gordon systems driven by a biharmonic force. By using a collective coordinate approach with two degrees of freedom, namely the center of the soliton, X(t), and its width, l(t), we show, first, that energy is inhomogeneously pumped into the system, generating as result a directed motion; and, second, that the breaking of the time shift symmetry gives rise to a resonance mechanism that takes place whenever the width l(t) oscillates with at least one frequency of the external ac force. In addition, we show that for the appearance of soliton ratchets, it is also necessary to break the time-reversal symmetry. We analyze in detail the effects of dissipation in the system, calculating the average velocity of the soliton as a function of the ac force and the damping. We find current reversal phenomena depending on the parameter choice and discuss the important role played by the phases of the ac force. Our analytical calculations are confirmed by numerical simulations of the full partial differential equations of the sine-Gordon and phi4 systems, which are seen to exhibit the same qualitative behavior. Our results show features similar to those obtained in recent experimental work on dissipation induced symmetry breaking.     

Dynamics of dark-bright solitons in cigar-shaped Bose-Einstein condensates

We explore the stability and dynamics of dark-bright (DB) solitons in two-component elongated Bose-Einstein condensates by developing effective one-dimensional vector equations and solving the three-dimensional Gross-Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the DB soliton on the atom number of its components is found; importantly, the wave may become dynamically unstable even in the 1D regime. As the atom number in the dark-soliton-supporting component is further increased, spontaneous symmetry breaking leads to oscillatory dynamics in the transverse degrees of freedom. Moreover, the interactions of two DB solitons are investigated with an emphasis on the importance of their relative phases. Experimental results showcasing multiple DB soliton oscillations and a DB-DB collision in a Bose-Einstein condensate consisting of two hyperfine states of ⁸⁷Rb confined in an elongated optical dipole trap are presented.     

Lie Symmetries,Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik – Novikov – Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation, the (2+1)-dimensional generalization of the nonlinear Schrödinger equation by Fokas as well as the (2+1)-dimensional generalized sine-Gordon equation of Konopelchenko and Rogers. We show that in all these cases the Lie symmetry algebra is infinite-dimensional; however, in the case of the breaking soliton equation they do not possess a centerless Virasorotype subalgebra as in the case of other typical integrable (2+1)-dimensional evolution equations. We work out the similarity variables and special similarity reductions and investigate them.     

Hyperon beta-decay and axial charges of the lambda in view of strongly distorted baryon wave-functions

Within the collective coordinate approach to chiral soliton models we suggest that breaking of SU(3) flavor symmetry mainly resides in the baryon wave-functions while the charge operators maintain a symmetric structure. Sizable symmetry breaking in the wave-functions is required to reproduce the observed spacing in the spectrum of the  baryons. The matrix elements of the flavor symmetric charge operators nevertheless yield g_A/g_V ratios for hyperon beta-decay which agree with the empirical data approximately as well as the successful F&D parameterization of the Cabibbo scheme. Demanding the strangeness component in the nucleon to vanish in the two flavor limit of the model, determines the structure of the singlet axial charge operator and yields the various quark flavor components of the axial charge of the Λ-hyperon. The suggested picture gains support from calculations in a realistic model using pion and vector meson degrees of freedom to build up the soliton.     

Highly asymmetric soliton complexes in parabolic optical lattices

We introduce multipole soliton complexes in optical lattices induced by nondiffracting parabolic beams. Despite the symmetry breaking dictated by the curvature of the lattice channels, we find that complex, asymmetric higher-order states can be stable. The unique topology of parabolic lattices affords new types of soliton motion: single solitons launched into the lattice with nonzero transverse momentum perform periodic oscillations along parabolic paths.     

Weak magnetism in chiral quark models

We discuss symmetry breaking in the weak magnetism form factors for the semileptonic octet baryon decays. In the chiral quark model, the symmetry breaking can be accounted for in the masses and the quark spin polarizations can take on more general values due to Goldstone boson depolarization. Here we clarify some features of the chiral quark model prediction for the weak magnetism and compare to the corresponding result of the chiral quark soliton model. Received: 29 June 1999 / Revised version: 15 September 1999 / Published online: 8 December 1999