The Nth-order Darboux transformation,vector dark solitons and breathers for the coupled defocusing Hirota system in a birefringent nonlinear fiber

Under investigation in this paper is the coupled defocusing Hirota system, which describes the propagation of ultra-short pulses in a birefringent nonlinear fiber. With respect to the amplitudes of the pulse envelopes, we construct the Nth-order Darboux transformation, which is different from those in the existing literatures, where N is a positive integer. The first- and second-order solutions are obtained via the Darboux transformation. Pulse envelopes including the gray-gray/antidark-gray/dark-bright solitons, vector Akhmediev breathers, temporal cavity solitons and (time, space)-periodic breathers are acquired. As the relative width of the spectrum that arises due to the quasi monochromocity decreases, we find that: the velocity and width of the dark-bright solitons increase; the width of the temporal cavity soliton decreases; the temporal period of the vector (time, space)-periodic breather becomes longer while its spacial period becomes shorter. Relative width of the spectrum that arises due to the quasi monochromocity does not affect the amplitudes of the pulse envelopes obtained. Elastic interactions between the breathers and solitons, and inelastic interactions between the two gray-gray solitons are obtained.     

Quasi-periodic breathers in Hamiltonian networks of long-range coupling

This work is concerned with Hamiltonian networks of weakly and long-range coupled oscillators with either variable or constant on-site frequencies. We derive an infinite dimensional KAM-like theorem by which we establish that, given any N-sites of the lattice, there is a positive measure set of small amplitude, quasi-periodic breathers (solutions of the Hamiltonian network that are quasi-periodic in time and exponentially localized in space) having N-frequencies which are only slightly deformed from the on-site frequencies.     

Linear stability of breathers of the discrete NLS

We consider real breather solutions of the discrete cubic nonlinear Schrödinger equation near the limit of vanishing coupling between the lattice sites and present leading order asymptotics for the eigenvalues of the linearization around the breathers. The expansion is given in fractional powers of the intersite coupling parameter and determines the linear stability of the breathers. The method we use relies on normal form ideas and applies to one and higher-dimensional lattices. We also present some examples.     

Invariant tori in the discrete NLS with small amplitude diffraction management

We consider the question of persistence of breather solutions of the discrete NLS equation under time-periodic perturbations corresponding to small amplitude diffraction management. The question is formulated as a problem of continuation of tori in an infinite-dimensional Hamiltonian system with symmetries and we show that one-peak breathers of the discrete NLS with zero residual diffraction can be continued to periodic or quasiperiodic solutions of the discrete NLS with small residual diffraction and small amplitude diffraction management, provided that a nonresonance condition is satisfied. We also present numerical evidence that a similar continuation should be possible for certain single-, and multi-peak breathers of the discrete NLS with small diffraction.     

Energy transfer in weakly coupled nonlinear oscillator chains: Transition from a wandering breather to nonlinear self-trapping

We present analytical and numerical studies of phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear oscillator chains. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic wandering of the low-amplitude breather between the chains, and the one-chain-localization of the high-amplitude breather. These two modes of coupled breathers can be mapped exactly onto two solutions of a pendulum equation, detached by a separatrix mode. We also show that these two regimes of the coupled breathers are similar, and are described by a similar pair of equations, to the two regimes in the nonlinear tunneling dynamics of two weakly coupled Bose-Einstein condensates. On the basis of this analogy, we predict a new tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around π/2 modulo π.     

Symmetry breaking bifurcations in a circular chain of N coupled logistic maps

A circular chain of N cells with logistic dynamics, coupled together with symmetric nearest neighbor coupling and periodic boundary conditions is investigated. For certain coupling parameters we observe bifurcation of a stable state into two types of period two solutions. By using the symmetry of this Coupled Map Lattice model, we show that the bifurcated system only can have periodic solutions with symmetry group corresponding to certain subgroups of the full symmetry group of the system.     

Strong coupling 1/N c expansion in the gluonic sector of lattice QCD: The next order

Continuing our previous work we determined the gluonic vacuum state up to sixth order and the lowest states with external quark-antiquark and (unscreened) gluon-gluon sources up to fourth order in the strong coupling 1/N _c expansion on the lattice. Unlike previously, we used here the colour electric flux operators on the links to define the colour electric energy. Additional remarks concerning the screening of external gluon sources and the analytic continuation toN _c =3 and zero lattice spacing are also included.     

Strong Coupling Problems of S-Matrix Equations

The analytic continuation of the solutions of nonlinear S-matrix equations for large values of a coupling parameter is investigated.     

Multibreather solitons in the diffraction managed NLS equation

We study analytically and numerically localized breather solutions in the averaged discrete nonlinear Schrödinger equation (NLS) with diffraction management, a system that models coupled waveguide arrays with periodic diffraction management geometries. Localized breathers can be characterized as constrained critical points of the Hamiltonian of the averaged diffraction managed NLS. In addition to local extrema, we find numerically more general solutions that are saddle points of the constrained Hamiltonian. An interesting class of saddle points are “multi-bump” solutions that are close to superpositions of translates of simpler breathers. In the case of zero residual diffraction and small diffraction management, the existence of multibumps can be shown rigorously by a continuation argument.     

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S₃-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z₃-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.     

Modulation of breathers in cigar-shaped Bose-Einstein condensates

We present new solutions to the nonautonomous nonlinear Schrödinger equation that may be realized through convenient manipulation of Bose-Einstein condensates. The procedure is based on the modulation of breathers through an analytical study of the one-dimensional Gross-Pitaevskii equation, which is known to offer a good theoretical model to describe quasi-one-dimensional cigar-shaped condensates. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Numerical simulations confirm stability of the modulated breathers against random perturbation on the input profile of the solutions.     

Laplace transform approach for the dynamics of N qubits coupled to a resonator

An approach to use the method of Laplace transform for the perturbative solution of the Schrödinger equation at any order of the perturbation for a system of N qubits coupled to a cavity with n photons is suggested. We investigate the dynamics of a system of N superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations.     

On a2(1) reflection matrices and affine Toda theories

We construct new non-diagonal solutions to the boundary Yang-Baxter equation corresponding to a two-dimensional field theory with U_q(a₂⁽¹⁾) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a₂⁽¹⁾ affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions.     

Obstructions to the continuation of analytic integrals of Hamiltonian systems under non-Hamiltonian perturbations

In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non-Hamiltonian perturbation controlled by a small parameter ε. An obstruction to the analytic continuation of the integrals of motion of the unperturbed system with respect to ε is developed for sufficiently small perturbations. The theory is applied to a perturbed system of Morse oscillators.     

Bifurcation structures and transient chaos in a four-dimensional Chua model

A four-dimensional four-parameter Chua model with cubic nonlinearity is studied applying numerical continuation and numerical solutions methods. Regarding numerical solution methods, its dynamics is characterized on Lyapunov and isoperiodic diagrams and regarding numerical continuation method, the bifurcation curves are obtained. Combining both methods the bifurcation structures of the model were obtained with the possibility to describe the shrimp-shaped domains and their endoskeletons. We study the effect of a parameter that controls the dimension of the system leading the model to present transient chaos with its corresponding basin of attraction being riddled.     

Dissipative discrete breathers: periodic,quasiperiodic, chaotic,and mobile

The properties of discrete breathers in dissipative one-dimensional lattices of nonlinear oscillators subject to periodic driving forces are reviewed. We focus on oscillobreathers in the Frenkel-Kontorova chain and rotobreathers in a ladder of Josephson junctions. Both types of exponentially localized solutions are easily obtained numerically using adiabatic continuation from the anticontinuous limit. Linear stability (Floquet) analysis allows the characterization of different types of bifurcations experienced by periodic discrete breathers. Some of these bifurcations produce nonperiodic localized solutions, namely, quasiperiodic and chaotic discrete breathers, which are generally impossible as exact solutions in Hamiltonian systems. Within a certain range of parameters, propagating breathers occur as attractors of the dissipative dynamics. General features of these excitations are discussed and the Peierls-Nabarro barrier is addressed. Numerical scattering experiments with mobile breathers reveal the existence of two-breather bound states and allow a first glimpse at the intricate phenomenology of these special multibreather configurations.     

Non-Abelian Chern-Simons particles in an external magnetic field

The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-) holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to these of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels.     

Interactions of Localized Wave Structures on Periodic Backgrounds for the Coupled Lakshmanan–Porsezian–Daniel Equations in Birefringent Optical Fibers

Nonlinear waves on periodic backgrounds play an important role in physical systems. In this study, nonlinear waves that include solitons, breathers, rogue waves, and semi-rational solutions on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations are investigated. Moreover, the interactions between different types of nonlinear waves are examined and their dynamic behaviors are studied. In particular, it is observed that bright-dark rogue waves interact with bright-dark breathers or solitons on periodic backgrounds, four-petaled breathers interact with two eye-shaped breathers on periodic backgrounds, and a four-petal rogue wave interplays with a rogue wave on periodic backgrounds. Furthermore, it is found that the value of the parameter γ₃ affects the weak and strong interactions of these nonlinear waves. These results may be useful in the study of nonlinear wave dynamics in coupled nonlinear wave models.     

Fast convergence of path integrals for many-body systems

We generalize a recently developed method for accelerated Monte Carlo calculation of path integrals to the physically relevant case of generic many-body systems. This is done by developing an analytic procedure for constructing a hierarchy of effective actions leading to improvements in convergence of N-fold discretized many-body path integral expressions from 1/N to 1/Np for generic p. In this Letter we present explicit solutions within this hierarchy up to level p=5. Using this we calculate the low lying energy levels of a two particle model with quartic interactions for several values of coupling and demonstrate agreement with analytical results governing the increase in efficiency of the new method. The applicability of the developed scheme is further extended to the calculation of energy expectation values through the construction of associated energy estimators exhibiting the same speedup in convergence.     

Quasiclassical symmetry properties II

Proofs are given that the quasiclassical approach proposed previously is able to work in conjunction with the 1/N method. Accordingly, the expansion parameter k = 2l + N − a of the shifted 1/N method should be chosen, order by order, such that the sum of corrections to the zeroth-order result vanishes. This interconnection criterion leads to order-dependent algebraic equations for the parameter k. In turn, the underlying phase-space quantum becomes just half the parameter k implied in this way. Alternative fixing conditions, based on the selection of a dominant potential term, can also be proposed. Quasiclassical symmetry transformations leaving corresponding equivalence classes of Hamiltonian forms invariant are established. Then energy levels and couplings characterizing such Hamiltonians become subject to mutual conversions. Scaling properties of the phase-space quantum are discussed. In addition, this quantum exhibits a covariance behaviour with respect to the quasiclassical symmetry transformations mentioned above. Critical coupling for several short-range potentials are given to first order. Generalizations to resonances and nonlinear quantum-mechanical potentials are also made. Except for dyons, we restrict ourselves to spherically symmetric nonrelativistic Hamiltonians.