Lattice sine-Gordon Research Articles

Nonlinear band gap transmission in optical waveguide arrays.

The effect of nonlinear transmission in coupled optical waveguide arrays is theoretically investigated and a realistic experimental setup is suggested. The beam is injected in a single boundary waveguide, linear refractive index of which (n(0)) is larger than refractive indexes (n) of other identical waveguides in the array. Particularly, the effect holds if omega(n(0)-n)/c>2Q, where Q is a linear coupling constant between array waveguides, omega is a carrier wave frequency, and c is a light velocity. Numerical experiments show that the energy transfers from the boundary waveguide to the waveguide array above a certain threshold intensity of the injected beam. This effect is due to the creation and the propagation of gap solitons in full analogy with a similar phenomenon in sine-Gordon lattice [Phys. Rev. Lett. 89, 134102 (2002)]].

A new barrier to the existence of moving kinks in Frenkel–Kontorova lattices

An explanation is offered for an observed lower bound on the wave speed of travelling kinks in Frenkel–Kontorova lattices. Kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves (phonons). However, they fail to exist even in this codimension-one sense whenever there is more than one phonon branch in the dispersion relation; inside such bands only quasi-kinks with nondecaying oscillatory tails are possible. The results are presented for a discrete sine-Gordon lattice with an onsite potential that has a tunable amount of anharmonicity. Novel numerical methods are used to trace kinks with topological charge Q=1 and 2 in three parameters representing the propagation speed, lattice discreteness and anharmonicity. Although none of the analysis is presented as rigorous mathematics, numerical results suggest that the bound on allowable wavespeeds is sharp. The results also explain why the vanishing, at discrete values of anharmonicity, of the Peierls–Nabarro barrier between stationary kinks as discovered by Savin et al., does not lead to bifurcation of kinks with small wave speed.

Construction of a new class of quantum integrable inhomogeneous models

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrodinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.

Simple theories of complex lattices

While the theory of solitons has been very successful for continuous systems, very few nonlinear discrete lattices are amenable to an exact analytical treatment. In these “complex lattices” discreteness can be hostile to the solitons, preventing them to move due to the lack of translational invariance or even to exist as localized excitations. On the other hand, lattice discreteness can sometimes be very helpful. It can stabilize solutions that otherwise would split apart as in the discrete sine-Gordon lattice, or even allow the existence of localized oscillatory modes as exact solutions in systems where they would decay in the continuum limit. It is interesting that many of these phenomena can be understood qualitatively, and sometimes quantitatively, with very simple theories that rely on the usual concepts of linear wave propagation, resonances, linear stability of waves, for instance. There are, however, phenomena specific to discrete nonlinear lattices which allow the build up of large amplitude localized excitations, sometimes out of thermal fluctuations, which are more resistant to simple approaches and could deserve further interest because they may be relevant for various physical systems.

On the fundamental L operator for the quantum lattice W algebra

Studied are remarkable properties of the fundamental L-operators for the quantum lattice WN algebra. Using operators which constitute the extended Weyl algebra, we show that the fundamental L-operators satisfy the Yang-Baxter equation. Briefly discussed is the construction of the generalized lattice sine-Gordon equation.

Zero curvature representation for classical lattice sine-Gordon model via quantum R matrix

Local M operators for the classical sine-Gordon model in discrete space-time are constructed by convolution of the quantum trigonometric 4×4 R matrix with certain vectors in its “quantum” space. Components of the vectors are τ functions of the model. This construction generalizes the known representation of continuous-time M operators in terms of the classical r matrix.

Integrals of motion for the classical sine-Gordon lattice system

Higher-order local integrals of motion for the classical sine-Gordon system are calculated. For the kth integral of motion, the general formula is obtained. It is discussed how the integrals of motion can be used for constructing classical lattice Wn-algebras. A direct consideration is given for the case of W2.

Connections of the Liouville model and XXZ spin chain

The quantum theory of the Liouville model with imaginary field is considered using the Quantum Inverse Scattering Method. An integrable structure with non-trivial spectral-parameter dependence is developed for lattice Liouville theory by scaling the L-matrix of lattice sine-Gordon theory. This L-matrix yields Bethe ansatz equations for Liouville theory, by the methods of the algebraic Bethe ansatz. Using the string picture of excited Bethe states, the lattice Liouville Bethe equations are mapped to the corresponding spin- 1 2 XXZ chain equations. The well developed theory of finite-size corrections in spin chains is used to deduce the conformal properties of the lattice Liouville Bethe states. The unitary series of conformal field theories emerge for Liouville couplings of the form γ = πν/( ν + 1), corresponding to root of unity XXZ anisotropies. The Bethe states give the full spectrum of the corresponding unitary conformal field theory, with the primary states in the Kač table parameterized by a string length K, and the remnant of the chain length mod ( ν + 1).

Breathing solitary waves in a Sine-Gordon two-dimensional lattice.

We study theoretically and numerically the dynamical behavior of a two-dimensional sine-Gordon lattice. We show that, via modulational instability, an initial-low-amplitude plane wave can evolve spontaneously into moving localized modes with large amplitude. These nonlinear modes, with dimensions depending on the characteristic wavelengths of the instability, behave like breathing solitary waves and present particlelike properties.

Integrals of motion of the classical lattice sine-Gordon system

We compute the local integrals of motions of the classical limit of the lattice sine-Gordon system, using a geometrical interpretation of the local sine-Gordon variables. Using an analogous description of the screened local variables, we show that these integrals are in involution. We present some remarks on relations with the situation at roots of 1 and results on another latticisation (linked to the principal subalgebra of $\widehat{s\ell}_{2}$ rather than the homogeneous one). Finally, we analyse a module of ``screened semilocal variables'', on which the whole $\widehat{s\ell}_{2}$ acts.

ALGEBRAIC ASPECTS OF THE BETHE ANSATZ

In this article an introduction to the algebraic aspects of the Bethe ansatz is given. The applications to the seminal spin 1/2 XXX model are discussed in detail and the generalization to higher spin as well as XXZ and the lattice sine-Gordon model are indicated. The origin of quantum groups and their appearance in CFT models are explained. The text can be considered as a guide to the research papers in this field.

Experimental and numerical study of dynamic regimes in a discrete sine-Gordon lattice.

We investigate fluxon dynamics in underdamped one-dimensional parallel arrays of small Josephson tunnel junctions. The current-voltage characteristics of the arrays show various resonant steps depending upon the temperature of the sample and the externally applied magnetic field. Experimental results on fluxon propagation in arrays are compared with that for continuous Josephson transmission lines fabricated on the same chip. By modeling fluxon dynamic states in a one-dimensional array as a propagation of kinks in a discrete sine-Gordon lattice, we find consistency between numerical results and the experimental data. This consistency indicates that the concept of fluxons as moving relativistic particles can be still used even for strongly discrete lines. However, two important classes of phenomena are found which do not have any counterparts in the continuum case. These are concerned with the data that we obtain in the short-wavelength limit (determined by line discreteness) and with damping requirements that are necessary in order to stabilize kink propagation.

Kink Casimir energy in a lattice sine-Gordon model.

The Casimir energy of quantum fluctuations about the classical kink configuration is computed numerically in the weak coupling approximation for a recently proposed lattice sine-Gordon model. This energy depends periodically on the kink position and is found to be approximately sinusoidal.

Coloured FRT algebra and its Yang-Baxterization leading to integrable models with non-additive R-matrices

A 'colour' representation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra, which, in contrast to the standard case, is related to the coloured braid group representation with generic values of q is presented. Explicit realizations of L(+or-)-matrices, occurring in this coloured variant of FRT algebra, are also obtained for the Uq(gl(2)) quantized algebra. Though these realizations are found to depend manifestly on the colour parameters, the underlying quantum group structure and associated co-product are interestingly free from such dependence. This allows us to perform the Yang-Baxterization of the coloured FRT algebra successfully, which leads to the construction of an ancestor Law operator associated with a new non-additive-type quantum R-matrix. Through different realizations of this Lax operator, a new class of quantum integrable models representing 'colour' generalizations of the well known models, such as the lattice sine-Gordon model, the Ablowitz-Ladik model lattice and the derivative nonlinear Schrodinger model etc, is generated.

Integrability of multidimensional discrete systems

The method of singularity confinement is applied to the study of the integrability of the (discrete) Hirota-Miwa equation. We show that this equation passes the integrability test that is here adapted to the bilinear formalism. Starting from the Hirota-Miwa equation we construct a novel form for the discrete mKdV and use it to derive the discrete KdV. We obtain a trilinear form for the latter, which, however, can be reduced to a bilinear one. A novel trilinear equation is also obtained from the discrete sine-Gordon lattice.

Renormalization group flow of a hierarchical Sine-Gordon model by partial differential equations

We use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice Sine-Gordon field theory in the Kosterlitz-Thouless phase.

Intrinsic Anharmonic Localized Modes in thed-Dimensional Sine-Gordon Lattice, the ϕ4-Lattice and the ϕ6-Lattice

Intrinsic anharmonic localized modes (or in-band resonant modes) are predicted for the sine-Gordon (SG) equation, the ϕ 4 -equation, and the ϕ 6 -equation defined in a d -dimensional version of the simple cubic lattice. The localization properties of a stationary mode is similar to those of breathers in the SG equation for d =1 in the continuum limit but the former is much more ubiquitous than the breathers and kinks. This is due to the softening near the maximum points of the on-site potential in the system, where conventional exact soliton solutions do not necessarily exist. The SG lattice is taken as an example to study the localization properties, and a simple numerical calculation is done for d =3 to estimate the critical value of the anharmonicity.

A classical version of the lattice sine-Gordon model

A completely integrable regularized lattice sine-Gordon model is considered by the method of the inverse scattering problem. It is shown that the variables of action-angle type for the lattice and continuous models coincide. Solitons of the lattice model are constructed, and their scattering is described.

Lattice sine-Gordon model with local Hamiltonian

This paper obtains the excitation spectrum in the quantum lattice sine-Gordon model with Hamiltonian describing the interaction of two nearest neighbors on a lattice. This lattice model is one of the possible regularizations of the quantum field sine-Gordon model that preserve the property of complete integrability. It is shown that in the quantum field model regularized in this manner, there are phase transitions at the points gamma =..pi.. (n + 1) (n is an integer), and these are explained by a change in the structure of the vacuum state.

Thermodynamics of the lattice quantum sine-Gordon model and the lattice massive Thirring model

The thermodynamics is studied for two exactly soluble models by means of the Bethe ansatz method: the lattice quantum sine-Gordon model and the lattice massive Thirring model proposed by Izergin and Korepin (1981-82), and Weiss and Schotte (1983), respectively. The specific heat is analysed for each of these models as regards the effect of the cut-off parameter. The classical limit of the free energy is also discussed for the lattice quantum Sine-Gordon model.