Discrete sine-Gordon Equation Research Articles

On the Differential-Difference Sine-Gordon Equation with an Integral Type Source

ABSTRACT In this work, we study the integration of the differential-difference sine-Gordon equation with an integral type source. We deduce the time performance of the scattering data of the spectral problem which is associated with the discrete sine-Gordon equation. Using the inverse scattering method, we integrate the Cauchy problem for the differential-difference sine-Gordon equation with the integral type source in the class of the rapidly decreasing functions.

Discrete sine-Gordon equation on metric graphs: A simple model for Josephson junction networks

We consider discrete sine-Gordon equation on branched domains. The latter is modeled in terms of the metric graphs with discrete bonds having the form of the branched 1D chains. Exact analytical solutions of the problem are obtained for special case of the constraints given in terms of a simple sum rule. Numerical solution is obtained when the constraint is not fulfilled. A simple model of a Josephson junction network is proposed using the obtained results.

Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits

Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits

Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits

Изучаются локальная и нелокальная комплексные редукции дискретного уравнения Абловица-Каупа-Ньюэлла-Сигура отрицательного порядка. Для получающихся в результате редукции локального и нелокального комплексных дискретных уравнений синус-Гордона построены решения типа матрицы Коши, в том числе солитонные решения и решения типа жордановой матрицы. Проанализирована и показана на графиках динамика односолитонного решения. Обсуждаются непрерывные пределы локального и нелокального комплексных дискретных уравнений синус-Гордона.

Stationary multi-kinks in the discrete sine-Gordon equation

We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein–Gordon models. The multi-kinks are constructed using Lin’s method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.

Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces

We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of 
 linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator
 interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation
 in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2\times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_\Theta$. We suppose that the weight $\Theta$ satisfies some regularity assumption.
 Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2\left((-T, T); l^2)$ belongs to $C^2\left((-T, T); l^2_\Theta\right)$. 
 And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.

Dynamics and invariant measures of multi-stochastic sine-Gordon lattices with random viscosity and nonlinear noise

We investigate mean dynamics and invariant measures for a multi-stochastic discrete sine-Gordon equation driven by random viscosity, stochastic forces, and infinite-dimensional nonlinear noise. We first show the existence of a unique solution when the random viscosity is bounded and the nonlinear diffusion of noise is locally Lipschitz continuous, which leads to the existence of a mean random dynamical system. We then prove that such a mean random dynamical system possesses a unique weak pullback mean random attractor in the Bochner space. Finally, we show the existence of an invariant measure. Some difficulties arise from dealing with the term of random viscosity in all uniform estimates (including the tail-estimate) of solutions, which lead to the tightness of a family of distribution laws of solutions.

A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography

Cryptography is the science and study of protecting data in computer and communication systems from unauthorized disclosure and modification. An ordinary difference equation (a map) can be used in encryption–decryption algorithms. In particular, the Arnold’s cat and the sine-Gordon linear maps can be used in cryptographic algorithms for encoding digital images. In this article, a two-dimensional linear mKdV map derived from an ordinary difference mKdV equation will be used in a cryptographic encoding algorithm. The proposed encoding algorithm will be compared with those generated using sine-Gordon and Arnold’s cat maps via the correlations between adjacent pixels in the encrypted image and the uniformity of the pixel distribution. Note that the mKdV map is derived from the partial discrete mKdV equation with Consistency Around the Cube (CAC) properties, whereas the sine-Gordon map is derived from the partial discrete sine-Gordon equation, which does not have CAC properties.

Non-Reciprocal Supratransmission in Mechanical Lattices with Non-Local Feedback Control Interactions

We numerically investigate the supratransmission phenomenon in an active nonlinear system modeled by the 1D/2D discrete sine-Gordon equation with non-local feedback. While, at a given frequency, the typical passive system exhibits a single amplitude threshold marking the onset of the phenomenon, we show that the inclusion of non-local feedback manifests additional thresholds that depend upon the specific boundary from which supratransmission is stimulated, realizing asymmetric (i.e., non-reciprocal) dynamics. The results illustrate a new means of controlling nonlinear wave propagation and energy transport for, e.g., signal amplification and mechanical logic.

Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

AbstractWe study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

Homoclinic traveling waves in discrete sine-Gordon equation with nonlinear interaction on 2D--lattice

Homoclinic traveling waves in discrete sine-Gordon equation with nonlinear interaction on 2D--lattice

Embedded solitons in the double sine-Gordon lattice with next-neighbor interactions.

Topological solitons can propagate without radiation in discrete media. These solutions are known as embedded solitons (ES). They come as isolated solutions and exist despite their resonance with the linear spectrum of the respective lattices. In this paper the properties of embedded solitons in the discrete double sine-Gordon equation with next-neighbor and second-neighbor interactions are investigated. Depending on the sign of these interactions they can be either destructive or favorable for the ES creation. The ES existence area depends on the width of the linear spectrum: narrowing of the spectrum widens the ES existence range and vice versa. The application to the Josephson junction arrays is discussed.

Integrable discretizations and numerical simulation for a modified coupled integrable dispersionless equation

In this paper, we propose an integrable semi-discrete and full-discrete analogues of the modified coupled integrable dispersionless (mCID) equation that was derived recently as a two-component generalization of CID equation. The key of the integrable discretization is to find the relation between the mCID equation and the sine-Gordon equation. We complete the integrable discrete analogues based on the discrete sine-Gordon equation. Numerical simulation of one-soliton, two-soliton and breather solution are conducted based on the integrable semi-discrete scheme. We also present the Casorati determinant solution in parametric form of N-soliton solutions for the semi-discrete analogue of the mCID equation. We show that the continuum limit of the full-discrete mCID equation leads to the semi-discrete mCID equation.

The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice

The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice

A discrete negative AKNS equation: generalized Cauchy matrix approach

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit we obtain a semidiscrete negative AKNS equation and after a second continuum limit we derive the nonlinear negative AKNS equation. The reductions to discrete, semi-discrete and continuous sine-Gordon equations are also discussed.

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations, we derive new vector Yang–Baxter map and integrable discrete vector sine-Gordon equation on a sphere.

Envelope solitons in a left-handed nonlinear transmission line with Josephson junction

We consider a nonlinear left-handed transmission line that incorporates an array of Josephson junctions in its periodic lattice structure. We show that the system dynamics is described by a discrete sine-Gordon-like equation, where the left-handedness of the lattice manifests in the form of a non-standard second-time- derivative term. Since this modified discrete sine-Gordon equation has not yet been extensively studied in the literature, this paper opens up the possibility of additional mathematical analysis. It is also intriguing that by means of a semi-discrete approximation we can derive a nonlinear Schrödinger equation and thus show that the system supports both bright and dark envelope soliton solutions depending on the choice of carrier frequency. The left-handedness of the network is explicitly confirmed in numerical simulations which demonstrate the backward propagation of the bright and dark soliton, in good agreement with analytical predictions.

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

We consider a chain of torsionally-coupled, planar pendula shaken horizontally by an external sinusoidal driver. It has been known that in such a system, theoretically modeled by the discrete sine-Gordon equation, intrinsic localized modes, also known as discrete breathers, can exist. Recently, the existence of multifrequency breathers via subharmonic driving has been theoretically proposed and numerically illustrated by Xu et al. (2014) [21]. In this paper, we verify this prediction experimentally. Comparison of the experimental results to numerical simulations with realistic system parameters (including a Floquet stability analysis), and wherever possible to analytical results (e.g. for the subharmonic response of the single driven–damped pendulum), yields good agreement. Finally, we report the period-1 and multifrequency edge breathers which are localized at the open boundaries of the chain, for which we have again found good agreement between experiments and numerical computations.

Fluxon mobility in an array of asymmetric superconducting quantum interference devices.

Fluxon dynamics in the dc-biased array of asymmetric three-junction superconducting quantum interference devices (SQUIDs) is investigated. The array of SQUIDs is described by the discrete double sine-Gordon equation. It appears that this equation possesses a finite set of velocities at which the fluxon propagates with the constant shape and without radiation. The signatures of these velocities appear on the respective current-voltage characteristics of the array as inaccessible voltage intervals (gaps). The critical depinning current has a clear minimum as a function of the asymmetry parameter (the ratio of the critical currents of the left and right junctions of the SQUID), which coincides with the minimum of the Peierls-Nabarro potential.