Dimensional sine-Gordon System Research Articles

Bistability of a two-dimensional Klein-Gordon system as a reliable means to transmit monochromatic waves: A numerical approach

Departing from a reliable computational method to approximate solutions of dissipative, nonlinear wave equations, we study the bistability of a (2+1) -dimensional sine-Gordon system, spatially defined on a bounded square of the first quadrant of the Cartesian plane, and subject to Dirichlet boundary data in the form of harmonic driving on the coordinate axes, oscillating at a frequency in the forbidden band gap of the medium. It is shown numerically that, as its spatially discrete counterpart, the continuous (2+1) -dimensional sine-Gordon equation presents the process of nonlinear supratransmission, and that this phenomenon is independent of the discretization procedure. Moreover, our simulations show that a bistable region, where a conducting state and an insulating state may coexist, is present in this system, even in the presence of external damping. As an application, it is shown that the bistable regime may be properly employed in order to transmit certain monochromatic waves through these media.

General multi-linear variable separation approach to solving low dimensional nonlinear systems and localized exitations

Multi-linear variable separation approach based on the corresponding Bcklund transformation (BT-MLVSA) is a useful method to solve nonlinear systems. General multi-linear variable separation approach (GMLVSA) is its extension and there are four ways to realize it. The first one is to expand the nonlinear systems according to multi-arbitrary functions, the second one is to expand the variable separation ansatz. The third one is the MLVSA based on the Darboux transformation (DT-MLVSA) and the last one is the derivative-dependent functional variable separation method. By using the first kind of GMLVSA, the solutions can be obtained for the (2+1)-dimensional mNNV system and sine-Gordon system. In this paper, the first kind of GMLVSA is extended to solve some two-dimensional nonlinear systems which are derived from the (2+1)-dimensional sine-Gordon system by using symmetry reduction method. Namely, the applicability of the method is retained from high dimensional systems to low dimensional systems in the symmery reduction sense. This also provide a way of deducing low dimensional systems which can be solved by GMLVSA from high dimensional systems.

New quasi-periodic waves of the ( [formula omitted])-dimensional sine-Gordon system

New exact solutions of the well-known ( 2 + 1 )-dimensional sine-Gordon system are studied by introducing the modified mapping relations between the cubic nonlinear Klein–Gordon and sine-Gordon equations. Two arbitrary functions are included into the Jacobi elliptic function solutions. By proper selections of the arbitrary functions, new quasi-periodic wave solutions are obtained and displayed graphically.

Solutions Generated from the Symmetry Group of the (2 + 1)-Dimensional Sine-Gordon System

Abstract Applying a symmetry group theorem on a two-straight-line soliton, some types of new localized multiply curved line excitations including the plateau-basin type ring solitons are obtained.

New Variable Separated Solutions and Ghoston Structure for the(2+1)-Dimensional Sine-Gordon System

The Bäcklund transformation and variable separation approachare developed for sine-Gordon systems. Three new types of variableseparated solutions with some arbitrary functions have beenobtained. A new kind of ghoston structure, which is invisible atmost of time and can only be detected when it meets with a foldon,is found. This new structure shows a novel interesting andmysterious phenomenon.

Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems

The multilinear variable separation approach and the related “universal” formula have been applied to many (2+1)-dimensional nonlinear systems. Starting from the universal formula, abundant (2+1)-dimensional localized excitations have been found. In this paper, the universal formula is extended in two different ways. One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the (2+1)-dimensional symmetric sine-Gordon system. The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved. Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons. The FSWs and foldons may be “folded” in quite complicated ways and possess quite rich structures and multiplicate interaction properties.

Localized excitations of the (2 1)-dimensional sine-Gordon system

The (2 + 1)-dimensional sine-Gordon system is investigated via a generalized bilinear operator representation. Plateau, basin, bowl and saddle-type ring solitons are thereby constructed. It is indicated how iterated Moutard transformations may be employed to extend the range of the method.

Symmetry analysis and exact solutions of the 2+1 dimensional sine-Gordon system

According to the space variable exchange symmetry, we change the 2+1 dimensional sine-Gordon system obtained by Konopelchenko and Rogers (KR) to a variant form. Some types of similarity reductions are obtained by using some Lie symmetry analysis. From these similarity reductions, we find that the soliton structure of the system possesses quite a rich structure. The line solitons parallel to the lines x+y=0 and x−y=0 may have an arbitrary shape. In addition to the well-known dromion solution, which is constructed by two line solitons, one may find many other kinds of soliton solutions localized in all directions. For instance, some kinds of ring type (basin-like, plateau-like and bowl-like) and instanton type soliton solutions, can be found directly by selecting the arbitrary functions included in the “single” soliton solution. The dromion solutions may also be constructed by straight line and curved line solitons.

Product and Rational Decompositions of Theta Functions Representations for Nonlinear Periodic Waves

A class of periodic solutions of nonlinear evolution equations is expressed as products and rational expressions of theta/elliptic functions. Examples of equations treated include a coupled system of nonlinear Schrodinger (NLS) equations, the (2+1) dimensional sine Gordon system and the Sasa-Satsuma equation. Coupled modified Korteweg-de Vries and NLS systems show that these product periodic waves can be expanded as an infinite sum of solitary waves arising from the coupling. Brief consideration of discrete evolution equations show similar trends but some quantitative difference with the continuous counterpart.

Dromion Structures of (2+1)-Dimensional sine-Gordon System**The project supported by National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province of China

Starting from n line soliton solutions of an integrable (2+1)-dimensional sine-Gordon system, one can find a dromion solution which is localized in all directions for a suitable potential. The dromion structures for a special (2+1)-dimensional sine-Gordon equation are studied in detail. The interactions among dromions are not elastic. In addition to a phase shift, the “shape” and the velocity of these dromions may also be changed after interaction.

On the geometry of an integrable (2+1)–dimensional sine–Gordon system

It is recorded that Darboux9s method of linking the classical Lame system governing triply orthogonal systems of surfaces with an integrable (2+1)–dimensional sine–Gordon equation may be extended and applied to the integrable two–component generalization of the latter introduced by Konopelchenko and Rogers. Thus, in a reinterpretation, this (2+1)–dimensional sine–Gordon system is shown to define particular (integrable) motions of surfaces.

Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)–dimensional sine–Gordon system

Superposition principles, both linear and nonlinear, associated with the Moutard transformation are found. On suitable reinterpretation, these constitute an integrable discrete nonlinear system and its associated linear system. Further, it is shown that, in a particular form, this system is an integrable discretization of a (2+1)–dimensional sine–Gordon system. Solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the Moutard transformation. Included in these solutions are discrete analogues of the kink solutions of the continuous system.

Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system

We investigate the classical and non-classical reductions of the (2 + 1)-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalization of the sine-Gordon equation. A family of solutions obtained as a non-classical reduction involves a decoupled sum of solutions of a generalized, real, pumped Maxwell-Bloch system. This implies the existence of families of solutions, all occurring as a decoupled sum, expressible in terms of the second, third and fifth Painleve transcendents, and the sine-Gordon equation. Indeed, hierarchies of such solutions are found, and explicit transformations connecting members of each hierarchy are given. By applying a known Backlund transformation for the system to the new solutions found, we obtain further families of exact solutions, including some which are expressed as the argument and modulus of sums of products of Bessel functions with arbitrary coefficients. Finally, we show that the sine-Gordon system satisfies the necessary conditions of the Painleve PDE test due to Weiss et al which requires the usual test to be modified, and derive a non-isospectral Lax pair for the generalized, real, pumped Maxwell-Bloch system.

On localized solitonic solutions of a (2+1)-dimensional sine-Gordon system

Exponentially decaying solutions of an integrable sine-Gordon system in 2+1 dimensions are presented. A simple superposition formula is given.

A.c. Conductivity of nearly commensurate charge density waves and application to NbSe 3 and TaS 3

The a.c. conductivity of an overdamped one dimensional sine-Gordon system with a low density of kinks is evaluated. This corresponds to a nearly commensurate charge or spin density wave with a kink-lattice representing the deviation from commensurability. The results explain the unusually broad crossover regime of NbSe 3 as compared with the similar but more commensurate compound TaS 3. When TaS 3 becomes commensurate ( T ⪅ 130 K) we predict that its a.c. response increases and its shape slightly sharpens.

Commensurate-Incommensurate Transition in a Two Dimensional Classical Sine-Gordon System

The critical line for the commensurate· incommensurate transition in the two dimensional classical sine-Gordon system is determined with the use of the Bethe ansatz equations for the massive Thirring model with finite hole density. Except slight differences the behavior of the critical line is the same as that obtained on the basis of the self-consistent harmonic phonon approximation.

On the possibility to create nonthermal solitions in a one dimensional magnetic sine-Gordon system I

On the possibility to create nonthermal solitions in a one dimensional magnetic sine-Gordon system I