Bounded States for Breathers–Soliton and Breathers of Sine–Gordon Equation

The Wronskians solutions to the sine-Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. A novel expression N-soliton solution with a nonzero background is presented which is used to construct the soliton and breather solutions. Due to the existences of abundant structures of the solitons and breathers, it is possible to search for the coherent structures, or bounded states of solitons and breathers. By introducing the velocity resonant conditions, the sG equation is proved to possess the bounded state for breathers-soliton, or breather-soliton molecules (BSMs) and the bounded state for breathers, or breather molecules (BMs) by different parameters selections. In addition, an approximately bounded state for solitons is demonstrated. The interesting thing is the interactions among the BSMs, BMs and solitons, breathers may be nonelastic by the particular meaning the sizes of the BSMs and BMs change.

Numerical Solution of Fractional Sine-Gordon Equation Using Spectral Method and Homogenization

In this paper, a new method for the numerical solution of fractional sine-Gordon (SG) equation is presented. Our method consists of two steps, in first step: the main equation is converted to a homogeneous one using interpolation. In second step: two-dimensional approximation of functions by shifted Jacobi polynomials is used to reduce the problem into a system of nonlinear algebraic equations. The archived system is solved by Newton’s iterative method. Our method is stated in general case on rectangular [a,b] × [0,T] which is based upon Jacobi polynomial by parameters (α,β). Several test problems are employed and results of numerical experiments are presented and also compared with analytical solutions. Also, we verify the numerical stability of the method, by applying a disturbance in the problem. The obtained results confirm the acceptable accuracy and stability of the presented method.

The breathers in nematic liquid crystals under applied electric field

In this paper, we study the twist of the nematic liquid crystal molecules under the applied electric field. The dynamic equation of the twisted molecules is derived. It is proved to be a kind of sine-Gordon (SG) equation. We obtain the breather solution of the equation and confirm that the deflection angles of the twisted molecules can distribute in the form of breathers. We give the relationship between the molecular deflection angle and the breather frequency, and discuss the effect of electric field on breather shape and breather frequency.

Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices

The sine-Gordon (SG) partial differential equation (PDE) with an arbitrary perturbation is initially considered. Using the method of Kuzmak–Luke, we investigate the conditions, which the perturbation must satisfy, for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion. The motivation for this study is that the mathematical modelling of physical systems often leads to the discrete SG system of ordinary differential equations, which are then approximated in the long wavelength limit by the continuous SG PDE. Such limits typically produce fourth-order spatial derivatives as correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi-continuum SG equation and not the damped SG system.

Exact Traveling Wave Solutions for the Modified Double Sine-Gordon Equation

With some elementary methods, a number of new travelling solutions of the modified double Sine-Gordon (SG) equation are obtained,including different types of exact solion solutions and exact periodic solutions.

Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space 𝔰𝔭(n)/𝔲(n) and its bracket relations within the symmetric Lie algebra (𝔲(n), 𝔰𝔭(n)).

STOCHASTIC DYNAMICS OF DC AND AC DRIVEN DISLOCATION KINKS

Dynamics of a pinned dislocation kink controlled by the acting DC and AC forces is studied analytically. The motion of the kink, described by sine-Gordon (sG) equation, is explored within the framework of McLaughlin–Scott perturbation theory. Assuming weakness of the acting AC force, the equation of motion of the dislocation kink in the pinning potential is linearized. Based on the equations derived, we study stochastic behavior of the kink, and determine the probability of its depinning. The dependencies of the depinning probability on DC and AC forces are analyzed in detail.

Soliton Creation and Annihilation in Josephson Junction in the Presence of Periodic Perturbation

The soliton creation and annihilation processes are demonstrated numerically, in the long overlap Josephson junction in the first, second and third Zero-Field Step (ZFS) cases, using the perturbed sine-Gordon (sG) equation in the presence of periodic point-like weak inhomogeneities. In all the ZFS cases, the created soliton is found to be in a bunched (congealed) mode with the other solitons. The I–V characteristics, corresponding to the kink dynamics under a dc bias, at which the creation and annihilation phenomena demonstrated is compared with that of an anti-kink dynamics under the same bias condition.

On polynomial symmetries of the sine-Gordon equation

The sine-Gordon (SG) equation uxt = sin u arises from many branches of physics, and now is one of the most important equations in soliton theory. There have been many works concerning its soliton solutions, Backhand transformations, symmetries and conservation laws and other properties. In this paper we prove that every polynomial symmetry of the SG equation is Hamiltonian, that is, takes the form of D-l δh/δu.