Nonlocal sine-Gordon Research Articles

Multiple mixed solutions of the nonlocal sine-Gordon equation

In this paper, we propose the nonlocal sine-Gordon equation from the AKNS system by the Parity and the Time symmetries. Moreover, the main work is to construct the Darboux transformation for the nonlocal sine-Gordon equation. In particular, various types of solutions are obtained by taking a seed solution, such as, soliton solutions, kink solutions, mixed solutions.

Linear superposition for a sine-Gordon equation with some types of novel nonlocalities

Some nonlocal sine-Gordon (SG) systems with some different types of nonlocalities are derived from the usual local SG equation by means of the consistent correlated bang approach. The nonlocal SG systems are Lax integrable. Two types of N-soliton solutions and six types of periodic solutions of the usual SG equation are presented. Some types of solutions of the nonlocal integrable SG systems are obtained by using the symmetric-antisymmetric separation approach. Usually, the linear superposition principle can not hold as a general principle in the presence of nonlinearity. In this paper, it is shown that for a special type of nonlocal SG equations, a linear superposition theorem can be survived for some special types of exact solutions.

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation.

Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time, and reverse space-time nonlocal integrable equations, including the nonlocal NLS equation, nonlocal sine-Gordon equation, nonlocal Davey-Stewartson equation, etc. Moreover, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS, which is a direct discrete version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for the nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for the NLS equation.

Soliton Solutions of Nonlinear and Nonlocal Sine-Gordon Equation Involving Riesz Space Fractional Derivative

AbstractIn this article, a novel approach comprising modified homotopy analysis method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation (SGE) ${u_{tt}}\; - \;{}^RD_x^\alpha u\; + \;{\rm{sin}}u\; = \;0,$ where $^RD_x^\alpha $ is the Riesz space fractional derivative, 1 ≤ α ≤ 2. For α=2, it becomes classical SGE utt−uxx + sinu=0, and corresponding to α=1, it becomes nonlocal SGE utt−Hu + sinu=0, which arises in the Josephson junction theory, where H is the Hilbert transform. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α=2) and nonlocal SGE (corresponding to α=1). Here, the approximate solution of fractional SGE is derived by using modified homotopy analysis method with the Fourier transform. Then, we analyse the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

Effective sine-Gordon model for the static properties of narrow window junctions

We show that the maximum tunneling current for a long thin window Josephson junction in the presence of an external magnetic field is given to a good approximation by the solution of an effective sine-Gordon model with appropriate rescalings. This model is obtained via a reduction of the two-dimensional system to a one-dimensional integro-differential equation (nonlocal sine-Gordon). The passive region introduces an effective Josephson characteristic length. Using numerical simulations we show that for a narrow junction the cases of inline and overlap current feed are well described by the effective model. We discuss the influence of the width of the junction and find that the model gives a good approximation for the behavior of large area windows in the inline geometry, while it deviates for the overlap.

On multikink states described by the nonlocal sine-Gordon equation

It was found recently that nonlocal non-dissipative generalizations of the sine-Gordon model may describe coherent multikink states. In this paper the problem of classification of such nonlocal states is considered for a model kernel of the integral operator. Rigorous results concerning a number of possible forms of these multikink states are set forth and the coding for such states is suggested. The exact statements are confirmed by results of numerical analysis.

Solitary wave solutions of nonlocal sine-Gordon equations.

In this paper a nonlocal generalization of the sine-Gordon equation, u(tt)+sin u=( partial differential / partial differential x) integral (- infinity ) (+ infinity )G(x-x('))u(x(') )(x('),t)dx(') is considered. We present a brief review of the applications of such equations and show that involving such a nonlocality can change features of the model. In particular, some solutions of the sine-Gordon model (for example, traveling 2pi-kink solutions) may disappear in the nonlocal model; furthermore, some new classes of solutions such as traveling topological solitons with topological charge greater than 1 may arise. We show that the lack of Lorenz invariancy of the equation under consideration can lead to a phenomenon of discretization of kink velocities. We discussed this phenomenon in detail for the special class of kernels G(xi)= summation operator (j=1) (N)kappa(j)e(-eta(j)mid R:ximid R:), eta(j)>0, j=1,2, em leader,N. We show that, generally speaking, in this case the velocities of kinks (i) are determined unambiguously by a type of kink and value(s) of kernel parameter(s); (ii) are isolated i.e., if c(*) is the velocity of a kink then there are no other kink solutions of the same type with velocity c in (c(*)- varepsilon,c(*)+ varepsilon ) for a certain value of varepsilon. We also used this special class of kernels to construct approximations for analytical and numerical study of the problem in a more general case. Finally, we set forth results of the numerical investigation of the problem with the kernel that is the McDonald function G(xi) approximately K(0)(mid R:ximid R:/lambda) (lambda is a parameter) that have applications in the Josephson junction theory. (c) 1998 American Institute of Physics.

Small-amplitude solitons in a nonlocal sine-Gordon model

It is shown that small amplitude solitons of a nonlocal sine-Gordon model corresponding to different frequencies of the carrier wave can create coupled states. The effect is due to a change of the dispersion orginated by a nonlocal nonlinearity. Within the framework of the multiscale expansion such pulses are described by a system of nonlinear Schrödinger equations which possesses coupled mode solutions in the form of running localized waves (breathers). Such breathers consist of modes with different frequencies and are characterized by two internal frequencies.

Monte Carlo simulations for a nonlocal sine-Gordon theory, vortex fluctuations, and connections to superconducting films.

Monte Carlo simulation results are presented for a two-dimensional Euclidian nonlocal sine-Gordon lattice field theory. This particular field theory is identical to a statistical-mechanical model called the Ginzburg-Landau Coulomb gas, a model designed to describe thermal vortex fluctuations for a two-dimensional superfluid. The Monte Carlo data obtained are through a chain of relations compared to the experimental data for the scaling function describing the resistive tail for thin superconducting films. The agreement found strongly supports a Ginzburg-Landau Coulomb-gas vortex-fluctuation explanation of the resistive tail.