Degenerate multisolitons in the sinegordon equation. Among other physical applications all these sinegordon s are significant to the theory of intense ultrashort optical pulse propagation. We study the stability and convergence of the method and show the convergence of our method. Read, download and publish sinegordon equation magazines. Numerical experiments are studied to show the efficiency of the technique. Implementation of the sine gordon expansion method. Coupled sine gordon equations as nonlinear second order evolution equations nakagiri, shinichi and ha, junhong, taiwanese journal of mathematics, 2001. We thus say that the derivative of sine is cosine, and the derivative of cosine is. The identifiability for nonlinear sine gordon equation remains an open question. It is known that the sine gordon has soliton solutions in the 1d and 2d cases, but such solutions are not known to exist in the 3d case.
In this section, we give the basic theory of the sine gordon equation and the variety of its travelingwave solutions, as spatiotemporal models of nonlinear excitations in complex physical systems. Specifically, a long josephsonjunction is described by the continuous sinegordon equation and a josephsonjunction parallel array by its discrete counterpart. In this work, we use a reproducing kernel method for investigating the sinegordon equation with initial and boundary conditions. Numerical solution of kleinsinegordon equations by. We refer to the references therein for more discussion on the dynamical properties of the hyperbolic sine gordon equation. New travelling wave solutions for sine gordon equation sun, yunchuan, journal of applied mathematics, 2014. Pdf on the integrability of the sinegordon equation. Due to the klein gordon equation s enormous popularity, the sge was named so as a wordplay. It was originally introduced by edmond bour 1862 in the course of study of surfaces of constant negative curvature as the gausscodazzi equation for surfaces of curvature. The sinegordon equation in josephsonjunction arrays.
Exact solutions of coupled sine gordon equations using the simplest equation method zhao, yunmei. To present the present methods effectiveness many examples are given. It was originally introduced by edmond bour in the course of study of surfaces of constant negative curvature as the gausscodazzi equation for surfaces of curvature. Home courses mathematics advanced partial differential equations with applications lecture notes. The sinegordon equation is integrated by the inverse scattering method. These results indicate that the reproducing kernel method is very effective. The elliptic sinegordon equation originates from the static case of the hyperbolic sinegordon equation modeling the josephson junction in superconductivity. It exhibits discrete conserved quantities and extended particle states, with the proper behavior regarding covariance, stability, etc. The sine gordon equation posed in the upper half plane was considered in 49, but the dirichlet to neumann map theorem 5. All web surfers are welcome to download these notes, watch the youtube videos.
View enhanced pdf access article on wiley online library html view download pdf for offline viewing. We also calculate explicit expressions of single, double, triple, quad soliton solutions as well as single and double breather solutions of discrete sine gordon equation. However, the elliptic sinegordon boundary value problem as. Pdf and epub downloads, according to the available data and abstracts views on vision4press platform. Fully discrete spectral methods for solving time fractional nonlinear sine gordon equation with smooth and nonsmooth solutions. Gordon equation in one and two spatial dimensions ameneh taleei department of applied mathematics, faculty of mathematics and computer science, amirkabir university of technology, no. Mathematical and theoretical degenerate multisolitons in the sine gordon equation julia cen 1, francisco correa2 and andreas fring,3 1 department of mathematics, city, university of london, northampton square. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sinegordon equation. In particular, one obtains explicit expressions for the finitegap solutions of the sinegordon equation in terms of functions on the corresponding abelian varieties. It is known that the sine gordon has soliton solutions in the 1d and 2d cases, but such. The equation, as well as several solution techniques, were known in the nineteenth century in the course of study of various problems of differential geometry. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services.
We propose a method to deal with the general sine gordon equation. In contrast to the classic sine gordon expansion method, it involves a more general ansatz that is a rational function, rather than a polynomial one, of the solutions of the auxiliary equation. A new rational sinegordon expansion method and its. The dirichlettoneumann map for the elliptic sinegordon. Numerical simulation to the above equation shows that i there exist onekink states in which initial shape and kink velocity are preserved during long period of time for 21.
Numerical solution of kleingordon and sinegordon equations by meshless method of lines. In the current subsection, the application of the sine gordon expansion method to the four dimensional nleqzk equation is presented. Among other results we show that near the equilibrium point, the hamiltonian of the sine gordon sg equation on the circle can be viewed as an element in the poisson algebra of the modified. Chapter 5 sinegordon equation the sine gordon equation is a nonlinear hyperbolic partialdifferential equation involving the dalembert operator and the sine of the unknown function. Notes on the sine gordon equation david gablinger january 31, 2007 abstract in this seminar, we will introduce the sinegordon equation, and solve it using a baecklund transfomation. Stochastic nonlinear thermoelastic system coupled sine gordon equation driven by jump noise cheng, shuilin, guo, yantao, and tang, yanbin, abstract and applied analysis, 2014. Shokria numerical method for solution of the twodimensional sine gordon equation using the radial basis functions. Compared with previous solutions, our solutions are more general than some of the previous. In this paper we establish the identifiability for a linearized sine gordon problem.
From a more physical perspective, at the birth of quantum mechanics, the klein gordon equation was found when trying to. Pdf numerical solution of kleingordon and sinegordon. They arise from a remarkably simple nonlocal reduction of the well. A note on the elliptic sinegordon equation goong chen, zhonghai ding, chiaren hu, weiming ni, and jianxin zhou abstract. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are exponentially asymptotic to integer multiples of 2. New travelling wave solutions for sinegordon equation. The periodic problem for the sinegordon equation can be studied by means of an algebraicgeometric method similar to the case of the kortewegde vries equation.
Numerical simulation of kink dynamics in sinelattice. The sinegordon equation is a nonlinear partial differential equation. Discrete singular convolution for the sinegordon equation. Exact solutions to the sinegordon equation mafiadoc. A serious implication is that at least some previous reports of chaos, in fact, are numerical artifacts. Zero curvature representation for classical lattice sine. In section 5 we also explore the relationship between the lyapunov equations and the sylvester equation and show how their solutions are related to each other in a simple but interesting way. Lecture notes massachusetts institute of technology. Several new exact travelling wave solutions with the form of jacobiamplitude function are derived for the general sine gordon equation by using some reasonable transformation. In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving klein gordon and sine gordon equations.
The acquired results are compared with the exact solutions and results obtained by different methods. Identifiability for linearized sinegordon equation. Solutions to the sine gordon equation with initial data specified on invariant algebraic manifolds of conserved quantities can be written explicitly in. Pdf approximate solutions for solving the kleingordon. For other exact solutions of the sinegordon equation, see the nonlinear klein gordon equation with fw bsinw. A new approach for onedimensional sinegordon equation. Exceptional discretisations of the sinegordon equation.
For general background and applications of b acklund transformation, we refer to 43,44. In this paper, we study the legendre spectral element method for solving the sine gordon equation in one dimension. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. Firstly, we discretize the equation by legendre spectral element in space and then discretize the time by the secondorder leapfrog method. This makes the approach introduced capable of capturing more exact solutions than that standard sine gordon. Solitary waves for the kdv equation, the sine gordon equation. Legendre spectral element method for solving sinegordon. This problem might affect the mathematical modeling of many real. The lecture notes below are courtesy hristina hristova, a student in the class. Zero curvature representation for classical lattice sine gordon equation via quantum rmatrix a. The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with chebyshev wavelets for the approximate solutions of klein gordon and sine gordon equations. A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sinegordon equation.
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