Numerical solution of kleinsinegordon equations by. The maxwell equation determines the strength of the torque vector along the zaxis which agrees with the conventional mechanical interpretation of the sine gordon equation as a continuum limit of the in nite chain of coupled pendulum equations, i. Orthogonal basis are used to establish computational algorithm. The threedimensional solutions depend on arbitrary functions f. Secondorder nonlinear hyperbolic partial differential equations exact solutions keywords hyperbolic, nonlinear, partial, differential, equations, klein gordon, wave, sine gordon, secondorder, equation. It belongs to nonlinear model of dna which is close to realistic model. The cauchy problem for the sine gordon equation can be solved using the inverse scattering transform. For completeness, without proof, we want to mention some other analytic solutions of the sge. The first terms on their righthand sides correspond to particles of mass and. In this paper, we develop spectral function method that allows l 2 projection of an operator onto a nite dimensional hilbert space to solve sine gordon equation numerically.
Then combining equation 6 and 7 we write numerical solution of sine gordon equation by reduced differential transform method y. Therefore, spectral collocation methods based on chebyshev. As a lorentz transformed sinus remains a sinus, the equation still remains a sinegordon equation, thus it is lorentz invariant. Github makes it easy to scale back on context switching. R 3 be a smooth function on an simply connected domain d in the uv plane satisfying the sinegordon equation 6.
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. Introduction to superconductivity hs14 exercise sheet 9. The name sinegordon equation is a pun on the wellknown kleingordon equation in physics. We find the equation for the sum of a general cosine function and a general sine function, having the same frequencies, by recalling. This type of the nonlocal sine gordon equation is shown to possess one, two and nsolitonic solutions which are a nonlocal deformation of the corresponding classical solutions of the sine gordon equation. For general background and applications of b acklund transformation, we refer to 43,44. The sine gordon equation is one of the basic equations in modern nonlinear wave theory. One interesting model in this regard is the sine gordon model of dna as proposed by salerno. The sinh gordon equation appears in integrable quantum field theory, kink dynamics, fluid dynamics, and in many other scientific applications. Since there are multiple solutions for the sine gordon equation, we will examine a few of them. The sinegordon equation in josephsonjunction arrays.
Numerical computation of the mass of a quantum sinegordon. However, the elliptic sinegordon boundary value problem as. We will also use di erent trial functions since some of the solutions. 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. Of particular interest is the domain wall collision to 2d sine gordon equation which to the authors knowledge have not been presented in the literature. We refer to the references therein for more discussion on the dynamical properties of the hyperbolic sine gordon equation. Simple transformations, parabolic pde with quadratic nonlinearity, viscous burgers equation and the colehopf transformation. Sinegordon equation in view of the able remark prop erties kdv equation. An unbound antikinked and a kinked soliton solution of the sine gordon equation at di. Supported by ras under grant po sardegna 200720, l. Based on nonlinear spectral analysis, exact nonreflecting boundary conditions ar. Sinegordon expansion method for exact solutions to. We obtain exact solutions ux, y, z, t of the threedimensional sinegordon equation in a form that lamb previously proposed for integrating the twodimensional sinegordon equation. In this work we propose a highorder and accurate method for solving the onedimensional nonlinear sine gordon equation.
Localization of the sine gordon equation solutions request pdf. After a brief survey of the sinegordon equation sge, this report presents a fourthorder numerical approximation. Lyngby, denmark in this exercise we are considering a horizontal along z, periodic distance a, linear. The distributed feedback control is developed to support propagation of localized nonlinear waves for the double sine gordon equation and the dispersive sine gordon equation previously obtained. Degenerate multisolitons in the sinegordon equation. In this paper, we develop and analyze an energyconserving local discontinuous galerkin ldg method for the twodimensional. 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. Abstract reduced differential transform method rdtm, which does not need small parameter in the equation is implemented for solving the sinegordon equation. Numerical solution of sinegordon equation by reduced. 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.
A direct method for solving the generalized sinegordon equation. Pdf on the integrability of the sinegordon equation. We will use di erent trial functions to nd di erent sets of solutions. Numerical simulation of the solution to the sine gordon equation on the whole real axis is considered in this paper. The approximate analytical solution of the equation is calculated in the form of a series with easily computable components. An unbound antikinked and a kinked soliton solution of the sinegordon equation at di. Lecture notes massachusetts institute of technology. Its name is a reference to the more basic kleingordon eld theory, but since the potential for this theory includes a sine or a cosine instead of the usual. We develop two novel classes of energypreserving algorithms for the sine gordon sg equation subject to neumann boundary conditions. The sine gordon equation is a nonlinear hyperbolic partial differential equation involving the dalembert operator and the sine of the unknown function.
One and two soliton solutions for the sinhgordon equation. The sinegordon equation is generally posed as an initial value problem on ir for t. Pdf numerical solution of kleingordon and sinegordon. Some applications of the sine gordon equation include. Introduction to superconductivity exercise sheet 9 hs14 dr.
Computationalengineeringwithmatlabsinegordonsoliton. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative curvature. Meanwhile, it is pointed out that the present method can be slightly adapted to the numerical solutions of other types of sine gordon equations, for example, sine gordon equation, doubled sine. Chapter 5 sinegordon equation the sinegordon equation is a nonlinear hyperbolic partialdifferential equation involving the dalembert operator and the sine of the unknown function. 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. This makes the approach introduced capable of capturing more exact solutions than that standard sine gordon.
The solution of the twodimensional sinegordon equation. A growing body of literature has been focusing on the question how individuals in organizations combine exploitation and exploration so as to. Introduction to solitary waves and solitons, water waves, solitary waves for the kdv equation, the sinegordon equation. Simulating solitons of the sinegordon equation using. In this section, we give the basic theory of the sinegordon equation and the variety of its travelingwave solutions, as spatiotemporal models of nonlinear excitations in complex physical systems. Institute for studies in theoretical physics and mathematics, p. Notes on the sine gordon equation david gablinger january 31, 2007 abstract in this seminar, we will introduce the sine gordon equation, and solve it using a baecklund transfomation. Numerical solution of the sine gordon equation guo benyu department of mathematics science and technology university of shanghai shanghai, china and pedro j. Travelling waves in a singularly perturbed sinegordon equation. The equa tion, as well as several solution techniques, were known in the nineteenth century in the course of study of various problems of differential geometry. Read rendered documentation, see the history of any file, and collaborate with contributors on projects across github.
The cosine pseudospectral method is first employed for spatial discretization under two different meshes to obtain two structurepreserving semidiscrete schemes, which are recast into a finitedimensional hamiltonian system and. Exact solutions to the sine gordon equation francesco demontis based on a joint work with t. Pdf new exact solutions for the sinegordon equations. Solutions of the threedimensional sinegordon equation. As it happ ens, b oth ers answ are es y for the ery v t di eren lo oking don sinegor quation e, a semilinear pde h whic in one. Applications and connections the sine gordon equation is used in fields as wide as. Static sine gordon equation with an imaginary mass term by reinaldo baretti machin uprhumacao home page. Experimental results for the sinegordon equation in arrays. Linear and nonlinear wave equations are important in various aspects of mathematics and physics, including the study of black holes. Lyngby, denmark in this exercise we are considering a horizontal along. From this it follows that the unperturbed linearised sinegordon equation has a double eigenvalue at zero and continuous spectrum on the imaginary axis. In the quasiclassical quantization of the field, a fundamental role is played by the above formulas for and. We introduced solitons as the solutions to a nonlinear wave equation, where the nonlinearity and the dispersion balance each other out, so that there exists a stable but nontrivial solution.
Pdf the sinegordon equation and its application to. For other exact solutions of the sinegordon equation, see the nonlinear klein gordon equation with fw bsinw. Abstract using a simple ansatz for the solutions of the three dimensional generalization of the. Pdf exact solutions of coupled sinegordon equations. Another transformation of pseudospherical surfaces is the lie transform introduced by sophus lie in 1879, which corresponds to lorentz boosts in terms of lightcone coordinates, thus the sinegordon equation is lorentz invariant. Numerical solution to the sinegordon equation defined on the. A note on the three dimensional sinegordon equation ahmad shariati institute for advanced studies in basic sciences, p. A note on the elliptic sinegordon equation goong chen, zhonghai ding, chiaren hu, weiming ni, and jianxin zhou abstract. Combining these two quantities we can construct a third lorentz invariant quantity. A new rational sinegordon expansion method and its. 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. This is simply the wave equation with a nonlinear source term. The representation on solutions of the sinegordon and klein. Superconducting josephsonjunctions are excellent experimental systems for the general study of nonlinear phenomena and nonlinear localised excitations.
Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sine gordon equation. Backlund transformations for the sinegordon equations. By inspiring the work done in 19 20, we use the chebyshev wavelet as basis function coupled with spectral method for solving nonlinear klein gordon and sine gordon equations. Furthermore, one of the distinguished features of the ch equation is that it possesses certain nonsmooth travelingwave solutions called peakons. The sinegordon equation is integrated by the inverse scattering method. Among other physical applications all these sinegordon s are significant to the theory of intense ultrashort optical pulse propagation. Multiple waves never merge, rather, they pass through each other. We verify that it satisfies the sine gordon equation in11. In particular, one obtains explicit expressions for the finitegap solutions of the sinegordon equation in terms of functions on the corresponding abelian varieties. Discrete singular convolution for the sinegordon equation.
V x, t in this numerical research, we examine the kleingordon and sinegordon equations n x. The sine gordon equation posed in the upper half plane was considered in 49, but the. Specifically, a long josephsonjunction is described by the continuous sinegordon equation and a josephsonjunction parallel array by its discrete counterpart. Sinegordon equation neqwiki, the nonlinear equations. There is a growing interest in the study of the sinh gordon equation, the double sinh gordon. On kinks and other travellingwave solutions of a modified sine. Nonabelian sinegordon theory and its application to.
Unperturbed sinegordon equation has exact solution. This paper gives three new solutions to solve the 2d sine gordon equation. A note on the three dimensional 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. A summary on solitons in quantum field theory diva portal. For other exact solutions of the sinegordon equation, see the nonlinear kleingordon equation with fw bsinw. The homogeneous balance procedure gives the order of the predicted polynomialtype solution that is inspired from wellknown sine gordon. Static sine gordon equation with an imaginary mass term. The elliptic sinegordon equation originates from the static case of the hyperbolic sinegordon equation modeling the josephson junction in superconductivity. The dirichlettoneumann map for the elliptic sinegordon.
The sinegordon equation is the eulerlagrange equation for this lagrangian. Properties of solutions to the diffusion equation with a foretaste of similarity solutions pdf conversion of nonlinear pdes to linear pdes. Numerical solution of the sinegordon equation sciencedirect. The sinegordon equation is a nonlinear hyperbolic partialdifferential equation in volving the dalembert operator and the sine of the unknown function. We shall now investigate in more detail the sinegordon equation as an. The sine gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in regularized long waverlwclass. A serious implication is that at least some previous reports of chaos, in fact, are numerical artifacts. Using the numerical method of nonlinear we use the approximation of rbfs in the methods and compare the obtained results.
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