Computationalengineeringwithmatlabsinegordonsoliton. Chapter 5 sinegordon equation the sinegordon equation is a nonlinear hyperbolic partialdifferential equation involving the dalembert operator and the sine of the unknown function. Bright soliton solutions yoshimasa matsunosmooth and singular multisoliton solutions of a modified camassaholm equation with cubic nonlinearity and linear dispersion yoshimasa matsunorecent citations. The sinegordon equation is integrated by the inverse scattering method. Using the numerical method of nonlinear we use the approximation of rbfs in the methods and compare the obtained results. A note on the three dimensional sinegordon equation ahmad shariati institute for advanced studies in basic sciences, p. Pdf on the integrability of the sinegordon equation. Therefore, spectral collocation methods based on chebyshev. 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. As a lorentz transformed sinus remains a sinus, the equation still remains a sinegordon equation, thus it is lorentz invariant. R 3 be a smooth function on an simply connected domain d in the uv plane satisfying the sinegordon equation 6. A ear y later, b lam and witz, ablo kaup, ell, new segur produced erse v in scattering metho ds to e solv the equation, reducing.
The solution of the twodimensional sinegordon equation. Sinegordon equation neqwiki, the nonlinear equations. This is simply the wave equation with a nonlinear source term. Numerical solution of sinegordon equation by reduced. Localization of the sine gordon equation solutions request pdf.
One and two soliton solutions for the sinhgordon equation. Furthermore, one of the distinguished features of the ch equation is that it possesses certain nonsmooth travelingwave solutions called peakons. The sine gordon equation posed in the upper half plane was considered in 49, but the. There is a growing interest in the study of the sinh gordon equation, the double sinh gordon. The dirichlettoneumann map for the elliptic sinegordon. A summary on solitons in quantum field theory diva portal. 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. This paper gives three new solutions to solve the 2d sine gordon equation. The representation on solutions of the sinegordon and klein. Pdf numerical solution of kleingordon and sinegordon. Degenerate multisolitons in the sinegordon equation. 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.
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. We will also use di erent trial functions since some of the solutions. Sinegordon expansion method for exact solutions to. Read rendered documentation, see the history of any file, and collaborate with contributors on projects across github. Numerical computation of the mass of a quantum sinegordon. Combining these two quantities we can construct a third lorentz invariant quantity. For other exact solutions of the sinegordon equation, see the nonlinear klein gordon equation with fw bsinw. We find the equation for the sum of a general cosine function and a general sine function, having the same frequencies, by recalling.
This is surprising in that as figure 1 indicates, the work of duckworth et al. Lyngby, denmark in this exercise we are considering a horizontal along z, periodic distance a, linear. A growing body of literature has been focusing on the question how individuals in organizations combine exploitation and exploration so as to. We develop two novel classes of energypreserving algorithms for the sine gordon sg equation subject to neumann boundary conditions. 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 sine gordon equation is a nonlinear hyperbolic partial differential equation involving the dalembert operator and the sine of the unknown function. The threedimensional solutions depend on arbitrary functions f. 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. In the quasiclassical quantization of the field, a fundamental role is played by the above formulas for and.
Pdf the sinegordon equation and its application to. Introduction to solitary waves and solitons, water waves, solitary waves for the kdv equation, the sinegordon equation. However, the elliptic sinegordon boundary value problem as. Institute for studies in theoretical physics and mathematics, p.
A direct method for solving the generalized sinegordon equation. Pdf new exact solutions for the sinegordon equations. Introduction to superconductivity hs14 exercise sheet 9. From this it follows that the unperturbed linearised sinegordon equation has a double eigenvalue at zero and continuous spectrum on the imaginary axis. Numerical solution of kleinsinegordon equations by. We shall now investigate in more detail the sinegordon equation as an. Numerical simulation of the solution to the sine gordon equation on the whole real axis is considered in this paper. Simulating solitons of the sinegordon equation using. Exact solutions to the sine gordon equation francesco demontis based on a joint work with t. In this work we propose a highorder and accurate method for solving the onedimensional nonlinear sine gordon 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. Properties of solutions to the diffusion equation with a foretaste of similarity solutions pdf conversion of nonlinear pdes to linear pdes. Since there are multiple solutions for the sine gordon equation, we will examine a few of them. The sine gordon equation is one of the basic equations in modern nonlinear wave theory.
Note that the largest range of u is at x0 where u varies between p and p. Abstract using a simple ansatz for the solutions of the three dimensional generalization of the. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative curvature. One interesting model in this regard is the sine gordon model of dna as proposed by salerno. Pdf a standard cranknicolson finitedifference scheme and a dufortfrankel finitedifference scheme are introduced to solve twodimensional damped. Among other physical applications all these sinegordon s are significant to the theory of intense ultrashort optical pulse propagation. Backlund transformations for the sinegordon equations. Orthogonal basis are used to establish computational algorithm. After a brief survey of the sinegordon equation sge, this report presents a fourthorder numerical approximation. Specifically, a long josephsonjunction is described by the continuous sinegordon equation and a josephsonjunction parallel array by its discrete counterpart. We refer to the references therein for more discussion on the dynamical properties of the hyperbolic sine gordon equation.
Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sine gordon equation. The approximate analytical solution of the equation is calculated in the form of a series with easily computable components. Github makes it easy to scale back on context switching. Sinegordon equation in view of the able remark prop erties kdv equation. Static sine gordon equation with an imaginary mass term. Experimental results for the sinegordon equation in arrays. The name sinegordon equation is a pun on the wellknown kleingordon equation in physics.
V x, t in this numerical research, we examine the kleingordon and sinegordon equations n x. 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. 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. Static sine gordon equation with an imaginary mass term by reinaldo baretti machin uprhumacao home page. A serious implication is that at least some previous reports of chaos, in fact, are numerical artifacts. Multiple waves never merge, rather, they pass through each other. We will use di erent trial functions to nd di erent sets of solutions. 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. The sinegordon equation is generally posed as an initial value problem on ir for t. Superconducting josephsonjunctions are excellent experimental systems for the general study of nonlinear phenomena and nonlinear localised excitations.
Then combining equation 6 and 7 we write numerical solution of sine gordon equation by reduced differential transform method y. 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. Solutions of the threedimensional sinegordon equation. Travelling waves in a singularly perturbed sinegordon equation. The sine gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in regularized long waverlwclass. 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. The smooth real solutions constructed of the sinegordon equation are. 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. Some applications of the sine gordon equation include.
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. A new rational sinegordon expansion method and its. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The sinh gordon equation appears in integrable quantum field theory, kink dynamics, fluid dynamics, and in many other scientific applications. A note on the three dimensional sinegordon equation. Supported by ras under grant po sardegna 200720, l. An unbound antikinked and a kinked soliton solution of the sine gordon equation at di. 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. The cauchy problem for the sine gordon equation can be solved using the inverse scattering transform. Lyngby, denmark in this exercise we are considering a horizontal along. 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 of the sinegordon equation sciencedirect. In this paper, we develop and analyze an energyconserving local discontinuous galerkin ldg method for the twodimensional.
Secondorder nonlinear hyperbolic partial differential equations exact solutions keywords hyperbolic, nonlinear, partial, differential, equations, klein gordon, wave, sine gordon, secondorder, equation. We verify that it satisfies the sine gordon equation in11. Linear and nonlinear wave equations are important in various aspects of mathematics and physics, including the study of black holes. 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. 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. 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.
The sinegordon equation is a nonlinear hyperbolic partialdifferential equation in volving the dalembert operator and the sine of the unknown function. The elliptic sinegordon equation originates from the static case of the hyperbolic sinegordon equation modeling the josephson junction in superconductivity. Numerical solution to the sinegordon equation defined on the. Lecture notes massachusetts institute of technology. For general background and applications of b acklund transformation, we refer to 43,44. On kinks and other travellingwave solutions of a modified sine. For completeness, without proof, we want to mention some other analytic solutions of the sge. In particular, one obtains explicit expressions for the finitegap solutions of the sinegordon equation in terms of functions on the corresponding abelian varieties. The sinegordon equation in josephsonjunction arrays. For other exact solutions of the sinegordon equation, see the nonlinear kleingordon equation with fw bsinw. Simple transformations, parabolic pde with quadratic nonlinearity, viscous burgers equation and the colehopf transformation. This makes the approach introduced capable of capturing more exact solutions than that standard sine gordon. An unbound antikinked and a kinked soliton solution of the sinegordon equation at di. A note on the elliptic sinegordon equation goong chen, zhonghai ding, chiaren hu, weiming ni, and jianxin zhou abstract.
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. Unperturbed sinegordon equation has exact solution. Numerical solution of the sine gordon equation guo benyu department of mathematics science and technology university of shanghai shanghai, china and pedro j. 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. It belongs to nonlinear model of dna which is close to realistic model. 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. 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. The homogeneous balance procedure gives the order of the predicted polynomialtype solution that is inspired from wellknown sine gordon. Based on nonlinear spectral analysis, exact nonreflecting boundary conditions ar. The method of lines, which was based on rational approximants in a threetime level recurrence relation, has been proposed for the numerical solution of the nonlinear twodimensional sine gordon equation. The first terms on their righthand sides correspond to particles of mass and. Applications and connections the sine gordon equation is used in fields as wide as. 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.
Pdf exact solutions of coupled sinegordon equations. Numerical solution of sinegordon equation by spectral method. Discrete singular convolution for the sinegordon equation. Abstract reduced differential transform method rdtm, which does not need small parameter in the equation is implemented for solving the sinegordon equation. Introduction to superconductivity exercise sheet 9 hs14 dr. The sinegordon equation is the eulerlagrange equation for this lagrangian. Nonabelian sinegordon theory and its application to. It has applications in many areas of physics and mathematics.
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