The equivalence of the methods for solving both the goursat problem and the cauchy problem for the sinegordon equation is discussed. Its name is a wordplay on its similar form to the kleingordon. The sinegordon equation on time scales sciencedirect. The sinegordon sg equation is a partial differential equation, which is defined as utt. Sinegordon equation in encyclopaedia of mathematics sinegordon equation in eqworld, the world of mathematical equations. Collision of wave packets in the sinegordon equation universiteit. Numerical solution of the sine gordon equation guo benyu department of mathematics science and technology university of shanghai shanghai, china and pedro j. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sine gordon equation. Meanwhile, it is pointed out that the present method can be slightly adapted to the numerical solutions of other types of sinegordon equations, for example, sinegordon equation, doubled sine.
Exact solutions of the coupled sine gordon equations. In this section, we solve the coupled sinegordon equations by the simplest equation method. An unbound antikinked and a kinked soliton solution of the sinegordon equation at di. For a singleparticle in a cosine potential, the slightest amount of tunneling between two cosine minima leads to conduction bands, for example, and restores the translational symmetry. In mathematics, the sine is a trigonometric function of an angle. Sufficient conditions on the initial data for which each proble. The sine gordon equation is a nonlinear hyperbolic partial differential equation involving the dalembert operator and the sine of the unknown function. It is obvious from the table that the present method is more accurate, simple and fast. The acquired results are compared with the exact solutions and results obtained by different methods. For other exact solutions of the sinegordon equation, see the nonlinear kleingordon equation with fw bsinw. How to construct pseudospherical surfaces from sinegordon. We focus, in particular, on two of their prototypical solutions, namely the kinklike heteroclinic connections and the timeperiodic, exponentially localized in space breather waveforms. Exact solutions of the coupled sinegordon equations. Plotting the timedevelopement of a kinksolution of the.
Furthermore, one of the distinguished features of the ch equation is that it possesses certain nonsmooth travelingwave solutions called peakons. 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. Sinh is the hyperbolic sine function, which is the hyperbolic analogue of the sin circular function used throughout trigonometry. Cambridge core structural geology, tectonics and geodynamics. Mar 29, 2014 this is a plot of how the kinksolition solution of the sine gordon equation evolves in time. Numerical solution of nonlinear sinegordon equation by. Soliton operators for the quantized sinegordon equation. The sine gordon equation in the semiclassical limit. The backlund transformation and sinegordon equations books. The goursat and cauchy problems for the sinegordon equation. The purpose of this book is to present a summary of recent developments in this. In the present chapter, we consider two prototypical kleingordon models. The first terms on their righthand sides correspond to particles of mass and to particles of the ground field, respectively cf.
The representation on solutions of the sinegordon and. 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. 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. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. How to construct pseudospherical surfaces from sinegordon solutions. The solution of the sinegordon equation using method of lines. 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. Solutions of the threedimensional sinegordon equation.
The sinegordon model and its applications from pendula and. The simplest and the most frequently used model which describes the dynamics of the josephson phase in ljj is the socalled perturbed sine gordon equation. Buckingham, department of mathematical sciences, university of cincinnati, po box 210025, cincinnati, ohio 45221 and peter d. Sinegordon solitons, kinks and breathers as physical models of. Linear and nonlinear waves wileyinterscience, 1999. Pdf sinegordon equation and its application to tectonic stress. This is a statement from giamarchis book on quantum physics in 1d. 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. The operators satisfy the anticommutation relations and field equations of the massive thirring model.
In the quasiclassical quantization of the field, a fundamental role is played by the above formulas for and. The results of coleman are thus reestablished without the use of perturbation theory. A partial differential equation which appears in differential geometry and relativistic field theory. The initial value problem for the sinegordon equation is solved by the inversescattering method. The representation on solutions of the sinegordon and klein. In the present chapter, we consider two prototypical klein gordon models. The method is based on collocation of modified cubic bsplines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The sinegordon equation is the eulerlagrange equation for this lagrangian. The sg equation is a relativistic integrable field theory 1 having applications in differential geometry, relativistic field theory and also in propagation of fluxons in. Miller, department of mathematics, university of michigan, east hall, 530 church st.
Numerical solution of kleinsinegordon equations by. Sufficient conditions on the initial data for which each problem may be solved by the inverse scattering transform are given. Numerical solution of the sinegordon equation sciencedirect. It belongs to nonlinear model of dna which is close to realistic model. A new approach for onedimensional sinegordon equation. Operators for the creation and annihilation of quantum sinegordon solitons are constructed. 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. We focus, in particular, on two of their principal solutions, namely the kinklike heteroclinic connections and the timeperiodic, exponentially localized in space breather waveforms. 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 is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola. The sinegordon model and its applications from pendula. Considering the amount of mathematical work that has been done on partial differential equations, one might have thought that a vast range of different. These results indicate that the reproducing kernel method is very effective.
The book consists of wellselected thematic chapters on diverse mathematical and physical aspects of the equation carefully chosen and assigned. The threedimensional solutions depend on arbitrary functions f. This leads to slightly more complicated behavior than the other equationsthough the pattern it generates still has a simple repetitive form. The data was produced with a semidiscrete solver based on regular spatial central difference and time. One interesting model in this regard is the sinegordon model of dna as proposed by salerno. Exact solutions of coupled sinegordon equations using the. This is due to the fact that it was formulated on the basis of the non relativistic energy momentum relationship mathe \dfrac p22mmath, with the usual subst.
In this work, we use a reproducing kernel method for investigating the sinegordon equation with initial and boundary conditions. The sinegordon equation is integrated by the inverse scattering method. Solitary waves, cnoidal waves, conservation laws, the initialvalue problem for the kortewegde vries equation, the lax method, the sinegordon equation and backlund transformations are treated. Lyngby, denmark in this exercise we are considering a horizontal along. The sinegordon equation is a nonlinear hyperbolic partial differential equation involving the dalembert operator and the sine of the unknown function. Pdf exact solutions of coupled sinegordon equations. 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 kleingordon and sinegordon equations. Sinegordon equation neqwiki, the nonlinear equations. Unperturbed sinegordon equation has exact solution. The essence of the book is an introduction to the method of inverse scattering. The sinegordon equation in the semiclassical limit. The kortewegde vries equation is of historical interest, since it was the. Lyngby, denmark in this exercise we are considering a horizontal along z, periodic distance a, linear.
The equivalence of the methods for solving both the goursat problem and the cauchy problem for the sine gordon equation is discussed. In this section, we give the basic theory of the sinegordon equation and the. In order to solve, we introduce a new unknown function by the formula so that. It is demonstrated that the sinegordon equation is applicable to the description of series of the. In this section, we solve the coupled sine gordon equations by the simplest equation method.
The numerical solution of sine gordon equation has presented in table 5 which shows the comparison of the errors of the present method with the exact solution. 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 question whether solution of sinegordon equation still exhibit soliton like behavior under an external forcing has been challenging as it is extremely difficult to obtain an exact solution. 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. 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. Numerical experiments are studied to show the efficiency of the technique. Plotting the timedevelopement of a kinksolution of the sine. The sine of an acute angle is defined in the context of a right triangle. Apr 24, 2008 the solution to the 3d sine gordon equation.
The backlund transformation and sinegordon equations book. Solitons, such as some solutions to the sinegordon equation, that emerge because of topological constraints. Sinh may also be defined as, where is the base of the. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative curvature. The sinegordon model is a ubiquitous model of mathematical physics with a wide range of applications extending from coupled torsion pendula and josephson junction arrays to gravitational and highenergy physics models. N2 the initial value problem for the sine gordon equation is solved by the inversescattering method. Due to the kleingordon equation s enormous popularity, the sge was named so as a wordplay. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sinegordon equation. Sinegordon equation solved by sov mathematics stack exchange. The condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature.
The goursat and cauchy problems for the sinegordon. The sine gordon equation is a particular case of klein gordon equation, which plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory, given by where is a nonlinear force and is a constant. Numerical solution of kleinsinegordon equations by spectral. This is a plot of how the kinksolition solution of the sinegordon equation evolves in time.
The first terms on their righthand sides correspond to particles of mass and. Modified cubic bspline collocation method is discussed for the numerical solution of onedimensional nonlinear sinegordon equation. The given equation is decomposed into a system of equations and. 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.
However, our sinegordon problem is a twodimensional one space one time problem. According to it the sinegordon equation is modified by 6 u t t. Numerical solution of the sinegordon equation guo benyu department of mathematics science and technology university of shanghai shanghai, china and pedro j. Due to my not being very skilled in differential geometry, i want to ask if there is a reference book, paper, etc. One example is a skyrmion, which is the solitarywave solution of a nuclear model whose topological charge is the baryon number.
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