In spite of the long history the sinegordon equation and its numerous extensions and generalizations still attract attention. In the continuumlimit approximation the fk model reduces to the exactly integrable sinegordon equation or sg equation which allows for soliton solutions. 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. This leads to slightly more complicated behavior than the other equations though the pattern it generates still has a simple repetitive form. We focus, in particular, on two of their prototypical solutions, namely the kinklike heteroclinic connections and the timeperiodic, exponentially localized in space breather waveforms. Linear and nonlinear waves wileyinterscience, 1999. 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. Sine gordon equation in josephsonjunction arrays, some selected thoughts old. Sinegordon equation in encyclopaedia of mathematics sinegordon equation in eqworld, the world of mathematical equations. The sinegordon equation is the euler lagrange equation for this lagrangian. Equations 1 and 2 are actually the solutions to these pdes. In particular, soliton solutions to the one dimensional sg equation have been investigated theoretically and numerically. Gpu programs for fourier pseudospectral simulations of the navierstokes, cubic nonlinear schrodinger and sine gordon equations visualization with paraview coprocessing programming languages edit. Sufficient conditions on the initial data for which each problem may be solved by the inverse scattering transform are given.
The equivalence of the methods for solving both the goursat problem and the cauchy problem for the sine gordon equation is discussed. It is demonstrated that the sinegordon equation is applicable to the description of series of the. 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. The sinegordon model is a toy model that helps to understand fundamental notions like duality in a simplified setup. And with this equation, the initial lump of gray shown just breaks into two identical pieces which propagate to the left and right without change. Sinegordon equation in view of the able remark prop erties kdv equation. Geickeon the reflection of radially symmetrical sinegordon kinks in the origin. The backlund transformation and sinegordon equations book. 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. Numerical solution of kleinsinegordon equations by. Unperturbed sinegordon equation has exact solution. It may be used to great effect as a course textbook, research reference, or selfstudy guide. Is there any solution of the sine gordon equation in the form 1. As it happ ens, b oth ers answ are es y for the ery v t di eren lo oking don sine gor quation e, a semilinear pde h whic in one.
As examples, we gave all traveling wave solutions to some equations such as mkdv equation, sine gordon equation, double sine gordon equation, triple sine gordon equation, fujimotowatanabe equation, coupled harrydym equation and coupled kdv equation and so on. The first terms on their righthand sides correspond. We study algebrogeometric finitegap and elliptic solutions of fully discretized kp or 2d toda equations. The sinegordon equation is a nonlinear hyperbolic partial differential. Direct integral method, complete discrimination system for. Nonlinear partial differential equations for scientists. Large time behavior of solutions to the initial boundary value problems for the damped sine gordon equation are found which guarantee that solutions of the sine gordon equation decay to certain solutions of the corresponding static problem.
We study the cauchy problem for the sinegordon equation in the semiclassical limit with pureimpulse initial data of sufficient strength to generate both highfrequency rotational motion near the peak of the impulse profile and also highfrequency librational motion in the tails. Elliptic solutions to difference nonlinear equations and. Stability theory for the damped sinegordon equation. Many basic features of such solitons can be studied in the sinegordon model in a simplified setup. The efficacy and accuracy of the method is confirmed by taking test problem 4. The sinegordon equation in josephsonjunction arrays. The goursat and cauchy problems for the sinegordon. Sinegordon equation from wikipedia, the free encyclopedia the sinegordon equation is a nonlinear hyperbolic partial. The solution of the twodimensional sinegordon equation using the.
Among other physical applications all these sine gordon s are significant to the theory of intense ultrashort optical pulse propagation. We received the system of differential equations that shows the soliton. In the present chapter, we consider two prototypical klein gordon models. 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. Perturbation theory for the doublesinegordon equation. The purpose of this book is to present a summary of recent developments in this field, incorporating both introductory. In addition, the equations of the theory permit topological nontrivial solutions called solitons. Its name is a wordplay on its similar form to the klein gordon. However, our sine gordon problem is a twodimensional one space one time problem. Nonlinear partial differential equations for scientists and engineers, third edition, improves on an already highly complete and accessible resource for graduate students and professionals in mathematics, physics, science, and engineering. All single traveling wave solutions to theses equations can be obtained. Localization of the sine gordon equation solutions. Due to my not being very skilled in differential geometry, i want to ask if there is a reference book, paper, etc. Sinegordon equation neqwiki, the nonlinear equations.
In the quasiclassical quantization of the field, a fundamental role is played by the above formulas for 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. This chapter constitute a revision of the physics of such superconducting systems in the light of the sine gordon equation. Pdf sinegordon equation and its application to tectonic stress.
The solution of the twodimensional sinegordon equation using the method of lines. The equations for stationary configurations of the fk model reduce to those of the standard map or chirikovtaylor map of stochastic theory. How to construct pseudospherical surfaces from sinegordon. Parallel spectral numerical methods wikibooks, open. Show that the transformation wz expifxyields a painleve equation. For this reason the fk model is also known as the discrete sinegordon or periodic kleingordon equation. The sinegordon model and its applications from pendula and. 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. Computation and stability of fluxons in a singularly. Exact solutions to the double sinh gordon equation by the tanh method and a variable separated ode method, computers and mathematics with applications, 50 2005 16851696. Forward controllability of a random attractor for the nonautonomous stochastic sine gordon equation on an unbounded domain shuang yang and yangrong li 2019 doi. Sufficient conditions on the initial data for which each proble. Again, both nearperiodicity and nonlinearity of the time evolution are apparent.
Numerical simulation of two dimentional sinegordon. Exceptional discretisations of the sinegordon equation. This is a statement from giamarchis book on quantum physics in 1d. The animations of various soliton solutions of the sinegordon equation are very nice, though a bit mysterious with all the arrows and twisting matter we cannot assume the reader is familiar with dodds elasticribbon analogy. Cambridge core structural geology, tectonics and geodynamics. 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. 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. The sinegordon equation on time scales sciencedirect. Soliton solution of certain nonlinear differential equations. Specifically, a long josephsonjunction is described by the continuous sine gordon equation and a josephsonjunction parallel array by its discrete counterpart. This paper presents the perturbation theory for the double sinegordon equation. Under the same boundary conditions, the sge with the plus sine gives about 20 times higher amplitude waves, which are all above the zero plane and decrease their width with time.
Discrete singular convolution for the sinegordon equation. A serious implication is that at least some previous reports of chaos, in fact, are numerical artifacts. The first terms on their righthand sides correspond to particles of mass and to particles of the ground field. 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 partial differential equation which appears in differential geometry and relativistic field theory. The book consists of wellselected thematic chapters on diverse mathematical and physical aspects of the equation carefully chosen and assigned. 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.
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