Coleman discovered an equivalence between the thirring and the sine gordon models. In the quasiclassical quantization of the field u, a fundamental role is played by the above formulas for p 0 and p 1. Experimental results for the sine gordon equation in arrays of coupled torsion. Chang in10, introduced an algorithm to calculate the differential transform of nonlinear functions. Considering the amount of mathematical work that has been done on partial differential equations, one might have thought that a vast range of different. It is manifestly invariant under spacetime translations and lorentz boosts, 2x, t. Correlations between chaos in a perturbed sinegordon equation. An effective slow modess theory is derived and rescaled to obtain the models flow equations. The sinegordon equation has conserved quantity e112.
As a penny to the empirical, you can have the many for oblivion. In addition, the equations of the theory permit topological nontrivial solutions called solitons. This unification contains the sine gordon equation, discrete. Sinegordon solitons, kinks and breathers as physical models of. Jul 17, 2006 1992 an initialboundary value problem for the sine gordon equation in laboratory coordinates. Attempts to understand the origin of the inverse scattering lax representation of. In this section, we give the basic theory of the sine gordon equation and the. It exhibits discrete conserved quantities and extended particle states, with the proper behavior regarding covariance, stability, etc. The name sinegordon equation is a pun on the wellknown klein gordon equation in physics. Two limits of the discrete variants of these models are.
We focus, in particular, on two of their prototypical solutions, namely the kinklike heteroclinic connections and the timeperiodic, exponentially localized in space breather waveforms. Cambridge core structural geology, tectonics and geodynamics. A partial differential equation which appears in differential geometry and relativistic field theory. Mahmood, approximate solutions for solving the klein gordon and sine gordon equations, journal of the association of arab. This chapter computes a numerical solution by the method of lines mol, including detailed discussion of the matlab routines and the numerical and graphical output. The book is well designed for graduate students in physics, engineering, and applied mathematics sequences, and scientists and engineers whose projects require the solution of partial. The backlund transformation and sinegordon equations book.
The sinegordon equation is the theory of a massless scalar field in one space and one time dimension with interaction density proportional to cos, where. How to construct pseudospherical surfaces from sinegordon. Sinegordon equation an overview sciencedirect topics. Yet, the method cannot be used to solve timedependent partial differential equations that involve more than one partial derivative with respect to the temporal variable t when they are of the same order, as in the case of the fornbergwhitham type equations. Quantum sinegordon equation as the massive thirring model. Jun 16, 2014 specifically, a long josephsonjunction is described by the continuous sinegordon equation and a josephsonjunction parallel array by its discrete counterpart. Many basic features of such solitons can be studied in the sinegordon model in a simplified setup. We verify that it satisfies the sine gordon equation in11.
In the case of the sinegordon equation, ru obeys the linear wave. This leads to slightly more complicated behavior than the other equations though the pattern it generates still has a simple repetitive form. The sinegordon model is a toy model that helps to understand fundamental notions like duality in a simplified setup. Different from the previous symplectic methods dealing with pdes, our method is that to view the pdes as a hamilton system in banach space, then to apply the generating functions method to the hamilton system.
Due to my not being very skilled in differential geometry, i want to ask if there is a reference book, paper, etc. Wave motion occurs as driven by the potential term on the left, which is attributed to an external or internal origin. The first terms on their righthand sides correspond to particles of mass m and to particles of the ground field, respectively cf. The sinegordon equation and solitons sixtysec youtube. The sinegordon equation is the eulerlagrange equation for this lagrangian. In this paper, taking the sine gordon equation as an example, we present a new method to construct the symplectic schemes for hamilton pdes. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. Let us look for travelling wave solutions of the sinegordon equation. Pdf sinegordon equation and its application to tectonic stress. It is known that the obstruction to the asymptotic stability of the sine gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. In this post we talk about the sine gordon kink solution. History media compliance careers affiliate program. The riemannhilbert problem and inverse scattering siam.
Note that the largest range of u is at x0 where u varies between p and p. Booknonlinear partial differential equations in engineering and ap. The sinegordon equation in josephsonjunction arrays. Spatially extended relativistic particles out of traveling front. View elliptic solutions of the sine gordon equation. The lagrangians and equations of motion are derived for the junction phase differences for a family of coupled josephson junction devices. Here we are interested in large amplitude solutions of eq. Coupled longjosephson junctions and the n sinegordon equation. The purpose of this book is to present a summary of recent developments in this. Sep 18, 2020 then i will discuss the asymptotic stability of the sine gordon equation in weighted energy space. Numerical solution to the sinegordon equation defined on the. Oct, 2020 the differential transform method is a powerful analytical method that can solve nonlinear partial differential equations. A perturbed sinegordon equation is considered under the restrictions on the model parameters corresponding to the single equilibrium in the.
Radiation phenomena, resonances, pinning and how to avoid it. Nonlinear systems and complexity ser the sinegordon model. Semilinear evolution equations book chapter iopscience. Numerical solution of fractional sinegordon equation using.
On robust stability of sinegordon equation ieee conference. Nonlinear partial differential equations for scientists and. Based on nonlinear spectral analysis, exact nonreflecting boundary conditions are derived at two artificially introduced boundary points. Method for solving the sinegordon equation university of. In the early stages of the history of solitons kacmoody lie algebras appeared rather artificially. We formulate and discuss integrable analogue of the sine gordon equation on arbitrary time scales. This conservation law is called topological chargeof solution. The sinegordon equation on time scales sciencedirect. Homotopy perturbation and elzaki transform for solving sine gorden and kleingorden equations, iranian j. The sine gordon equation is the classical wave equation with a nonlinear sine source term.
In a static solution, it is localized around the origin in the xy plane. May 04, 2018 book search tips selecting this option will search all publications across the scitation. In this paper, by means of a proper orthogonal decomposition pod we mainly reduce the order of the classical cranknicolson finite difference ccnfd model for the fractionalorder parabolictype sine gordon equations foptsges. General background, physical motivations, inverse scattering, and solitons. 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. For general background and applications of b acklund transformation, we refer to 43,44.
That is am to purchase not understanding the view elliptic solutions of the sine gordon equation from fact to labourintensive europe to help more awardwinning to the plan. Numerical simulation of the solution to the sinegordon equation on the whole real axis is considered in this paper. The resulting kosterlitzthouless phase diagram is obtained. 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.
It is demonstrated that the sine gordon equation is applicable to the description of series of the. The sine gordon model is a ubiquitous model of mathematical physics with a wide. Despite the fact that the latter is a pure boson model, massless thirring fermions are equivalent to free bosons. Feb 16, 2012 the sinegordon model is discussed and analyzed within the framework of the renormalization group theory. Sine gordon equation the sine gordon equation is a nonlinear hyperbolic partialdifferential equation involving the dalembert operator and the sine of the unknown function. High order symplectic schemes for the sinegordon equation. World heritage encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Debnaths book comprehensively surveys methods for solving partial differential equations as well as a complete and thorough analysis of their solutions. Application of differential transform method to the sine. The fornbergwhitham equation solved by the differential. A perturbative renormalization group procedure is carried out through a decomposition of the sine gordon field in slow and fast modes. Positive numerical solutions of the dirichlet case, which are global minima of the variational problem, are computed on a dumbbellshaped 2d domain for visualization. The sinegordon model and its applications springerlink. Longtime dynamics of the sinegordon equation mathematics.
A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine gordon equation. This article is within the scope of wikiproject mathematics, a collaborative effort to improve the coverage of mathematics on wikipedia. Let us look for travelling wave solutions of the sine gordon equation. The sine gordon equation is the theory of a massless scalar field in one space and one time dimension with interaction density proportional to cos, where. Sinegordon equation neqwiki, the nonlinear equations. The initial value problem for the sinegordon equation is solved by the inversescattering method. 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.
Mar 25, 2014 in the present chapter, we consider two prototypical klein gordon models. 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. This phenomenon is more general in two dimensions and is called bosonization. This chapter constitute a revision of the physics of such superconducting systems in the light of the sinegordon equation.
Then a numerical scheme of second order is proposed to approximate the solution. The purpose of this book is to present a summary of recent developments in this field, incorporating both introductory background material, but also with a strong view towards modern applications, recent experiments, developments regarding the existence, stability, dynamics. The sinegordon model and its applications from pendula and. The sine gordon equation is a nonlinear hyperbolic partial differential equation in 1. Nonlocal nonlinear schrodinger equations and their soliton. We refer to the references therein for more discussion on the dynamical properties of the hyperbolic sine gordon equation. An optimized cranknicolson finite difference extrapolating. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sinegordon equation. Sinegordon equation project gutenberg selfpublishing. In spite of the long history the sine gordon equation and its numerous extensions and generalizations still attract attention. 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 discuss the bogomolnyi bound for this case and its non linear realization. The sine gordon 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.
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