In this paper, we are interested to study the Sine-Gordon equation in generalized function theory, we give a result of existence and uniqueness of generalized solution with initial data are distributions (elements of the Colombeau algebra). Then we study the association concept with the classical solution.
This sine-Gordon model falls into a class of solid-on-solid (SOS) models, which display the BKT transition. In the context of the SOS models, this is a roughening transition. The relationship between roughening transitions in crystal facets and the XY universality class was first described in [7, 8]. The variable φ(x) is viewed as a height variable above a two-dimensional (2d) surface—the facet of a crystal. Hence above the critical t, where φ(x) becomes highly nonuniform, it is described as the rough surface that grows at high temperatures. The critical line is described by t c (g), where g then
numerical solutions. The finite difference (FD) scheme (see ) is one of the most valid numerical methods to solve the tow-dimensional (2D) nonlinear Sine-Gordon equation. However, the classical FD scheme for the 2D nonlinear Sine- Gordon equation is a macroscale system of equations including lots of unknowns so as to undertake very large calculating load in the real-life applications. Therefore, a vital problem is to lessen the unknowns (i.e., degrees of freedom) of the classical FD scheme so that it can save the computing time but its numerical solutions keeps sufficiently high accuracy.
Even though some nonlinear partial differential equations are integrable, it may be not easy to integrate them. Instead, a predicted solution with parameters are assumed to be a solution of governing equations and the relations among the parameters are investigated. The logic is simply based on the similarity with exponential-type solutions to the ordinary differential equations with constant co- efficients. These predicted solutions are of various forms covering exponential, hyperbolic, trigonometric or rational functions, and more. Parallel to the recent developments in computer algebra in the last four decades, a tendency has been observed to determine exact solutions to nonlinear PDEs by following the proce- dure that starts with a predicted solution. Recently, this tendency has focused on exact solutions to fractional nonlinear partial differential equations. Many of tech- niques implemented to nonlinear PDEs to find exact solutions have been adapted for fractional nonlinear PDES [1–10]. We also derive exact solutions to some conformable fractional equations in RLW-class modeling various wave phenomena both in nature or technology implementations. Different from previous studies, we adapt Sine-Gordon expansion approach to determine exact solutions to governing equations in fractional RLW-class.
C is the velocity of the solitary wave, and the boundary conditions are given according. The problem is solved for C = 0.5 , ∆ = t 0.125 , and ∆ = t 0.04 . This example was chosen to demonstrate that the EBNUM can be used to solve the Sine-Gordon equation. The results produced by the EBNUM are accurate as shown by the graphical evidence given in Figure 6.
In the present manuscript we provide a general expression (see theorem 4.1) of a differ- ent kind, which solves all the consistency requirements. It is very generic by construction and, roughly speaking, captures the vectorial nature of the form factors by means of “off-shell” Bethe ansatz states and the pole structure by particular contour integrals. We exemplify this general expression for the form factors of the Sine-Gordon model involving an odd number of states, which was hitherto unknown. For the even case similar expres- sions may be found in [17, 18]. We present a detailed analysis of the three particle form factor.
The goal of this paper is to find exact solutions to strongly perturbed sine-Gordon (SG) type equations. Recent works in the literature proposed ana- lytical and numerical solutions to this problem    . In this work, we find exact solutions by means of the Ansatz method. This research is a direct continuation of the research done in     and . The research done in those papers was primarily to find solutions to the sine-Gordon equa- tion and its variations under small, adiabatic perturbations. The solutions found in this paper are for strong perturbations to those same equations. In Section 2 we describe the Ansatz method, and in Section 3 we propose exact soliton solu- tions to sine-Gordon type equations and higher order dispersive versions. We conclude this work with a summary of our methods, the applicability of our re- sults, and possible avenues of future work.
This paper is structured as follows: Section 2 contains the basic definitions and the properties of the proposed method. Section 3 shows the theoretical approach of the proposed method on SineGordon equations. In Section 4, the Elzaki transform method and Adomian polynomial is applied to solve three problems in order to show its efficiency. Section 5 contains the discussion of results and the conclusion is presented in Section 6.
In this paper, we study a conservative diﬀerence scheme for the sine-Gordon equation (SGE) with the Riesz space fractional derivative. We rigorously establish the conservation property and solvability of the diﬀerence scheme. We discuss the stability and convergence of the diﬀerence scheme in the L ∞ norm. To reduce the computational complexity, we introduce a revised Newton method for implementing the diﬀerence scheme. Finally, we provide several numerical experiments to support the theoretical results.
energies, in region VI of Fig. 9a, although the mechanism is somewhat different than that for region III. This process is shown in Fig. 11k : after initially rebounding the antikink fails to escape the boundary and instead forms a breather upon colliding with the boundary a second time. This breather appears to collide with the boundary multiple times and may eventually escape the boundary, as in Fig. 11k, or fail to do so over the time we evolve the sine-Gordon equation. This behaviour can be traced to the phase dependence of breather/boundary collisions and is discussed further at the end of § 5. In this region we also often detect several very low energy breathers with frequency ω > 0.999.
The rest of the paper is as follows: In Section 2, Chebyshev wavelet and its properties are discussed. Operational matrix of derivative required for our subsequent develop- ment is presented in Section 3. Section 4 is devoted to present the Chebyshev wavelets spectral collocation method for solving Klein-Gordon and Sine-Gordon equations then approximate the unknown function. Section 5 deals with the illustrative examples and their solutions by the proposed approach compared with exact as well as with existing literature. Finally, concluding remarks are made in Section 6.
The equations of motion of this kind of theories for more general choices of the normal subgroup H were originally considered in the context of the, so called, reduced two- dimensional σ-models , although their Lagrangian formulation was not known until much later . The results of [3, 4] show that they fit quite naturally into the class of non- abelian affine Toda theories and, what is more important, that the condition of having a mass gap requires that H is either trivial or abelian. The simplest SSSG theories are the ubiquitous sine-Gordon field theory, which corresponds to G/G 0 = SU (2)/SO(2), and the
The sine-Gordon field theory and the associated massive Thirring model  are some of the best studied quantum field theories. In view of its connections to other important physical models, some of which in principle admit actual realizations in nature  , a huge mass of important exact results have been obtained for this fascinating integrable system -. However, no less fascinating are the remarkable mathematical and physical properties of its soliton (or “solitary wave”) solutions which have contributed, along the last decades, to turning the physics of solitons into a very active research topic.
For finite-gap solutions, for both sine-Gordon and KdV, it was noted that, for a given choice of distinct real branch points, there is a range of values of the defect parameter for which the phase shift becomes generically complex and the reality conditions for the field to the right of the defect cannot be satisfied. This appears to be a unique additional feature of the finite-gap solutions that has no analogue for soliton-defect scattering in either sine-Gordon or KdV. Here, only solutions for which the defect acts as a transition between a field to the left of a defect and its B¨acklund transformed partner field to the right have been considered. This fact was critical to being able to find solutions to the defect sewing equations at a point by using solutions that would in fact satisfy the B¨acklund transformation at every point. Therefore, the fact that these solutions also happen to solve the defect sewing equations might be viewed as incidental. However, it is not necessarily true that all solutions to the defect equations must have this property. For example, the bound state solution for the nonlinear Schr¨odinger equation in the presence of a type I defect [2, § 6] only solves the defect equations for x < x 0 (assuming
Abstract: We study the Sine-Gordon model with Minkowski signature in the framework of perturbative algebraic quantum field theory. We calculate the vertex operator algebra braiding property. We prove that in the finite regime of the model, the expectation value— with respect to the vacuum or a Hadamard state—of the Epstein Glaser S-matrix and the interacting current or the field respectively converge, both given as formal power series.
u with respect to the variable t; u xx is the two-order partial derivative of the u about the independent variable x. Subsequently, Zhu  considered the following problem: u tt − α u t − u xx + λ g ( sin u ) = f x t ( ) , (where Ω is a bounded domain of R 3 ) and he proved the existence of the global solution of the equation. For more research on the global solutions and global attractors of Kirchhoff and sine-Gordon equations, we refer the reader to -.
conditions on the fields obtained in this way were recognized as B¨ acklund transformations frozen at the location of the defect. This approach triggered a strong activity in the analysis of the defect in integrable classical field theories. The observation on frozen B¨ acklund transformations was fully exploited in  in conjunction with the Lax pair formulation of the general AKNS approach  to obtain a generating function of the entire set of modified conserved quantities. This settled the question of integrability in the sense of having an infinite number of local conserved charges. The approach also allowed to answer some questions left open in the Lagrangian formulation, like the formulation of the defect conditions directly in terms of the fields of the theory for models like KdV. The question of Liouville integrability, however, remained opened. The direct use of Poisson brackets to implement the method of the classical r-matrix is hindered by divergences due to the localized defect. Some regularization is needed. The sine-Gordon model was first studied in . Later on, a very nice series of papers tackled the question systematically for several models [16, 17, 18]. The procedure is based on the a priori assumption that the defect matrix satisfies appropriate Poisson bracket relations. A careful regularization then yields the so-called ”sewing conditions” between the fields in the bulk and those contained in the defect matrix. The consistency of the approach must then be checked a posteriori.
The paper is organised as follows. In Section 2, the two Poisson brackets at the basis the multi- symplectic formalism are introduced for the sine-Gordon model and it is shown that two completely equivalent Hamiltonian descriptions exist for this model. Conservation laws and the classical r ma- trix approach are discussed in the light of the two Poisson brackets, each one corresponding to an independent variable (called space and time variable). In Section 3, we review the sine-Gordon model with a defect from the Lagrangian and Lax pair points of view. We then go on to show how the new formalism allows to prove Liouville integrability of the model with defect directly, with no resort to sewing conditions. The key point is the fact that the defect conditions are reinterpreted as a canonical transformation with respect to the new Poisson structure. It turns out that the gener- ating functional for this canonical transformation is directly built on the defect Lagrangian density originally derived to study problems with defects. The new classical r matrix approach with defect is also derived there and the new monodromy matrix containing the generating function for the conserved quantities in involution is exhibited. Conclusions and outlooks are gathered in Section 4
The nonlinear one-dimensional sine-Gordon (SG) equation came into sight in the dif- ferential geometry and attracted a lot of attention because of the collisional behaviors of solitons that arise from this equation. Numerical solutions of the SG equation have been widely investigated in recent years [–]. Compact ﬁnite diﬀerence and diagonally implicit Runge-Kutta-Nyström (DIRKN) methods were used . The authors of  introduced a numerical method for solving the SG equation by using collocation and radial basis func- tions. The boundary integral equation technique is presented in . Bratsos [, ] has researhed a numerical technique for solving the one-dimensional SG equation and a third- order numerical technique for the two-dimensional SG equation. A numerical technique using radial basis functions for the solution of the two-dimensional SG equation has been shown in . Some authors advised spectral techniques and Fourier pseudospectral tech- nique for solving nonlinear wave equation taking a discrete Fourier series and Chebyshev orthogonal polynomials [–]. Ma and Wu  used a meshless technique by using a multiquadric (MQ) quasi-interpolation. In this paper, we investigate the one-dimensional nonlinear sine-Gordon equation
In  a certain physical picture for the quantum field theory of the Homogeneous Sine-Gordon models (HSG)  was extracted from a thermodynamic Bethe ansatz analysis. The central aim of this manuscript is to inspect the picture for consistency by means of the form factor approach [3, 4].