# The sine-Gordon equation may be derived from the Lagrangian: (1.2) L = Z dx 1 2 ˚2 t − 1 2 ˚2 x +cos˚−1 ; 2000 Mathematics Subject Classi cation. Primary: 35J20, 58E05; Secondary: 35Q20, 65N30. Key words and phrases. Elliptic sine-Gordon equation, variational methods, numerical computation.

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Under certain restrictions these solutions For other exact solutions of the sine-Gordon equation, see the nonlinear Klein–Gordon equation with f(w) =bsin(‚w). 5–. The sine-Gordon equation is integrated by the inverse scattering method. References Steuerwald, R., Uber enneper’sche Fl¨ achen und B¨ ¨acklund’sche Transformation, Abh. Bayer. Akad.

All points on the uv -plane are singular points. References [6-1] ªÊ± q°|Â ØqD u ü% | q E, Å:¶ 18 | Ç 2021-04-07 · The sine-Gordon equation also appears in a number of other physical applications (Barone 1971; Gibbon et al. 1979; Bishop and Schneider 1981; Davydov 1985; Infeld and Rowlands 2000, pp. 202 and 240), including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendula attached to a stretched wire, and dislocations in crystals. Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atoms Submitted by Emmanuel Lemoine on Wed, 10/29/2014 - 11:46 Titre Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atoms For other exact solutions of the sine-Gordon equation, see the nonlinear Klein–Gordon equation with f(w) =bsin(‚w). 5–.

## 2020-04-01 · Abstract. In this paper, a (2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are derived from the well-known AKNS system. Based on the Hirota bilinear method and Lie symmetry analysis, kink wave solutions and traveling wave solutions of the (2+1)-dimensional sine-Gordon equation are constructed.

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### 1. INTRODUCTION. Our goal in this paper is to derive, in terms of a triplet of constant matrices, explicit formulas for exact solutions to the sine-Gordon equation.

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References Steuerwald, R., Uber enneper’sche Fl¨ achen und B¨ ¨acklund’sche Transformation, Abh. Bayer. Akad.

(The derivation presented in.

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### Numerical solutions of the perturbed Sine-Gordon equation in two space variables, arising from a Josephson junction are presented. The method proposed arises from a two-step, one parameter method for the numerical solution of second-order ordinary differential equations. Though implicit in nature, the method is applied explicitly. Global extrapolation in both space and time is used to improve

(1.1) The initial value problem posed on the full line can be solved by the Inverse Scattering The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics: φ t t − φ x x + φ = 0. {\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi \ =0.\,} The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by (x, t) ↦ (x − α, t − β) (x, t) ↦ (x cosh θ − tsinh θ, tcosh θ − x sinh θ) It shares this relativistic invariance property with the linear Klein–Gordon equation, which is obtained upon replacing sin ϕ by ϕ. (The name sine-Gordon equation is derived from this relation, and was introduced by Kruskal.) Unperturbed sine-Gordon equation has exact solution: φ(x,t) = 4arctanexp ±√x−ut 1−u2 This is a solitary wave or soliton.

## The nonlinear sine-Gordon equation (SGE), a type of hyperbolic partial differential equation, is often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, such as nonlinear waves, propagation of fluxons and dislocation of metals [ 1 – 4 ].

References Steuerwald, R., Uber enneper’sche Fl¨ achen und B¨ ¨acklund’sche Transformation, Abh. Bayer.

This dispersion tends to 3 May 2013 In this section, we give the basic theory of the sine–Gordon equation (and To derive dynamical equations of the sine–Gordon chain (SGC), 8 Sep 2020 time asymptotics of the solutions to the sine-Gordon equation whose initial cubic NLS and by Jenkins-Liu-Perry-Sulem [39] for the derivative 1. INTRODUCTION. Our goal in this paper is to derive, in terms of a triplet of constant matrices, explicit formulas for exact solutions to the sine-Gordon equation. Using this technique, we derive an equation for kink solutions, which is a travelling wave. We also derive a formula for breather solutions, which behave as wave Title: Critical velocity in kink solutions of the sine-Gordon equation 2.3 Variational derivation of a Hamiltonian ODE approximation . .