# This paper applies the sine-Gordon expansion method to the extended nonlinear solution was subsequently derived from that transformed bilinear equation. we apply SGEM to the Equation (1) for obtaining new traveling wave solutions.

Traveling wave solutions of the Camassa-Holm and Korteweg-de Vries for the elliptic sine-Gordon and the elliptic Ernst equations2020Ingår i: Journal of

In the case of rotational waves, the exact solution obtaining traveling wave solutions of nonlinear partial diﬀerential equa-tions. Applying this, exact traveling wave solutions for the coupled Sine-Gordon equations are constructed. Mathematics Subject Classiﬁcation: 35Q58; 37K50 Keywords: Coupled Sine-Gordon equations; Hyperbolic auxiliary func- wave equation φtt - φxx = φ(2φ2 - 1) has a family of solitary–wave solutions We show that sine–Gordon traveling waves can give us new insights even in such long–time Each traveling soliton solution of the SGE has the corresponding May 28, 2015 The (1+1)-dimensional Sine-Gordon equation passes integrability In (1+2) dimensions, each multi-front solution propagates rigidly at one velocity. the construction of travelling-wave solutions in (1+2) or (1+3) di rapidly to accurate solutions. Keywords: Sine-Gordon equation; Coupled sine- Gordon equation; Homotopy-perturbation method;. Traveling wave solution. Keywords: Coupled Sine-Gordon equations; Hyperbolic auxiliary func- tion; Travelling wave solution; Exact solution; Solitary wave solution.

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It is 2. The Proposed Method. Our method is based on two assumptions. Then, the main steps are as follows. Consider 3. The travelling wave solutions for a more general sine-Gordon equation: = + sin ( ). ( ) In this paper, a method will be employed to derive a set of exact travelling wave solutions with a JacobiAmplitude function form which has been employed to the Dodd-Bullough equation and some new travelling wave solutions have been derived [ ].

II The sine-Gordon Equation ] also presents some exact travelling wave solutions for a more general sine-Gordon equation: In this paper, a method will be employed to derive a set of exact travelling wave solutions with a JacobiAmplitude function form which has been employed to the Dodd-Bullough equation and some new travelling wave solutions have been derived [ 22 And [ ]alsopresentssomeexact travelling wave solutions for a more general sine-Gordon equation: = + sin ( ). of the sine-Gordon equation when the underlying wave is a travelling wave. This is related to the work done in [DDvGV03], where stability of a singularly perturbed subluminal kink wave solution was shown.

## The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S , 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925

Using various geometric techniques as well as some elementary methods from ODE theory, we find that the point spectrum of such an operator is purely imaginary provided the wave speed c of the travelling wave is not ±1. In this article, we construct the traveling wave and elliptic function solutions of some special nonlinear evolution equations which are arising in mathematical physics, solid-state physics, fluid flow, fluid dynamics, nonlinear optics, electromagnetic waves, quantum field theory etc.

### derivative term in the sine- Gordon equation allows one to generate the same Gordon equation does not possess exact bell-shaped traveling wave solution [1,

In [6], a generalized tanh method for obtaining multiple traveling wave solutions has been developed, where the solution of Riccati equation is used to replace the hyperbolic tan function in the tanh method. In [5], an extension to wider classes of evolution equations has been ex- amined.

Math. Comput. 215 (2009). 3777-3781 ] are analyzed. We have observed that fourteen solutions by Li from thirty do not satisfy the equation. The other sixteen exact solutions
Using a complete discrimination system for polynomials and elementary integral method, we obtain the travelling solutions for triple sine-Gordon equation. This method can be applied to similar problems and has general meaning.

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2.1. The sine-Gordon equation has conserved quantity E1=12π∫−∞+∞φxdx which equals integer number. This conservation law is called topological chargeof solution φ(x,t). In Section 3, the sine-Gordon expansion method is implemented to produce the exact traveling wave solutions of the Tzitzeica type equations in nonlinear optics; and finally, a short conclusion is provided in Section 4. 2.

Then the sine-Gordon equation will take the form (c 0 2 − 1) U θ θ + sin U = 0. Multyplying the latter equation by U θ and integrating with respect to θ one
We investigate the spectrum of the linear operator coming from the sine-Gordon equation linearized about a travelling kink-wave solution. Using various geometric techniques as well as some elementary methods from ODE theory, we find that the point spectrum of such an operator is purely imaginary provided the wave speed c of the travelling wave
For the (n + 1)-dimensional sine- and sinh-Gordon equations, by using the approach of dynamical systems to a class of travelling wave solutions, in 21 different regions of a five-parameter space
2.2. The tanh method The tanh method is a powerful solution method for the computation of ex- act traveling wave solutions [16–18].

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### May 23, 2020 (d) waht is the maximum transverse speed of an element of the string? check- circle. Text Solution. Solution : Let us compare the given equation

215 (2009). 3777-3781 ] are analyzed. We have observed that fourteen solutions by Li from thirty do not satisfy the equation. The other sixteen exact solutions 2020-09-24 2004-12-01 In this article, we construct the traveling wave and elliptic function solutions of some special nonlinear evolution equations which are arising in mathematical physics, solid-state physics, fluid flow, fluid dynamics, nonlinear optics, electromagnetic waves, quantum field theory etc.

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### Under the assumption that u ' is a function form of e inu , this paper presents a new set of traveling-wave solutions with JacobiAmplitude function for the generalized form of the double Sine-Gordon equation u tt = ku xx + 2 α sin ( nu ) + β sin ( 2 nu ) .

Mathematics Subject Classiﬁcation: 35Q58; 37K50 Keywords: Coupled Sine-Gordon equations; Hyperbolic auxiliary func- 2007-07-01 As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach. Discover the world's research 20 2020-05-18 Abstract. Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. 2008-12-01 Gordon equation with quintic nonlinearity can be thought of as an approximate version of the sine-Gordon equation if one relates the nonlinearity to the Taylor series expansion of the sine function. Wazwaz [11] used the tanh method to construct traveling wave solutions for the Klein–Gordon equation with quintic nonlinearity. 2005-12-01 For the (n + 1)-dimensional sine- and sinh-Gordon equations, by using the approach of dynamical systems to a class of travelling wave solutions, in 21 different regions of a five-parameter space 2020-06-17 New Doubly Periodic Solutions of (2+1)-Dimensional Nonlinear Wave Equations via the Generalized Sine-Gordon Equation Expansion Method Zhenya Yan∗ Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of … 2.2 Traveling Wave Reduction Traveling wave solutions of the sine-Gordon equation are of the form u(x;t) = f(x ct); (2.3) where cis a real valued constant often refered to as the wave speed and f: R !

## 2007-07-01

Double Sine–Gordon equation JacobiAmplitude Traveling wave solution Implicit solution abstract Under the assumption that u0 is a function form of einu, this paper presents a new set of traveling-wave solutions with JacobiAmplitude function for the generalized form of the double Sine–Gordon equation u tt ¼ ku xx þ2asinðnuÞþbsinð2nuÞ 2015-05-28 · Although the Sine-Gordon equation was derived for quite a few physical systems that have nothing to do with Special Relativity, the equation itself emerges as a non-linear relativistic wave equation. This is why in later years it has found applications in theoretical High-Energy Physics (e.g., in Relativistic Quantum Field Theory, and, in recent years, in String Theory). Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation, Appl. Math.

For a travelling kink wave solution of speed $c eq \pm 1$, the wave is spectrally stable.