# Apr 21, 2020 Traveling wave solutions of the sine-Gordon equation are of the form its corresponding solution χ to the Lax equations (3.1) and (3.2) with a

Time-saving video that explains how to find the equation of a sine or cosine wave given a graph. Examples show different transformations and apply a data set

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]. Various extension forms of the tanh method have been developed. First a power series in tanh was used as an ansatz to obtain analytical solutions of traveling wave type of certain nonlinear evolution equations. The double sine-Gordon equation, (2 + 1) dimensional sine-Gordon equation, and the coupled Schro- dinger–KdV equation were handled by using the generalized tanh method. 2.

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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 Some exact travelling wave solution of the discrete sine-Gordon equation are obtained in terms of hyperbolic function approach. The (+) branch of the kink-type solitary wave solution (25) at time in the form of traveling wave φ(x,t)=U(θ), θ=x−c0t. Then the sine-Gordon equation will take the form (c02−1)Uθθ+sinU=0.

In the paper, we ﬁrst make the travelling wave In this article, we have applied the Sine-Gordon expansion method for calculating new travelling wave solutions to the potential-YTSF equation of dimension (3+1) and the reaction-diffusion equation. We have found these solutions of the equation in the trigonometric, complex and hyperbolic function forms.

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The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift. Travelling wave solutions of the sine-Gordon equation are written in the form u(x, t) = f(x − ct), where c is the wave speed and f (x): R → R is the wave profile satisfying the following differential equation: Using the methods of dynamical systems for the (n + 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions are obtained.

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O. E. Martinez, J. P. Gordon and R. L. Fork.

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 ! R. Substituting (2.3) into (2.1) creates an Ordinary Diﬀerential Equation (ODE) reduction of the sine-Gordon equation …
but solving nonlinear equations is still an important task -[27][30]. In this paper, with the aid of Mathematica, a traveling wave solution for a class of Kaup-Kupershmidt Equation, 2 53 2 25 5 2 u u uu u u u u t x x xx x=++ +. In order to obtain the soliton solution of (1), I will use the modified sine-cosine to develop traveling wave so
, only the ﬁrst solution is meaningful. Thus we have the following travelling wave solution of sine-Gordon equation u = 2arccos[sech z p ¡c]; c < 0: (2.9) Using the identity tan2 u 4 = 1¡cos u 2 1+cos u 2; (2.10) we may write the solution (2.9) in the form u = 4arctan[exp(z p ¡c)]¡…: (2.11) The solutions (2.11) was also given in [7].

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Solution : Let us compare the given equation Feb 8, 2010 We have already seen such a wave as a solution of the constant coefficient transport equation ut + cux = 0, it can, however, appear in many other String Wave Solutions · Traveling Wave Solution for String · String Traveling Wave Velocity · Traveling Wave Parameters · Plane Wave Expressions.

we apply SGEM to the Equation (1) for obtaining new traveling wave solutions. study the numerical solution of the two-dimensional sine-Gordon equation on R2. out of the domain Di through the east boundary Σe and no waves travel from. PACS numbers: 02.30.Jr, 05.45.Yv, 02.30.Ik. Key words: discrete sine-Gordon equation, exact travelling wave solution, extended tanh-function approach.

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### 2020-06-17 · We derive exact solutions to the sine--Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) which decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution

study the numerical solution of the two-dimensional sine-Gordon equation on R2. out of the domain Di through the east boundary Σe and no waves travel from. PACS numbers: 02.30.Jr, 05.45.Yv, 02.30.Ik. Key words: discrete sine-Gordon equation, exact travelling wave solution, extended tanh-function approach. Feb 11, 2020 traveling wave solutions of the (2+1)-dimensional sinh-Gordon equation can also be provided in a Leibbrandt [12] studied solutions of the sine-Gordon equation in which gives us the following single kink wave solut 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, Sine-Gordon equations by using a reliable analytical method called New Travelling Wave Solutions for Time-Space Fractional the following solution sets . 26.