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

Traveling Wave Solutions of the Sine-Gordon and the Coupled Sine-Gordon Equations Using the Homotopy-Perturbation Method A. Sadighi1, D.D. Ganji1; and B. Ganjavi2 Abstract. In this research, the Homotopy-Perturbation Method (HPM) has been used for solving sine-Gordon and coupled sine-Gordon equations, which have a wide range of applications in

Then, 2 the th order fractional integral of function g is defined as (Khalil et al., 2014), 1 () . t ta a gx Igt dx x 3 2006-07-01 · Li and Chen studied bifurcations of travelling wave solutions of the following double Sine–Gordon equation (1.5) u xt = sin (u) + sin (2 u). In this paper, we consider the following general Sine–Gordon equation (1.6) u tt - u xx + α sin ( u ) + β sin ( 2 u ) = 0 , where α , β are constants and ( α , β ) ≠ (0, 0). equations (Tang, 2010), the solutions of the combined sine-cosine-Gordon equation were studied by the variable separated ODE method (Kuo, 2009). 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.

Sine gordon equation travelling wave solution

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The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. 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. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions 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).

Sine gordon equation travelling wave solution

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.

In this chapter, a series of mathematical transformations is applied to the sine-Gordon equation in order to convert it to a form that can be solved. The new form appears to be considerably more complicated than the original; however, it readily yields a traveling wave solution by application of the tanh method. For example, the travelling wave solutions of the (1+2)-dimensional Kadomtsev-Petviashvili II equation (KP II) are solitons, and those of the higher-dimensional Sine-Gordon equation are fronts. Still, localized structures, which emulate spatially extended particles, can be generated from such solutions in two or three space dimensions by a procedure that is a natural consequence of the In this paper, we use the generalized kudryashov method to seek the traveling wave solutions of the 2-dim sine Gordon and the double sine-Gordon and equations. travelling wave solutions for a more general sine-Gordon equation: = + sin ( ).

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dimensional sine-Gordon equation. 1. Introduction. In recent years, nonlinear It is denominated following its similar form to the Klein-Gordon equation. The equation, as well as several solution techniques, was known in the 19th century, but travelling wave moving with velocity c, and that the general solution to the equation utt − c2uxx is the NLS, and Sine-Gordon equation are also CPT- invariant.

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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2. Kink Waves Travelling wave solutions to the sine-Gordon equation for which the quantity c2 − 1 < 0 are called subluminal waves. When c2 − 1 > 0 they are called superluminal waves. We have the following theorem: Theorem 1. Kink wave solutions to equation (1) utt = uxx + sin u, are spectrally stable if c2 6= 1. 2.1. Subluminal kink waves.

2. The tanh method In this paper we will use the tanh method in its systematized form as pre- sented by Malﬂiet [1] and by Fan et al. [6].

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] 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

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]. Case(2) c0 = c2 2 4c4: New Travelling Wave Solutions for Sine-Gordon Equation 1. Introduction. The sine-Gordon equation appears in differential geometry and relativistic field theory. It is 2. The Proposed Method. Our method is based on two assumptions.