On the group of invariance of the (one-dimensional) sine-Gordon equation

1978 ◽  
Vol 22 (1) ◽  
pp. 17-20 ◽  
Author(s):  
B. Leroy
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
The One

scholarly journals Parametrically Driven One-Dimensional Sine-Gordon Equation and Chaos

1988 ◽  
Vol 43 (8-9) ◽  
pp. 727-733
Author(s):  
B. M. Herbst ◽  
W.-H. Steeb
Keyword(s):  
Driving Force ◽  
Gordon Equation ◽  
Zero Mode ◽  
Space Structure ◽  
Chaotic Behaviour ◽  
Force Frequency ◽  
Sine Gordon Equation ◽  
Mode Structure ◽  
One Dimensional ◽  
The One

AbstractThe chaotic behaviour of the parametrically driven one-dimensional sine-Gordon equation with periodic boundary conditions is studied. The initial condition is u(x, 0) = ƒ(x), ut (x, 0) = 0 where ƒ is the breather solution of the one-dimensional sine-Gordon equation at t = 0. We vary the amplitude of the driving force, the frequency of the driving force and the damping constant. For appropriate values of the driving force, frequency and damping constant chaotic behaviour with respect to the time-evolution of w(x = fixed, t) can be found. The space structure u(t = fixed, x) changes with increasing driving force from a zero mode structure to a breather-like structure consisting of a few modes.

scholarly journals Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

2022 ◽  
Vol 2022 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse
Keyword(s):  
Iterative Method ◽  
Gordon Equation ◽  
Approximate Analytical Solution ◽  
Test Problems ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
Nonlinear Part ◽  
Sumudu Transform ◽  
The One ◽  
The Given

In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

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