scholarly journals Discrete-Time Orthogonal Spline Collocation Method for One-Dimensional Sine-Gordon Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoquan Ding ◽  
Qing-Jiang Meng ◽  
Li-Ping Yin
Keyword(s):  
Discrete Time ◽  
Gordon Equation ◽  
Basis Functions ◽  
Spline Collocation ◽  
Time Level ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
Orthogonal Spline Collocation ◽  
Collocation Scheme ◽  
The One

We present a discrete-time orthogonal spline collocation scheme for the one-dimensional sine-Gordon equation. This scheme uses Hermite basis functions to approximate the solution throughout the spatial domain on each time level. The convergence rate with orderO(h4+τ2)inL2norm and stability of the scheme are proved. Numerical results are presented and compared with analytical solutions to confirm the accuracy of the presented scheme.

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.

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