A fourth-order AVF method for the numerical integration of sine-Gordon equation

Numerical inverse scattering for the sine-Gordon equation

“…Breather: z = ± exp(i(π/2 ± (π/2 − arcsin(γ(µ − 2p )))) with p ∈ Z + such that (γ(µ) − 1)/2 < n ≤ γ(µ)/2 , four points on the unit circle that are symmetric with respect to both the real and imaginary axis. Figure 6 shows the distribution of the eigenvalues in the upper-half z-plane and the spectral convergence of the difference between the numerical results using Hill's method and the zeros of the known formula (19).…”

“…The SG equation is a nonlinear partial differential equation which appears in differential geometry and various applications such as superconductivity and Josephson junctions [6]. Many numerical methods have been developed to solve the SG equation [17,19,24]. Using these methods, or other more traditional but less specialized methods, it is hard to obtain the solution accurately, especially for long time [3,4].…”

Line Integral Solution of Hamiltonian PDEs

“…The novelty in their use stems from the fact that they provide effective and arbitrarily high-order energy-conserving methods for the time integration of the Hamiltonian semi-discrete problems obtained from Hamiltonian PDEs. In fact, low order methods have been mainly considered for this purpose, so far (see, e.g., [41,50,51,59,61,64]). Further approaches can be found in [43-49, 52, 63].…”

Legendre spectral element method for solving sine-Gordon equation

“…Yousif an Mahmood [22] used the variational homotopy perturbation method for solving the Klein-Gordon and sine-Gordon equations. In [23] a new scheme, which has energy-preserving property, is proposed for solving the sine-Gordon equation with periodic boundary conditions. This method is obtained by the Fourier pseudo-spectral method and the fourth-order average vector field method.…”