Symplectic integration of Sine–Gordon type systems

Geometric Integrators for the Nonlinear Schrödinger Equation

“…The newly emerging class of multisymplectic integrators for nonlinear wave equations also proved extremely promising, although limits of applicability of this method are still to be precisely determined, as it includes simple and fast schemes with remarkable conservation properties for local as well as global invariants. As in [20], we begin by discretizing in space and apply an implicit s-stage RungeKutta scheme to the multisymplectic formulation of NLS (14) to obtain the spatial semidiscretization, This is conservation of symplecticity in time with respect to the state variables a = {a i,k } and b = {b i,k } and the wedge product da ∧Bdb, whereB is a diagonal matrix with entries {b i }.…”

“…Applying the centered cell discretization to (14), we obtain the following multisymplectic scheme for NLS:…”

“…One long standing approach to this problem (which has met with varying degrees of success) has been to introduce a Hamiltonian semi-discretization of the PDE using, for example, spectral methods or finite differencing and then applying a symplectic scheme to integrate the semi-discrete system in time; see e.g. [4,10,12,14,16,23]. The resulting numerical schemes preserve the symplectic structure of the semi-discretization but as the level of discretization is refined or the dimension of the system is increased, the advantage obtained using the symplectic integrator can occur only on a long enough timescale (see the numerical experiments in Section 5.1).…”

Chaotic behaviour of solutions to a perturbed Korteweg—de Vries equation

“…Our future work will focus on the development of kinetic description of chaotic behaviour for the obtained mapping [9], and with the derivation of new methods for the symplectic integration Figure 5 with ω = 30 of the KdV equation under hamiltonian perturbations following the framework of [13], where the same was done for sin-Gordon system.…”

Nonlinear Analysis of Surface EMG Signals

“…It can describe the system structure, especially nonlinear structure, very well. It has been used to study various nonlinear dynamical systems [50][51][52] since Feng Kang [53] has proposed a symplectic algorithm for solving symplectic differential. However, from the view of data analysis, few literatures have employed symplectic geometry theory to explore the dynamics of the system.…”