Narrow Results

In this paper, symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrödinger Equation (NLSE) in case of dark soliton motion. Firstly, by Ablowitz–Ladik model (A–L model), the NLSE is discretized into a non-canonical Hamiltonian system. Then, different kinds of coordinate transformations can be used to standardize the non-canonical Hamiltonian system. Therefore, the symplectic schemes and symmetric schemes can be employed to simulate the solitons motion and test the preservation of the invariants of the A–L model and the conserved quantities approximations of the original NLSE. The numerical experiments show that symplectic schemes and symmetric schemes have similar simulation effect, and own significant superiority over non-symplectic and non-symmetric schemes in long-term tracking the motion of solitons, preserving the invariants and the approximations of conserved quantities. Moreover, it is obvious that coordinate transformations with more symmetry have a better simulation effect.

In this article, we mainly deal with the boundary value problem for triharmonic function with value in universal Clifford algebra: where (j = 1, … , 5) ∂Ω is a Liapunov surface in Rⁿ, the Dirac operator are unknown functions with values in an universal Clifford algebra Cl(V_n,n). Under some hypotheses, it is proved that the boundary value problem has a unique solution.

In this paper, we study on elastodynamic fracture problems and present a symplectic scheme, combined with the spectral boundary integral method, for solving anti-plane spontaneous rupture problems via constructing a nonautonomous Hamiltonian system by introducing independent variables in extended phase space. Slip-weakening friction law is considered with the approach. Numerical experiments are employed to test the effectiveness of symplectic scheme, and the comparisons of numerical results show that the present method is of higher accuracy compared to the non-symplectic method. The results show that the method is a efficient tool for simulating quasi-static and dynamic earthquake processes.