A Multisymplectic Variational Framework for the Nonlinear Elastic Wave Equation

A multisymplectic variational internal energy corresponding equation, its associated local framework for the nonlinear elastic wave equation is presented. The modified to the approximate nonlinea.r elastic wave equation is derived, we obtain the energy and momentum conservation laws as well as the multisymplectic form simultaneously directly from the variational principle     

Euler-Lagrange Forms and Cohomology Groups on Jet Bundles

Using the language of jet bundles, we generalize the definitions of Euler-Lagrange one-form and the associated cohomology which were introduced by Guo et al. [Commun. Theor. Phys. 37 (2002) 1]. Continuous and discrete Lagrange mechanics and field theory are presented. Higher order Euler-Lagrange cohomology groups are also introduced.     

Multisymplectic Hamiltonian Formulation for a One-Way Seismic Wave Equation of High-order Approximation

Based on the Lagrangian density and covariant Legendre transform, we obtain the multisymplectic Hamiltonian formulation for a one-way seismic wave equation of high-order approximation. This formulation provides a new perspective for studying the one-way seismic wave equation. A multisymplectic integrator is also derived.     

Two Kinds of Square-Conservative Integrators for Nonlinear Evolution Equations

Based on the Lie-group and Gauss-Legendre methods, two kinds of square-conservative integrators for square- conservative nonlinear evolution equations are presented. Lie-group based square-conservative integrators are linearly implicit, while Gauss-Legendre based square-conservative integrators are nonlinearly implicit and iterative schemes are needed to solve the corresponding integrators. These two kinds of integrators provide natural candidates for simulating square-conservative nonlinear evolution equations in the sense that these integrators not only preserve the square-conservative properties of the continuous equations but also are nonlinearly stable. Numerical experiments are performed to test the presented integrators.     

射线追踪、辛几何算法与波场的数值模拟

论述了射线追踪、辛几何算法与波场的数值模拟之间的关系,说明了辛几何算法长时间守恒性质及运用辛几何算法进行射线追踪的必要性.对线性层状模型,利用辛几何算法和Maslov渐近理论对波场进行了数值模拟,并与解析解进行了比较.     

Multisymplectic Geometry and Its Appiications for the Schrodinger Equation in Quantum Mechanics

Multisymplectic geometry for the Schrodinger equation in quantum mechanics is presented. This formalism of multisymplectic geometry provides a concise and complete introduction to the Schrodinger equation. The Schrodinger equation, its associated energy and momentum evolution equations, and the multisymplectic form are derived directly from the variational principle. Some applications are also explored.