This paper focuses on studying the symmetry of a practical wave equation on new lattices.It is a new step in that the new lattice equation is applied to reduce the discrete problem of motion of an elastic thin homogeneous bar.The equation of motion of the bar can be changed into a discrete wave equation.With the new lattice equation,the translational and scaling invariant,not only is the infinitesimal transformation given,but the symmetry and Lie algebras are also calculated.We also give a new form of invariant called the ratio invariant,which can reduce the process of the computing invariant with the characteristic equation.
我们证明那是形式 m &summation 的任何线性多步方法 G<sub>1</sub><sup>τ</sup> ；k=0 α<sub>k</sub>Z<sub>k</sub>=τ
m &summation；有奇怪的顺序 u 的 k=0 β<sub>k</sub>J<sup>-1</sup>λ
H (Z<sub>k</sub>)(u ≥
3 ) 不能是结合到一个 symplectic 方法(顺序 w 的 G<sub>2</sub><sup>τ</sup>(w ≥
u ) 经由形式 m &summation 的任何概括线性多步方法 G<sub>3</sub><sup>τ</sup> ；k=0 α<sub>k</sub>Z<sub>k</sub> m &summation；k=0 β<sub>k</sub>J<sup>-1</sup>λ
H (m &summation；l=0 γ<sub>kl</sub>Z<sub>l</sub>) 。我们也给一个必要条件让这种概括线性多步方法是 conjugate-symplectic。当 G<sub>3</sub><sup>τ</sup> 是一个更一般的操作符时，我们也证明这些结果调用容易被扩大到盒子。
Based on the property of the discrete model entirely inheriting the symmetry of the continuous system,we present a method to construct exact solutions with continuous groups of transformations in discrete nonconservative systems.The Noether's identity of the discrete nonconservative system is obtained.The symmetric discrete Lagrangian and symmetric discrete nonconservative forces are defined for the system.Generalized quasi-extremal equations of discrete nonconservative systems are presented.Discrete conserved quantities are derived with symmetries associated with the continuous system.We have also found that the existence of the one-parameter symmetry group provides a reduction to a conserved quantity;but the existence of a two-parameter symmetry group makes it possible to obtain an exact solution for a discrete nonconservative system.Several examples are discussed to illustrate these results.
FU JingLi1,LI XiaoWei2,LI ChaoRong1,ZHAO WeiJia3 & CHEN BenYong4 1 Institute of Mathematical Physics,Zhejiang Sci-Tech University,Hangzhou 310018,China
By performing a particular spatial discretization to the nonlinear Schrodinger equation(NLSE),we obtain a non-integrable Hamiltonian system which can be decomposed into three integrable parts(L-L-N splitting).We integrate each part by calculating its phase flow,and develop explicit symplectic integrators of different orders for the original Hamiltonian by composing the phase flows.A 2nd-order reversible constructed symplectic scheme is employed to simulate solitons motion and invariants behavior of the NLSE.The simulation results are compared with a 3rd-order non-symplectic implicit Runge-Kutta method,and the convergence of the formal energy of this symplectic integrator is also verified.The numerical results indicate that the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical tool for solving the NLSE.
