Multi-symplectic structures and wave propagation
Multi-symplectic structures and wave propagation
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DOI：
10.1017/s0305004196001429
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发表时间：
1997-01
影响因子：
0.8
通讯作者：
T. Bridges
中科院分区：
文献类型：
--
作者：
T. Bridges
通过为一个或多个空间维上的哈密顿发展方程的每个无界空间方向和时间分配一个不同的辛算子，提出了一种哈密顿结构，它推广了经典的哈密顿结构。这种称为多辛结构的推广对于色散波传播问题来说是很自然的。应用多辛结构框架的抽象性质，得到了时空周期态的一个新的变分原理，它使人想起不变环面的变分原理，作用量和作用量概念的几何改写，Whitham调制方程所预言的不稳定性判据的严格证明，Noether理论的新辛分解，时空可逆性概念的推广，以及Lighthill关于一维周期波不稳定性的几何判据的证明。例如，非线性薛定谔方程和水波问题被描述为多辛结构上的哈密顿系统。此外，还讨论了具有理论意义和实际意义的广义辛结构的进一步分支。
A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian evolution equation on one or more space dimensions. This generalization, called multi-symplectic structures, is shown to be natural for dispersive wave propagation problems. Application of the abstract properties of the multi-symplectic structures framework leads to a new variational principle for space-time periodic states reminiscent of the variational principle for invariant tori, a geometric reformulation of the concepts of action and action flux, a rigorous proof of the instability criterion predicted by the Whitham modulation equations, a new symplectic decomposition of the Noether theory, generalization of the concept of reversibility to space-time and a proof of Lighthill's geometric criterion for instability of periodic waves travelling in one space dimension. The nonlinear Schrödinger equation and the water-wave problem are characterized as Hamiltonian systems on a multi-symplectic structure for example. Further ramifications of the generalized symplectic structure of theoretical and practical interest are also discussed.