Multisymplectic geometry, covariant Hamiltonians, and water waves.

*(English)*Zbl 0922.58029This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by T. J. Bridges. The authors’ goal is to use a variant of the multisymplectic Hamiltonian formalism to generalize and make intrinsic the seminal and extremely important work of T. J. Bridges on wave propagation, periodic pattern formation and linear stability. Roughly speaking the main result states that in the case of \(n\) distinct and possibly unbounded directions, the \(n+1\) pre-symplectic 2-forms introduced by T. J. Bridges are actually contained in a single higher degree multisymplectic \((n+2)\) form and that, in the presence of symmetries, these many forms can be assembled into this single canonical form.

Reviewer: Messoud Efendiev (Berlin)

##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

35Q35 | PDEs in connection with fluid mechanics |