412 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
of the multisymplectic and Noether theorem structure directly from the variational
principle we have described above entirely on the Lagrangian side, J1 (Y).
This is then the beginnings of the development of the basic differential-
geometric formulation of the mutlisymplectic structure that can be used for a variety
of purposes, including Hamiltonian and Lagrangian PDE's. Using this framework,
Marsden, Patrick and Shkoller [1998] produce multisymplectic variational integra-
tors of Veselov type and test these multisymplectic integrators numerically. For
example, in a test involving 5000 collisions of kink solitons (see Figure 5 .2) for the
sine-Gordon equation on a circle, the method does very well indeed.
Kink-antikink collision0.
0Figure 5.2. Collision of a kink-antikink pair for the sine-Gordon equation. {
I