Symplectic forms also appear as one tries to give a description of selfadjoint (also dissipative etc) boundary conditions for a differential operator. Already for ordinary differential operators, the standard Green-Lagrange formula defines a symplectic form on appropriate collections of boundary values of functions and their derivatives. Selfadjoint boundary value problems can be identified with Lagrangian linear manifolds. Modern theory of selfadjoint extensions is a far-reaching generalization of this observation and can be applied to some classes of partial differential equations. For an introduction see
M. L. Gorbachuk and V. I. Gorbachuk, Boundary value problems for operator differential equations. Dordrecht, Kluwer, 1991.
A review of recent results is given in the paper
J. Bruening, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, No. 1, 1-70 (2008).