  In the first part of the article we study Hamiltonian diffeomorphisms of
$\mathbb{R}^{2n}$ which are generated by sub-quadratic Hamiltonians and prove a
middle dimensional rigidity result for the image of coisotropic cylinders. The
tools that we use are Viterbo's symplectic capacities and a series of
inequalities coming from their relation with symplectic reduction. In the
second part we consider the nonlinear string equation and treat it as an
infinite-dimensional Hamiltonian system. In this context we are able to apply
Kuksin's approximation by finite dimensional Hamiltonian flows and prove a PDE
version of the rigidity result for coisotropic cylinders. As a particular
example, this result can be applied to the Sine-Gordon equation.
