  We derive an asymptotic equation for quasi-static, nonlinear surface plasmons
propagating on a planar interface between isotropic media. The plasmons are
nondispersive with a constant linearized frequency that is independent of their
wavenumber. The spatial profile of a weakly nonlinear plasmon satisfies a
nonlocal, cubically nonlinear evolution equation that couples its left-moving
and right-moving Fourier components. We prove short-time existence of smooth
solutions of the asymptotic equation and describe its Hamiltonian structure. We
also prove global existence of weak solutions of a unidirectional reduction of
the asymptotic equation. Numerical solutions show that nonlinear effects can
lead to the strong spatial focusing of plasmons. Solutions of the
unidirectional equation appear to remain smooth when they focus, but it is
unclear whether or not focusing can lead to singularity formation in solutions
of the bidirectional equation.
