Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

Families of Periodic Solutions for Some Hamiltonian PDEs

Abstract. We consider the nonlinear wave equation utt − uxx = ±u and the beam equation utt+uxxxx = ±u on an interval. Numerical observations indicate that time-periodic solutions for these equations are organized into structures that resemble branches and seem to undergo bifurcations. Besides describing our observations, we prove the existence of time-periodic solutions for various periods (a s...

Hamiltonian PDEs: deformations, integrability, solutions

We review recent classification results on the theory of systems of nonlinear Hamiltonian partial differential equations with one spatial dimension, including a perturbative approach to the integrability theory of such systems, and discuss universality conjectures describing critical behaviour of solutions to such systems. PACS numbers: 02.30.Ik, 02.30.Jr

Stability of periodic waves in Hamiltonian PDEs

Partial differential equations endowed with a Hamiltonian structure, like the Korteweg–de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectr...

Dual Curves and Pseudoholomorphic Curves

A notion of dual curve for pseudoholomorphic curves in 4–manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with, and yields many analogues of results in complex surface theory, using a description of the local geometry via Cartan’s method of equivalence. Duality then un...

Families of Periodic Solutions of Resonant PDEs