Peter Polacik

University of Minnesota

Positive quasiperiodic solutions of scalar field equations: an overview and recent progress

For a class of semilinear elliptic equations on the entire space, we consider positive solutions which have a predetermined behavior (decay to zero or periodicity) in all but one variable. Our concern is the behavior of such solutions in the remaining variable. Specifically, with a formal Hamiltonian structure of the equation in mind, our goal in a joint project with Dario Valdebenito has been to establish the existence of quasiperiodic solutions. In the lecture, after briefly discussing general methods based on spatial dynamics and KAM-type results that we use for proving the existence, I will focus on more specific equations where the nonlinearity is given by a power or two combined powers. In particular, for the power nonlinearity I will state a theorem that we have proved and a more interesting “theorem” that I would like to see proved.