Looking for nice solutions of partial differential equations

Nonlinear waves are particular solutions of partial differential
equations which have a well-defined spatio-temporal structure. The
focus of this talk is on multidimensional traveling waves which
describe propagation phenomena with constant speed. We present an
approach to the existence of such waves which relies upon methods from
the theory of infinite-dimensional dynamical systems (center manifold
reduction, normal form theory). During the last decades this approach
led to significant progress in the understanding of nonlinear
waves. In particular, it allowed to rigorously construct various types
of multidimensional waves for different types of problems, as for instance,
three-dimensional water-waves (doubly periodic waves, periodically
modulated solitary waves), corner defects in interface propagation, or
defects in periodic patterns (dislocations, grain boundaries).