Modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales (Q956626)

The authors prove the existence of modulating pulse solutions to a wide class of quadratic quasilinear Klein-Gordon equations. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave-train; they are also referred to as `moving breathers' since they are time-periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with three stable, three unstable and infinitely many neutral directions. By transforming part of the equation into a normal form with an exponentially small remainder term and using a generalisation of local invariant-manifold theory to the quasilinear setting, the authors establish the existence of small-amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse.