Modulating Pulse Solutions to Quadratic Quasilinear Wave Equations over Exponentially Long Length Scales

Abstract

This paper presents an existence proof for modulating pulse solutions to a wide class of quadratic quasilinear Klein-Gordon equations of the form $$\partial_t^2 u = \partial_x^2 u - u + f_1(u, \partial_x u, \partial_t u)\partial_x^2 u + f_2(u, \partial_x u, \partial_t u).$$ 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, we prove the existence of small-amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse.