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.