  We investigate sudden quenches across the critical point in the transverse
field Ising chain with a perturbing non-integrable next-nearest-neighbour
interaction. Expressions for the return (Loschmidt) amplitude and associated
rate function are derived to linear order in the next-nearest-neighbour
coupling. In the thermodynamic limit these quantities exhibit non-analytic
behaviour at a set of critical times, a phenomenon referred to as a dynamical
quantum phase transition. We quantify the effect of the integrability breaking
perturbation on the location and shape of these non-analyticities. Our results
agree with those of earlier numerical studies and offer further support for the
assertion that the dynamical quantum phase transitions exhibited by this model
are a generic feature of its post-quench dynamics and is robust with respect to
the inclusion of non-integrable perturbations.
