The parametrically modified sine-Gordon equation: spectral properties, collisions, resonances and their connection to integrability

In this work, we revisit kinks and their interactions in the parametrically modified sine-Gordon (pmsG) equation. The central motivating question of our work is the following: what is the mechanism which in the absence of internal modes generates phonon radiation and causes the inelasticity of collisions in a near-integrable system? We do not give a complete answer to this question but rather we study a case example where we examine the penetration of an internal mode into the phonon band, and identify its consequences in the spectrum, the shape of the eigenfunctions and resonances of the spectrum with an external ac drive. In particular, by studying the linear spectrum around the kink solution we show that in the absence of the internal mode, some odd phonon modes can act as a localized wavepacket of modes depending on the parameter r of the potential. We also show by using the phenomenological theory developed in [Physica D 9 (1983) 1] that these modes are related to the quasi-resonance process and bound states suggested in [Physica D 9 (1983) 33], hence during the collision process the energy of the translational modes goes directly to the “localized” phonons. Finally, we investigate another problem for which these “localized” phonon modes also act as internal modes do. In particular, we study the resonance phenomena (related with either the internal mode or with the phonon modes) that appear in this system when it is subject to external ac drive. We attempt to connect such properties of the phonons (as well as properties of the internal modes) to the non-integrability of the model for r≠0.