Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems

Abstract

While it is known that Hamiltonian systems may undergo a phenomenon of condensation akin to Bose-Einstein condensation, not all the manifestations of this phenomenon have been uncovered yet. In this work, we present a novel form of condensation in conservative Hamiltonian systems, which happens through coherent states and exploits the discreteness of our system. Both features markedly differ from well-known condensation processes in the literature. Our result is based on a deterministic approach to obtain exact explicit solutions representing the dynamical formation of condensates in finite time. We reveal a dual-cascade behavior during the process, featuring inverse and direct transfer of conserved quantities across the spectrum. The direct cascade yields the excitation of high modes in finite time, a phenomenon quantified through the blow-up of Sobolev norms. We provide a fully analytic description of all the processes involved.