Nonlinear dynamics on branched structures and networks

TL;DR

This paper investigates nonlinear Schrödinger equations on branched structures and networks, revealing how the nonlinearity power affects the boundedness of energy and the existence of ground states, with results differing from the line case.

Contribution

It provides a detailed analysis of the critical and supercritical nonlinearities on graphs, highlighting differences from the real line case and exploring ground state existence.

Findings

Critical nonlinearity power is six, affecting energy boundedness.

Ground states exist in subcritical cases but not necessarily on graphs.

Rich phenomenology in the critical case compared to the line.

Abstract

Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose-Einstein condensation. Whenever in a physical experiment a ramified structure is involved, it can prove useful to approximate such a structure by a metric graph, or network. For the Schroedinger equation it turns out that the sixth power in the nonlinear term of the energy is critical in the sense that below that power the constrained energy is lower bounded irrespectively of the value of the mass (subcritical case). On the other hand, if the nonlinearity power equals six, then the lower boundedness depends on the value of the mass: below a critical mass, the constrained energy is lower bounded, beyond it, it is not. For powers larger than six the constrained energy functional is never lower bounded, so that it is meaningless to speak about ground states (supercritical case). These results are the same as in the case of the nonlinear Schrodinger equation on the real line. In fact, as regards the existence of ground states, the results for systems on graphs differ, in general, from the ones for systems on the line even in the subcritical case: in the latter case, whenever the constrained energy is lower bounded there always exist ground states (the solitons, whose shape is explicitly known), whereas for graphs the existence of a ground state is not guaranteed. For the critical case, our results show a phenomenology much richer than the analogous on the line.