Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs

TL;DR

This paper studies the existence of ground states for the nonlinear Schrödinger equation on doubly periodic metric graphs, revealing a dimensional crossover phenomenon characterized by a continuum of critical exponents between 4 and 6.

Contribution

It introduces a new family of Gagliardo-Nirenberg inequalities explaining the continuum of critical exponents and the dimensional crossover in the NLS on metric graphs.

Findings

Ground states exist for all masses when nonlinearity power is below 4.

A threshold mass exists for powers between 4 and 6, indicating a criticality transition.

The crossover is due to coexistence of 1D and 2D Sobolev inequalities.

Abstract

We investigate the existence of ground states for the focusing nonlinear Schroedinger equation on a prototypical doubly periodic metric graph. When the nonlinearity power is below 4, ground states exist for every value of the mass, while, for every nonlinearity power between 4 (included) and 6 (excluded), a mark of $L^2$-criticality arises, as ground states exist if and only if the mass exceeds a threshold value that depends on the power. This phenomenon can be interpreted as a continuous transition from a two-dimensional regime, for which the only critical power is 4, to a one-dimensional behavior, in which criticality corresponds to the power 6. We show that such a dimensional crossover is rooted in the coexistence of one-dimensional and two-dimensional Sobolev inequalities, leading to a new family of Gagliardo-Nirenberg inequalities that account for this continuum of critical exponents.