One-dimensional versions of three-dimensional system: Ground states for the NLS on the spatial grid.
Riccardo Adami, Simone Dovetta

Abstract. We investigate the existence of ground states for the focusing Nonlinear Schrödinger Equation on the infinite three-dimensional cubic grid. We extend the result found for the analogous two-dimensional grid by proving an appropriate Sobolev inequality giving rise to a family of critical Gagliardo-Nirenberg inequalities that hold for every nonlinearity power from 10/3 to 6, namely, from the L²-critical power for the same problem in R³ to the critical power for the same problem in R. Given the Gagliardo-Nirenberg inequality, the problem of the existence of ground state can be treated as already done for the two-dimensional grid.

Rend. Mat. Appl. (7) 39 (2018) 181-194; pdf