Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential

Rémi Carles (1) , Chunmei Su (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Yau Mathematical Science Center and Beijing Institute of Mathematical Sciences and Applications

We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.

Mots clés: Nonlinear Schrödinger equation; logarithmic nonlinearity; ground states; instability

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Rémi Carles, Chunmei Su. Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential. Communications in Partial Differential Equations, 2022, 47 (6), pp.1176-1192. ⟨10.1080/03605302.2022.2050257⟩. ⟨hal-03293209v2⟩