R. Carles, Q. Chaulleur, G. Ferriere, and D. E. Pelinovsky

On the ground state of the nonlinear Schrodinger equation: asymptotic behavior at the endpoint powers

Abstract:
We consider the ground states of the nonlinear Schrodinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity, up to some rescaling to infer non-trivial limits. One case corresponds to the limit towards a Gaussian function called Gausson, which is the ground state of the stationary logarithmic Schrodinger equation. The other case, for dimension at least three, corresponds to the limit towards the Aubin-Talenti algebraic soliton. We prove strong convergence with explicit bounds for both cases, and provide detailed asymptotics. These theoretical results are illustrated with numerical approximations.

Keywords:
Nonlinear Schrodinger equation; ground state; gaussian solitons; algebraic solitons; asymptotic behavior.