Ground States and Energy Asymptotics of the Nonlinear Schrödinger Equation with a General Power Nonlinearity

Abstract

We study analytically the existence and uniqueness of the ground state of the nonlinear Schrödinger equation (NLSE) with a general power nonlinearity described by the power index σ≥0. For the NLSE under a box or a harmonic potential, we can derive explicitly the approximations of the ground states and their corresponding energy and chemical potential in weak or strong interaction regimes with a fixed nonlinearity σ. Besides, we study the case where the nonlinearity σ→∞ with a fixed interaction strength. In particular, a bifurcation in the ground states is observed. Numerical results in 1D and 2D will be reported to support our asymptotic results.