Abstract
We study the nonlinear Schrödinger equation with an “attractive” linear potential: iϕ t = -Δϕ + (V(x) − |ϕ| 2σ)ϕ, 0<σ<2 (n−2) (NLS) , which arises in the mathematical description of phenomena in nonlinear optics and plasma physics. Nonlinear bound states are finite energy localized solutions which, if dynamically stable, play an important role in the structure of general solutions of NLS. We discuss the existence and nonlinear orbital stability of nonlinear ground states of NLS. In particular, if - Δ + V supports a linear bound state, then NLS has stable nonlinear ground states in the supercritical and critical cases ( σ≥2/ n), where blow-up ( self-focusing or collapse) can occur. This is a phenomenon not present in the case where V≡0. In addition, islands of stability in the regime of large H 1 norm exist.
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