Global well-posedness for the cubic nonlinear Schrödinger equation with initial data lying in <i>L</i> <i>p</i>-based Sobolev spaces
Benjamin Dodson, Avy Soffer, Thomas Spencer
Abstract
In this paper, we continue our study [B. Dodson, A. Soffer, and T. Spencer, J. Stat. Phys. 180, 910 (2020)] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on R was proved for real analytic data. Here, we prove global well-posedness for the 1D NLS with initial data lying in Lp for any 2 &lt; p &lt; ∞, provided that the initial data are sufficiently smooth. We do not use the complete integrability of the cubic NLS.
Topics & Concepts
Sobolev spaceBounded functionNonlinear Schrödinger equationNonlinear systemInfinityMathematicsSchrödinger equationMathematical physicsInitial value problemMathematical analysisSmall dataPhysicsQuantum mechanicsData miningComputer scienceAdvanced Mathematical Physics ProblemsNonlinear Waves and SolitonsNonlinear Photonic Systems