Semilinear Schrodinger equation

[Many thanks to Kenji Nakanishi with valuable help with the scattering theory portion of this section. However, we are still missing many references and results, e.g. on NLS blowup. - Ed.]

The semilinear Schrodinger equation (NLS) is

$i\partial _{t}u+\Delta u=\pm |u|^{p-1}u$

for p>1. There are many specific cases of this equation which are of interest, but in this page we shall focus on the general theory. The $+$ sign choice is the defocusing case; $-$ is focussing. There are also several variants of NLS, such as NLS with potential or NLS on manifolds and obstacles; see the general page on Schrodinger equations for more discussion.

Theory

• Algebraic structure (Symmetries, conservation laws, transformations, Hamiltonian structure)
• Local and global well-posedness theory
• Scattering theory
• Stability of solitons
• Blowup
• Unique continuation

Specific semilinear Schrodinger equations

• Quadratic NLS
• Cubic NLS (on R, on R^2, on R^3, etc.)
• Quartic NLS
• Quintic NLS
• Septic NLS
• Mass critical NLS