Semilinear Schrodinger equation
From DispersiveWiki
Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | Hamiltonian |
Nonlinearity | semilinear |
Linear component | Schrodinger |
Critical regularity | |
Criticality | varies |
Covariance | Galilean |
Theoretical results | |
LWP | for (*) |
GWP | varies |
Related equations | |
Parent class | Nonlinear Schrodinger equations |
Special cases | quadratic, cubic, quartic, quintic, mass critical NLS |
Other related | NLS with potential, on manifolds |
The semilinear Schrodinger equation (NLS) is displayed on the right-hand box, where the exponent p is fixed and is larger than one. One can also replace the nonlinearity by a more general nonlinearity of power type. The sign choice is the defocusing case; is focusing. 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
The NLS has been extensively studied, and there is now a substantial theory on many topics concerning solutions to this equation.
- Algebraic structure (Symmetries, conservation laws, transformations, Hamiltonian structure)
- Well-posedness (both local and global)
- Scattering (as well as asymptotic completeness and existence of wave operators)
- Stability of solitons (orbital and asymptotic)
- Blowup
- Unique continuation
Specific semilinear Schrodinger equations
There are many special cases of NLS which are of interest:
- Quadratic NLS
- Cubic NLS (on R, on R^2, on R^3, etc.)
- Quartic NLS
- Quintic NLS (on R, on T, on R^2, and on R^3)
- Septic NLS
- Mass critical NLS
- Energy critical NLS