Semilinear Schrodinger equation
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 F(u) 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.
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)
- Unique continuation
Specific semilinear Schrodinger equations
There are many special cases of NLS which are of interest: