Semilinear Schrodinger equation

NLS

Description
Equation	$iu_{t}+\Delta u=\pm |u|^{p-1}u$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{d})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{{\frac {d}{2}}-{\frac {2}{p-1}}}(\mathbb {R} ^{d})$
Criticality	varies
Covariance	Galilean
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq \max(s_{c},0)$ (*)
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 $\pm |u|^{p-1}u$ 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.

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