# Semilinear Schrodinger equation

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NLS
Description
Equation ${\displaystyle iu_{t}+\Delta u=\pm |u|^{p-1}u}$
Fields ${\displaystyle u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C} }$
Data class ${\displaystyle u(0)\in H^{s}(\mathbb {R} ^{d})}$
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity ${\displaystyle {\dot {H}}^{{\frac {d}{2}}-{\frac {2}{p-1}}}(\mathbb {R} ^{d})}$
Criticality varies
Covariance Galilean
Theoretical results
LWP ${\displaystyle H^{s}(\mathbb {R} )}$ for ${\displaystyle s\geq \max(s_{c},0)}$ (*)
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 ${\displaystyle \pm |u|^{p-1}u}$ by a more general nonlinearity ${\displaystyle F(u)}$ of power type. The ${\displaystyle +}$ sign choice is the defocusing case; ${\displaystyle -}$ 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.

## Specific semilinear Schrodinger equations

There are many special cases of NLS which are of interest: