Schrodinger equations

Overview

There are many nonlinear Schrodinger equations in the literature, all of which are perturbations of one sort or another of the free Schrodinger equation. One general class of such equations takes the form

$i\partial _{t}u+\Delta u=f(u,{\overline {u}},Du,D{\overline {u}})$

where $D$ denotes spatial differentiation. In such full generality, we refer to this equation as a derivative non-linear Schrodinger equation (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a semilinear Schrodinger equation (NLS). These equations (particularly the cubic NLS) arise as model equations from several areas of physics.

One can generalize both the linear and nonlinear perturbations to these equations and consider the class of quasilinear Schrodinger equations or even fully nonlinear Schrodinger equations. Needless to say, these equations are significantly more difficult to analyse than the simpler model cases discussed above.

One can combine these nonlinear perturbations with a linear perturbation, leading for instance to the NLS with potential and the NLS on manifolds and obstacles.

The perturbative theory of nonlinear Schrodinger equations (and the semilinear Schrodinger equations in particular) rests on a number of linear and nonlinear estimates for the free Schrodinger equation.

Specific Schrodinger Equations

Monomial semilinear Schrodinger equations can indexed by the degree of the nonlinearity, as follows.

Quadratic NLS

NLS equations of the form

$i\partial _{t}u+\Delta u=Q(u,{\overline {u}})$

with $Q(u,{\overline {u}})$ a quadratic function of its arguments are quadratic nonlinear Schrodinger equations. They are mass-critical in four dimensions.

Cubic NLS

The cubic nonlinear Schrodinger equation is of the form

$i\partial _{t}u+\Delta u=\pm |u|^{2}u$

They are completely integrable in one dimension, mass-critical in two-dimensions, and energy-critical in four dimensions.

Quartic NLS

A nonlinear Schrodinger equation with nonlinearity of degree 4 is a quartic nonlinear Schrodinger equation.

Quintic NLS

NLS equations of the form

$i\partial _{t}u+\Delta u=\pm |u|^{4}u$

are quintic nonlinear Schrodinger equations. They are mass-critical in one dimension and energy-critical in three dimensions.

Septic NLS

NLS equations of the form

$i\partial _{t}u+\Delta u=\pm |u|^{6}u$

are septic nonlinear Schrodinger equations.

$L^{2}$-critical NLS

The nonlinear Schrodinger equation

$i\partial _{t}u+\Delta u=\pm |u|^{\frac {4}{d}}u$

posed for $x\in R^{d}$ is scaling invariant in $L_{x}^{2}$. This family of nonlinear Schrodinger equations is therefore called the mass critical nonlinear Schrodinger equation.

Higher order NLS

One can study higher-order NLS equations in which the Laplacian is replaced by a higher power. One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable cubic NLS on $R$. Another is the nonlinear Schrodinger-Airy system. A third class arises from the elliptic case of the Zakharov-Schulman system.

Schrodinger maps

A geometric derivative non-linear Schrodinger equation that has been intensively studied is the Schrodinger map equation. This is the Schrodinger counterpart of the wave maps equation.

Cubic DNLS on $R$

The deriviative cubic nonlinear Schrodinger equation has nonlinearity of the form $i\partial _{x}(|u|^{2}u).$

Hartree Equation

The Hartree equation has a nonlocal nonlinearity given by convolution, as does the very similar Schrodinger-Poisson system, and certain cases of the Davey-Stewartson system.

Maxwell-Schrodinger system

A Schrodinger-wave system closely related to the Maxwell-Klein-Gordon equation is the Maxwell-Schrodinger system.