Semilinear Schrodinger equation: Difference between revisions

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[Many thanks to Kenji Nakanishi with valuable help with the scattering theory portion of this section. However, we are still missing many references and results, e.g. on NLS blowup. - Ed.]
{{equation
| name = NLS
| equation = <math>iu_t + \Delta u = \pm |u|^{p-1} u</math>
| fields = <math>u: \R \times \R^d \to \mathbb{C}</math>
| data = <math>u(0) \in H^s(\R^d)</math>
| hamiltonian = [[Hamiltonian]]
| linear = [[free Schrodinger equation|Schrodinger]]
| nonlinear = [[semilinear]]
| critical = <math>\dot H^{\frac{d}{2} - \frac{2}{p-1}}(\R^d)</math>
| criticality = varies
| covariance = [[Galilean]]
| lwp = <math>H^s(\R)</math> for <math>s \geq \max(s_c, 0)</math> (*)
| gwp = varies
| parent = [[Schrodinger equations|Nonlinear Schrodinger equations]]
| special = [[Quadratic NLS|quadratic]], [[cubic NLS|cubic]], [[quartic NLS|quartic]], [[quintic NLS|quintic]],<br> [[mass critical NLS|mass critical]] NLS
| related = NLS [[NLS with potential|with potential]], [[NLS on manifolds and obstacles|on manifolds]]
}}


The '''semilinear Schrodinger equation''' (NLS) is


<center><math>i \partial_t u + \Delta u = \pm |u|^{p-1} u </math></center>
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 <math>\pm |u|^{p-1} u</math> by a more general nonlinearity
<math>F(u)</math> of [[power type]].  The <math>+</math> sign choice is the [[defocusing]] case; <math>-</math> 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.


for p>1.  There are many [[Schrodinger:specific equations|specific cases]] of this equation which are of interest, but in this page we shall focus on the general theory. The <math>+</math> sign choice is the ''defocusing'' case; <math>-</math> is ''focussing''. 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 ==


== Theory ==
The NLS has been extensively studied, and there is now a substantial theory on many topics concerning solutions to this equation.


* [[Algebraic structure of NLS|Algebraic structure]] (Symmetries, conservation laws, transformations, Hamiltonian structure)
* [[Algebraic structure of NLS|Algebraic structure]] (Symmetries, conservation laws, transformations, Hamiltonian structure)
* [[NLS wellposedness|Local and global well-posedness theory]]
* [[NLS wellposedness|Well-posedness]] (both local and global)
* [[NLS scattering|Scattering theory]]
* [[NLS scattering|Scattering]] (as well as asymptotic completeness and existence of wave operators)
* [[NLS stability|Stability of solitons]] (orbital and asymptotic)
* [[NLS blowup|Blowup]]
* [[NLS blowup|Blowup]]
* [[Unique continuation]]
* [[Unique continuation]]


== Specific semilinear Schrodinger equations ==
== Specific semilinear Schrodinger equations ==
There are many special cases of NLS which are of interest:


* [[Quadratic NLS]]
* [[Quadratic NLS]]
* [[Cubic NLS]] ([[cubic NLS on R|on R]], [[cubic NLS on R2|on R^2]], [[cubic NLS on R3|on R^3]], etc.)
* [[Cubic NLS]] ([[cubic NLS on R|on R]], [[cubic NLS on R2|on R^2]], [[cubic NLS on R3|on R^3]], etc.)
* [[Quartic NLS]]
* [[Quartic NLS]]
* [[Quintic NLS]]
* [[Quintic NLS]] ([[Quintic NLS on R|on R]], [[Quintic NLS on T|on T]], [[Quintic NLS on R2|on R^2]], and [[Quintic NLS on R3|on R^3]])
* [[Septic NLS]]
* [[Septic NLS]]
* [[Mass critical NLS]]
* [[Mass critical NLS]]
* [[Energy critical NLS]]


[[Category:Equations]]
[[Category:Equations]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]

Latest revision as of 05:44, 21 July 2007

NLS
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.

Specific semilinear Schrodinger equations

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