Cubic NLS: Difference between revisions
No edit summary |
No edit summary |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 10: | Line 10: | ||
| criticality = varies | | criticality = varies | ||
| covariance = [[Galilean]] | | covariance = [[Galilean]] | ||
| lwp = <math>H^s(\R)</math> for <math>s \geq \max(d/2-1, 0)</math> | | lwp = <math>H^s(\R^d)</math> for <math>s \geq \max(d/2-1, 0)</math> | ||
| gwp = varies | | gwp = varies | ||
| parent = [[NLS]] | | parent = [[NLS]] | ||
| Line 17: | Line 17: | ||
}} | }} | ||
The '''cubic NLS''' is displayed on the box on the right. The sign + is ''defocusing'', while the - sign is ''focusing''. This equation is traditionally studied on Euclidean domains <math>R^d</math>, but other domains are certainly possible. | The '''cubic NLS''' is displayed on the box on the right. The sign + is ''defocusing'', while the - sign is ''focusing''. This equation is traditionally studied on Euclidean domains <math>R^d</math>, but other domains are certainly possible. | ||
| Line 26: | Line 24: | ||
The cubic NLS can be viewed as an oversimplified model of the [[Schrodinger map]] equation. It also arises as the limit of a number of other | The cubic NLS can be viewed as an oversimplified model of the [[Schrodinger map]] equation. It also arises as the limit of a number of other | ||
equations, such as the [[mKdV|modified Korteweg-de Vries equation]] and [[Zakharov system]]. | equations, such as the [[mKdV|modified Korteweg-de Vries equation]] and [[Zakharov system]]. | ||
One can also consider variants of the cubic NLS in which the ([[Hamiltonian]], [[Galilean]]-invariant) nonlinearity <math>\pm |u|^2 u</math> is replaced by a non-Hamiltonian, non-Galilean-invariant cubic polynomial such as <math>u^3</math> or <math>\overline{u}^3</math>. Typically, for this variant the local theory remains unchanged (or even improves somewhat), but the global theory is lost (especially for large data) due to the lack of conservation laws. | |||
== Scaling analysis == | == Scaling analysis == | ||
Latest revision as of 21:55, 4 March 2007
| Description | |
|---|---|
| Equation | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle iu_t + \Delta u = \pm |u|^2 u} |
| Fields | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u: \R \times \R^d \to \mathbb{C}} |
| Data class | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(0) \in H^s(\R^d)} |
| Basic characteristics | |
| Structure | Hamiltonian |
| Nonlinearity | semilinear |
| Linear component | Schrodinger |
| Critical regularity | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot H^{d/2 - 1}(\R^d)} |
| Criticality | varies |
| Covariance | Galilean |
| Theoretical results | |
| LWP | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s(\R^d)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq \max(d/2-1, 0)} |
| GWP | varies |
| Related equations | |
| Parent class | NLS |
| Special cases | on R, on T, on R^2, on T^2, on R^3, on R^4 |
| Other related | Schrodinger maps, mKdV, Zakharov |
The cubic NLS is displayed on the box on the right. The sign + is defocusing, while the - sign is focusing. This equation is traditionally studied on Euclidean domains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^d} , but other domains are certainly possible.
In one spatial dimension the cubic NLS equation is completely integrable. but this is not the case in higher dimensions.
The cubic NLS can be viewed as an oversimplified model of the Schrodinger map equation. It also arises as the limit of a number of other equations, such as the modified Korteweg-de Vries equation and Zakharov system.
One can also consider variants of the cubic NLS in which the (Hamiltonian, Galilean-invariant) nonlinearity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm |u|^2 u} is replaced by a non-Hamiltonian, non-Galilean-invariant cubic polynomial such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u^3} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overline{u}^3} . Typically, for this variant the local theory remains unchanged (or even improves somewhat), but the global theory is lost (especially for large data) due to the lack of conservation laws.
Scaling analysis
On Euclidean domains at least, the cubic NLS obeys the scale invariance
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(t,x) \mapsto \frac{1}{\lambda} u(\frac{t}{\lambda^2}, \frac{x}{\lambda}).}
Thus the critical regularity is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = \frac{d}{2} - 1} .
Specific domains
- Cubic NLS on R (Mass and energy sub-critical; scattering-critical; completely integrable)
- Cubic NLS on the half-line and interval (Mass and energy sub-critical)
- Cubic NLS on T (Mass and energy sub-critical; completely integrable)
- Cubic NLS on R^2 (Mass-critical; energy-subcritical; scattering-subcritical)
- Cubic NLS on two-dimensional manifolds (Mass-critical; energy-subcritical)
- Cubic NLS on R^3 (Mass-supercritical; energy-subcritical; scattering-subcritical)
- Cubic NLS on three-dimensional manifolds (Mass-supercritical; energy-subcritical)
- Cubic NLS on R^4 (Mass-supercritical; energy-critical; scattering-subcritical)
- Cubic NLS on four-dimensional manifolds (Mass-supercritical; energy-critical)
- Cubic NLS on six-dimensional manifolds (Mass-supercritical; energy-supercritical)
