View source for Algebraic structure of NLS

The [[NLS]] is a Hamiltonian flow with the Hamiltonian

<center><math>H(u) = \int_{R^d} |\nabla u |^2  \pm  |u|^{p+1}/(p+1) dx</math></center>

and symplectic form

<center><math>\{u, v\} = Im \int_{R^d} u \overline{v} dx.</math></center>

From the phase invariance <math>u \to e^{iq}u</math> one also has conservation of the <math>L^2_x</math>norm. 

The scaling regularity is <math>s_c = d/2 - 2/(p-1)</math>. The most interesting values of p are the <math>L^2_x</math>-critical or [[Pseudoconformal Transformation|pseudoconformal]] power <math>p=1+4/d</math> and the <math>H^1_x</math>-critical power <math>p=1+4/(d-2)</math> for <math>d>2</math> (for <math>d=1,2</math> there is no <math>H^1</math> conformal power). The power <math>p = 1 + 2/d</math> is also a key exponent for the scattering theory (as this is when the non-linearity <math>|u|^{p-1}u</math> has about equal strength with the decay <math>t^{-d/2}</math>). The cases <math>p=3,5</math> are the most natural for physical applications since the non-linearity is then a polynomial. The [[cubic NLS]] in particular seems to appear naturally as a model equation for many different physical contexts, especially in dispersive, weakly non-linear perturbations of a plane wave. For instance, it arises as a model for dilute Bose-Einstein condensates. <br /><br />

{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"
| style="padding: .75pt .75pt .75pt .75pt" |
Dimension <math>d</math>
| style="padding: .75pt .75pt .75pt .75pt" |
Scattering power <math>1+2/d</math>
| style="padding: .75pt .75pt .75pt .75pt" |
<math>L^2</math> -critical power <math>1+4/d</math>
| style="padding: .75pt .75pt .75pt .75pt" |
<math>H^1</math>-critical power <math>1+4/(d-2)</math>
|- style="mso-yfti-irow: 1"
| style="padding: .75pt .75pt .75pt .75pt" |
1
| style="padding: .75pt .75pt .75pt .75pt" |
3
| style="padding: .75pt .75pt .75pt .75pt" |
5
| style="padding: .75pt .75pt .75pt .75pt" |
N/A
|- style="mso-yfti-irow: 2"
| style="padding: .75pt .75pt .75pt .75pt" |
2
| style="padding: .75pt .75pt .75pt .75pt" |
2
| style="padding: .75pt .75pt .75pt .75pt" |
3
| style="padding: .75pt .75pt .75pt .75pt" |
<math>\infty</math>
|- style="mso-yfti-irow: 3"
| style="padding: .75pt .75pt .75pt .75pt" |
3
| style="padding: .75pt .75pt .75pt .75pt" |
5/3
| style="padding: .75pt .75pt .75pt .75pt" |
7/3
| style="padding: .75pt .75pt .75pt .75pt" |
5
|- style="mso-yfti-irow: 4"
| style="padding: .75pt .75pt .75pt .75pt" |
4
| style="padding: .75pt .75pt .75pt .75pt" |
3/2
| style="padding: .75pt .75pt .75pt .75pt" |
2
| style="padding: .75pt .75pt .75pt .75pt" |
3
|- style="mso-yfti-irow: 5"
| style="padding: .75pt .75pt .75pt .75pt" |
5
| style="padding: .75pt .75pt .75pt .75pt" |
7/5
| style="padding: .75pt .75pt .75pt .75pt" |
9/5
| style="padding: .75pt .75pt .75pt .75pt" |
7/3
|- style="mso-yfti-irow: 6; mso-yfti-lastrow: yes"
| style="padding: .75pt .75pt .75pt .75pt" |
6
| style="padding: .75pt .75pt .75pt .75pt" |
4/3
| style="padding: .75pt .75pt .75pt .75pt" |
5/3
| style="padding: .75pt .75pt .75pt .75pt" |
2
|}

The [[pseudoconformal transformation]] of the Hamiltonian gives that the time derivative of

<center><math>\|(x + 2it \tilde{N})u \|^2_2 - 81t^2/(p+1)\|U\|{P+1}^{P+1}</math></center>

is equal to

<center><math>4dt\lambda(p-(1+4/d))/(p+1) \|u\|_{p+1}^{p+1}.</math></center>

This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. <math>xu(0)\,</math> in <math>L^2\,</math>), especially in the <math>L^2\,</math>-critical case <math>p=1+4/d\,</math> (when the above derivative is zero). The <math>L^2\,</math> norm of <math>xu(0)\,</math> is sometimes known as the ''pseudoconformal charge''.

The equation is invariant under Galilean transformations

<center><math>u(x,t) \rightarrow e^{(i (vx/2 - |v|^{2}t)} u(x-vt, t).\,</math></center>

This can be used to show ill-posedness below <math>L^2\,</math> in the focusing case [[KnPoVe-p]], and also in the defocusing case [[CtCoTa-p2]]. (However if the non-linearity is replaced by a non-invariant expression such as <math>\underline{u^2}\,,</math> then [[Quadratic NLS|one can go below L^2]]).

From scaling invariance one can obtain Morawetz inequalities, which usually estimate quantities of the form

<center> <math>\iint \frac{|u|^{p+1}}{|x|} dx dt</math></center>

in the defocussing case in terms of the <math>H^{1/2}\,</math> norm. These are useful for limiting the number of times the solution can concentrate at the origin; this is especially handy in the radially symmetric case.

[[Category:Schrodinger]]