Algebraic structure of NLS
From DispersiveWiki
The NLS is a Hamiltonian flow with the Hamiltonian
and symplectic form
From the phase invariance 
one also has conservation of the 
norm.
The scaling regularity is sc = d / 2 − 2 / (p − 1). The most interesting values of p are the -critical or pseudoconformal power p = 1 + 4 / d and the 
-critical power p = 1 + 4 / (d − 2) for d > 2 (for d = 1,2 there is no H1 conformal power). The power p = 1 + 2 / d is also a key exponent for the scattering theory (as this is when the non-linearity | u | p − 1u has about equal strength with the decay t − d / 2). The cases p = 3,5 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.
|
Dimension d |
Scattering power 1 + 2 / d |
L2 -critical power 1 + 4 / d |
H1-critical power 1 + 4 / (d − 2) |
|
1 |
3 |
5 |
N/A |
|
2 |
2 |
3 |
|
|
3 |
5/3 |
7/3 |
5 |
|
4 |
3/2 |
2 |
3 |
|
5 |
7/5 |
9/5 |
7/3 |
|
6 |
4/3 |
5/3 |
2 |
The pseudoconformal transformation of the Hamiltonian gives that the time derivative of
is equal to
This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. 
in 
), especially in the -critical case 
(when the above derivative is zero). The norm of is sometimes known as the pseudoconformal charge.
The equation is invariant under Galilean transformations
This can be used to show ill-posedness below 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 
then one can go below L^2).
From scaling invariance one can obtain Morawetz inequalities, which usually estimate quantities of the form
in the defocussing case in terms of the 
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.
