Algebraic structure of NLS
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 . The most interesting values of p are the critical or pseudoconformal power and the critical power for (for there is no conformal power). The power is also a key exponent for the scattering theory (as this is when the nonlinearity has about equal strength with the decay ). The cases are the most natural for physical applications since the nonlinearity 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 nonlinear perturbations of a plane wave. For instance, it arises as a model for dilute BoseEinstein condensates.
Dimension 
Scattering power 
critical power 
critical power 
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 illposedness below in the focusing case KnPoVep, and also in the defocusing case CtCoTap2. (However if the nonlinearity is replaced by a noninvariant 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.