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 . 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 non-linearity has about equal strength with the decay ). The cases 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

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 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.