Algebraic structure of NLS

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The NLS is a Hamiltonian flow with the Hamiltonian

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

and symplectic form

\{u, v\} = Im \int_{R^d} u \overline{v} dx.

From the phase invariance u \to e^{iq}u one also has conservation of the L^2_xnorm.

The scaling regularity is sc = d / 2 − 2 / (p − 1). The most interesting values of p are the L^2_x-critical or pseudoconformal power p = 1 + 4 / d and the H^1_x-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

\infty

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

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

is equal to

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

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

The equation is invariant under Galilean transformations

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

This can be used to show ill-posedness below L^2\, 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 \underline{u^2}\,, then one can go below L^2).

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

\iint \frac{|u|^{p+1}}{|x|} dx dt

in the defocussing case in terms of the H^{1/2}\, 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.

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