Algebraic structure of NLS

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 $L_{x}^{2}$norm.

The scaling regularity is $s_{c}=d/2-2/(p-1)$. The most interesting values of p are the $L_{x}^{2}$-critical or pseudoconformal power $p=1+4/d$ and the $H_{x}^{1}$-critical power $p=1+4/(d-2)$ for $d>2$ (for $d=1,2$ there is no $H^{1}$ conformal power). The power $p=1+2/d$ is also a key exponent for the scattering theory (as this is when the non-linearity $|u|^{p-1}u$ 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 $1+2/d$	$L^{2}$ -critical power $1+4/d$	$H^{1}$-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

$\iint {\frac {|u|^{p+1}}{|x|}}dxdt$

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.