# Algebraic structure of NLS

The NLS is a Hamiltonian flow with the Hamiltonian

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

and symplectic form

${\displaystyle \{u,v\}=Im\int _{R^{d}}u{\overline {v}}dx.}$

From the phase invariance ${\displaystyle u\to e^{iq}u}$ one also has conservation of the ${\displaystyle L_{x}^{2}}$norm.

The scaling regularity is ${\displaystyle s_{c}=d/2-2/(p-1)}$. The most interesting values of p are the ${\displaystyle L_{x}^{2}}$-critical or pseudoconformal power ${\displaystyle p=1+4/d}$ and the ${\displaystyle H_{x}^{1}}$-critical power ${\displaystyle p=1+4/(d-2)}$ for ${\displaystyle d>2}$ (for ${\displaystyle d=1,2}$ there is no ${\displaystyle H^{1}}$ conformal power). The power ${\displaystyle p=1+2/d}$ is also a key exponent for the scattering theory (as this is when the non-linearity ${\displaystyle |u|^{p-1}u}$ has about equal strength with the decay ${\displaystyle t^{-d/2}}$). The cases ${\displaystyle p=3,5}$ 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 ${\displaystyle d}$ Scattering power ${\displaystyle 1+2/d}$ ${\displaystyle L^{2}}$ -critical power ${\displaystyle 1+4/d}$ ${\displaystyle H^{1}}$-critical power ${\displaystyle 1+4/(d-2)}$ 1 3 5 N/A 2 2 3 ${\displaystyle \infty }$ 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

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

is equal to

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

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

The equation is invariant under Galilean transformations

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

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

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

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

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