#!/usr/bin/env python
# coding: utf-8

# # Nonlinear Schrödinger as a Dynamical System
# 
# #### [Ascona Winter School 2016](http://www.math.uzh.ch/pde16/index-Ascona2016.html), [(alternate link)](http://www.monteverita.org/en/90/default.aspx?idEvent=295&archive=)
# 
# 
# #### [J. Colliander](http://colliand.com) ([UBC](http://www.math.ubc.ca))

# ## Lectures
# 
# 1. [Introduction](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture1.ipynb)
# 2. **[Conservation](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture2.ipynb)**
# 3. [Monotonicity](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture3.ipynb)
# 4. [Research Frontier](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture4.ipynb)
# 
# ###  https://github.com/colliand/ascona2016
# 
# ***

# ## Overview of Lecture 2
# 
# * Conserved Quantities
# * Bourgain's High/Low Frequency Decomposition
# * $I$-Method
# * Multilinear Correction Terms
# * Applications

# ## Lecture Notes
# 
# ### [https://github.com/colliand/ascona2016](https://github.com/colliand/ascona2016)

# # Conserved Quantities
# 
# $$
# \begin{align*}
# {\mbox{Mass}}& = \| u \|_{L^2_x}^2 = \int_{\mathbb{R}^d} |u(t,x)|^2 dx. \\
# {\mbox{Momentum}}& = {\textbf{p}}(u) = 2 \Im \int_{\mathbb{R}^2} {\overline{u}(t)} \nabla u (t)
# dx. \\
# {\mbox{Energy}} & = H[u(t)] = \frac{1}{2} \int_{\mathbb{R}^2} |\nabla u(t) |^2 dx {\pm}  \frac{2}{p+1} |u(t)|^{p+1}  dx .
# \end{align*}
# $$

# ## Conserved ##
# $$ \partial_t Q[u] = 0.$$

#  
# ## Almost Conserved##
# $$\big| \partial_t Q[u] \big| ~\mbox{is small}.$$
# 

# $$ \sup_{t \in T_{lwp}} Q[u(t)] - \inf_{t \in T_{lwp}} Q[u(t)] < \epsilon$$

# $$ \int_0^{T_{lwp}} (\partial_t Q)[u(\tau)] d\tau < \epsilon $$ 

# ## Conservation of Mass

# $$ \partial_t |u(t)|^2 = \partial_t ( u \overline{u} ) = u_t \overline{u} + u \overline{u}_t$$
# 
# 
# 

# From the equation $ (i \partial_t  + \Delta) u = \pm |u|^{p-1} u $, we have:
# $$ u_t = i \Delta u  \mp i |u|^{p-1} u$$
# $$ {\overline{u}}_t = -i \Delta {\overline{u}} \pm i |u|^{p-1} {\overline{u}}$$

# Thus,
# $$ \partial_t |u(t)|^2 = [ i \Delta u \mp i |u|^{p-1} u] \overline{u} + u [  -i \Delta {\overline{u}} \pm i |u|^{p-1} {\overline{u}}]$$

# $$ \partial_t |u(t)|^2 = i[ \overline{u} \Delta u  - u\Delta {\overline{u}} ] \pm i [ |u|^{p+1}  - |u|^{p+1}  ] $$

# $$ \partial_t |u(t)|^2 = \nabla \cdot \Im [ \overline{u} \nabla u ]$$ 

# # Bourgain's High/Low Method

# ![bourgain-front-page](https://wwejubwfy.s3.amazonaws.com/1998_Bourgain_IMRN_FrontPage.pdf-2016-01-11-06-15-59.jpg)

# 
# 
# 
# Consider the Cauchy problem for defocusing cubic NLS on $\mathbb{R}^2$: 
# \begin{equation*}
# \tag{{$NLS^{+}_3 (\mathbb{R}^2)$}}
#  \left\{
#  \begin{matrix}
#  (i \partial_t  + \Delta) u = +|u|^{2} u \\
#  u(0,x) = u_{hi_0} (x).
#  \end{matrix}
#  \right.
# \end{equation*}
# We describe the first result to give global well-posedness below
# $H^1$.
# 
# *  $NLS_3^+ (\mathbb{R}^2)$ is GWP in $H^s$ for $s > \frac{2}{3}$.
# *  First use of *Bilinear Strichartz*
#  estimate was in this proof. 
# *  Proof cuts solution into low and high frequency parts.
# *  For $u_0 \in H^s,
#   ~s>\frac{2}{3},$ Proof gives (and *crucially exploits*),
# $$ u(t) - e^{it \Delta } u_{hi_0} \in H^1 (\mathbb{R}^2_x).$$
# 

# 
# 
# # Setting up; Decomposing Data
# 
# *  Fix a large target time $T$. 
# *  Let $N = N(T)$ be large to be determined.
# *  Decompose the initial data:
# $$
# u_0 = u_{low} + u_{high}
# $$
# where
# $$
# u_{low} (x) = \int_{{|\xi| < N}} ~e^{i x \cdot \xi }
# \widehat{u_0} (\xi) d \xi.
# $$ 
# *  Our plan is to evolve:
# $$
# u_0  =  u_{low} + u_{high}
# $$
# to
# $$ 
# u(t)  =  u_{{low}} (t) + u_{{high}} (t) .
# $$
# 

# 
# # Sizes of the Data Components
# 
#  
#  Low Frequency Data Size:
# 
# *  Kinetic Energy:
# \begin{align*}
#  \| \nabla u_{low} \|^2_{L^2}  &= \int_{|\xi| < N} |\xi|^{2}
#   | \widehat{u_0} (\xi)|^2 dx \ \\
# &= \int_{|\xi| < N} |\xi|^{2(1-s)}
#   |\xi|^{2s} |\widehat{u_0} (\xi)|^2 dx  \\
# & \leq  N^{2(1-s)} \| u_0 \|^2_{H^s} \leq C_0 N^{2(1-s)}.
# \end{align*}
# 

# *  Potential Energy:
# $
# \| u_{low} \|_{L^4_x} \leq \| u_{low} \|_{L^2}^{1/2} \|
# \nabla u_{low} \|_{L^2}^{1/2}
# $
# $$
# \implies H[ u_{low} ] \leq C N^{2(1-s)}.
# $$
# 

# High Frequency Data Size:
# $$\| u_{high}  \|_{L^2} \leq C_0 N^{-s}, ~\| u_{high} \|_{H^s} \leq C_0.$$

# 
# 
# # LWP of $u_{low}$ Frequency Evolution along NLS
# 
# The NLS Cauchy Problem for the low frequency data
# \begin{equation*}
# \tag{{${{NLS}}$}}
# %\tag{{$NLS^{+}_3 (\mathbb{R}^2)$}}
#  \left\{
#  \begin{matrix}
#  (i \partial_t  + \Delta) u_{{low}} = +|u_{{low}}|^{2} u_{{low}} \\
#  u_{{low}}(0,x) = u_{low} (x)
#  \end{matrix}
#  \right.
# \end{equation*}
# is well-posed on $[0, T_{lwp}]$ with $T_{lwp} \thicksim \|
# u_{low} \|_{H^1}^{-2} \thicksim N^{-2(1-s)}$.
# 
# We obtain, as a consequence of the local theory, that
# $$
# \| u_{{low}} \|_{L^4_{[0,T_{lwp}], x}} \leq \frac{1}{100}.
# $$
# 

# 
# # LWP of $u_{high}$ Evolution along DE
# 
# The NLS Cauchy Problem for the high frequency data
# \begin{equation*}
# %\tag{{$NLS^{+}_3 (\mathbb{R}^2)$}}
#  \left\{
#  \begin{matrix}
#  (i \partial_t  + \Delta) u_{{high}} = +2 |u_{{low}}|^2 u_{{high}} +
#  {\mbox{similar}} + |u_{{high}}|^{2} u_{{high}} \\
#  u_{{high}} (0,x) = u_{high} (x)
#  \end{matrix}
#  \right.
# \end{equation*}
# is also well-posed on $[0, T_{lwp}]$. 
# 
# 
# **Crucial Observation:** The LWP lifetime of $NLS$ evolution of $u_{{low}}$ AND
# the LWP lifetime of the $DE$ evolution of $u_{{high}}$ are controlled by
# $\| u_{{low}}(0)\|_{H^1}$.

# 
# 
# # Extra Smoothing of Nonlinear Duhamel Term
# 
# The high frequency evolution may
# be written
# $$
# u_{{high}} (t) = e^{it \Delta} u_{{high}} + w.
# $$
# The local theory gives $\| w(t) \|_{L^2}  \lesssim N^{-s}$. Moreover, 
# due to smoothing (obtained via bilinear Strichartz), we have that
# \begin{equation}
# \tag{SMOOTH!}
# w \in H^1, ~ \| w(t) \|_{H^1}
# \lesssim N^{1-2s+}. 
# \end{equation}
# Let's postpone the proof of (SMOOTH!).

# 
# 
# # Nonlinear High Frequency Term Hiding Step!
# 
# *  $\forall ~t \in [0, T_{lwp}]$, we have 
# $$
# u(t) = u_{{low}} (t) + e^{it \Delta } u_{high}  + w(t).
# $$
# * 
# At time $T_{lwp}$, we define data for the progressive scheme:
# $$
# u(T_{lwp} ) = u_{{low}} (T_{lwp}) + w(T_{lwp} ) + e^{iT_{lwp} \Delta}
# u_{high}.
# $$
# 
# $$
# u(t) = u^{(2)}_{{low}} (t) + u^{(2)}_{{high}} (t)
# $$
# for $ t > T_{lwp}$.

# 
# 
# # Hamiltonian Increment: $u_{low} (0) \longmapsto
#     u^{(2)}_{{low}} (T_{lwp})$
# 
# The Hamiltonian increment due to $w(T_{lwp})$ being added to low
# frequency evolution can be calcluated. Indeed, by Taylor expansion,
# using the bound  (SMOOTH!) and energy conservation
# of $u_{{low}}$ evolution, we have
# using 
# \begin{align*}
# H[u^{(2)}_{{l}} (T_{lwp})] &= H[u_{{l}} (0)] + (H[ u_{{l}} (T_{lwp}) +
# w(T_{lwp}) ] - H[u_{{l}} (T_{lwp})]) \\
# & \thicksim N^{2(1-s) } + N^{2 -3s+} \thicksim N^{2(1-s)}.
# \end{align*}
# 

# We can accumulate $N^s$ increments of size $N^{2-3s+}$
# before we double the size $N^{2(1-s)}$
# of the Hamiltonian. During the iteration, Hamiltonian of ``low
# frequency'' pieces remains of size $\lesssim N^{2(1-s)}$ so the LWP
# steps are of uniform size $N^{-2(1-s)}$. We advance the solution on a
# time interval of size:
# $$
# N^s N^{-2(1-s)} = N^{-2 + 3s}.
# $$
# For $s>\frac{2}{3}$, we can choose $N$ to go past target time $T. ~\blacksquare$
# 

# 
# # How do we prove (SMOOTH!)?
# 
# The proof follows from a **bilinear estimate**.
# 
# 

# 
# # Bilinear Strichartz Estimate
# 
# 
# 
# *  Recall the Strichartz estimate for $(i \partial_t + \Delta)$ on $\mathbb{R}^2$:
# $$
# \| e^{it \Delta} u_0 \|_{L^4 ( \mathbb{R}_t \times \mathbb{R}^2_x)} \leq C 
# \| u_0  \|_{L^2 (\mathbb{R}^2_x)}.
#  $$
# 

# *  We can view this trivially as a bilinear estimate by writing
# $$
#  \| e^{it \Delta} u_0  ~ e^{it \Delta} v_0 \|_{L^2 ( \mathbb{R}_t \times \mathbb{R}^2_x)} \leq C 
#  \| u_0 \|_{L^2 (\mathbb{R}^2_x)} \| v_0  \|_{L^2 (\mathbb{R}^2_x)} .
# $$
# *  Bourgain refined this trivial bilinear estimate for
# functions having certain Fourier support properties. 

# 
# 
# # Bilinear Strichartz Estimate
# 
# For (dyadic) $N \leq L$ and for $x \in \mathbb{R}^2$,
# $$
# \| e^{it\Delta} f_L e^{it\Delta} g_N \|_{L^2_{t,x}} \leq
# \frac{N^{\frac{1}{2}}}{L^{\frac{1}{2}}} \| f_L \|_{L^2_x} \| g_N
# \|_{L^2_x}.
# $$
# 
# 
# 
# *  Here $\mbox{spt}~(\widehat{f_L}) \subset \{ |\xi | \thicksim L\},
#   ~g_N$ similar.
# *  Observe that $\sqrt{\frac{N}{L}} \ll 1$ when $N \ll L$.

# # $I$-Method

# 
# # The $I$-Method of Almost Conservation
# 
# 
# 
# Let $H^s \ni u_0 \longmapsto u$ solve $NLS$ for $t \in [0, T_{lwp}], T_{lwp} \thicksim \|u_0 \|_{H^s}^{-2/s}.$
# 
# 
#  Consider two ingredients (to be defined):
#  
#  *  A **smoothing operator** $I = I_N: H^s \longmapsto H^1$. The $NLS$ evolution $u_0 \longmapsto u$ induces a **smooth reference evolution** $H^1 \ni Iu_0 \longmapsto Iu$ solving $I(NLS)$ equation on $[0,T_{lwp}]$.
#  *  A **modified energy** $\widetilde{E}[Iu]$ built using the reference evolution.

# 
# # First Version of the $I$-method: ${\widetilde{E}}= H[Iu]$
# 
# 
#  
# For $s<1, N \gg 1$ define smooth monotone $m: \mathbb{R}^2_\xi \rightarrow \mathbb{R}^+$ s.t.
# $$
# m(\xi) = 
# \left\{
# \begin{matrix}
# 1 & {\mbox{for}}~ |\xi | <N \\
# \left( \frac{|\xi|}{N} \right)^{s-1} &{\mbox{for}}~ |\xi | > 2N.
# \end{matrix}
# \right.
# $$
# 
# 
# The associated Fourier multiplier operator,
# ${\widehat{(Iu)}} (\xi) = m(\xi) \widehat{u} (\xi),$
# satisfies $I: H^s \rightarrow H^1 $.  Note that, pointwise in time, we have
# $$
# \| u \|_{H^s} \lesssim \| Iu \|_{H^1} \lesssim N^{1-s} \|u \|_{H^s}.
# $$
# 
# 
# Set $\widetilde{E}[Iu(t)] = H[Iu(t)]$. Other choices of $\widetilde E$
# are mentioned later.

# 
# 
# # AC Law Decay and Sobolev GWP index
# 
# 
# *  **Modified LWP.** Initial $v_0$ s.t. $\| \nabla I v_0
# \|_{L^2} \thicksim 1$ has $T_{lwp} \thicksim 1$. 
# 
# *  **Goal.**  $\forall ~u_0 \in H^s, ~\forall ~T > 0$,  construct 
# $u:[0,T] \times \mathbb{R}^2 \rightarrow \mathbb{C}.$
# 
# * $\iff$ **Dilated Goal.** Construct
# $
# u^\lambda: [0, \lambda^2 T] \times \mathbb{R}^2 \rightarrow \mathbb{C}.
# $
# 
# * **Rescale Data.** $\| I \nabla u_0^\lambda
#   \|_{L^2} \lesssim N^{1-s} \lambda^{-s} \| u_0 \|_{H^s} \thicksim 1$
# provided we choose $\lambda = \lambda (N) \thicksim
# N^{\frac{1-s}{s}} \iff N^{1-s} \lambda^{-s} \thicksim 1$.
# * **Almost Conservation Law.** $\| I \nabla u ( t ) \|_{L^2}
#   \lesssim H[Iu(t)]$ and
# $$
# \sup_{t \in [0, T_{lwp}]} H[Iu(t) ] \leq H [Iu(0)] + N^{-\alpha}.
# $$
# * **Delay of Data Doubling.** Iterate modified LWP $N^\alpha$ steps
#   with $T_{lwp}  \thicksim 1$. We obtain rescaled solution for $t \in
#   [0, N^\alpha]$.
# $$
# \lambda^2(N) T < N^\alpha \iff T < N^{\alpha + \frac{2(s-1)}{s}}
# ~{\mbox{so}}~ s > \frac{2}{2+\alpha}~{\mbox{suffices}}.
# $$  
# 

# 
# # Almost Conservation Law for $H[Iu]$
# 
# 
# 
# Given $s > \frac{4}{7}, N \gg 1,$ and initial data 
# $u_0 \in C^{\infty}_0(\mathbb{R}^2)$  with $E(I_N u_0) \leq 1$, then
# there exists a $ T_{lwp}\thicksim 1$  so that the solution
# \begin{align*}
# u(t,x) & \in C([0,T_{lwp}], H^s(\mathbb{R}^2))
# \end{align*}
# of $NLS_3^+ (\mathbb{R}^2)$ satisfies
# \begin{equation*}
# \label{increment}
# E(I_N u)(t)  =  E(I_N u)(0) + O(N^{- \frac{3}{2}+}),
# \end{equation*}
# for all $t \in [0, T_{lwp}]$.
# 

# 
# 
#  # Ideas in the Proof of Almost Conservation
#  
#  *  Standard Energy Conservation Calculation:
# \begin{align*}
#  \partial_t H(u) &= \Re \int_{\mathbb{R}^2} \overline{u_t} (|u|^2 u -
# \Delta u) dx \\
# & = \Re \int_{\mathbb{R}^2} \overline{u_t} ( |u|^2 u -
# \Delta u - i u_t) dx = 0.
# \end{align*} 
# 

# 
# *  For the smoothed reference evolution, we imitate....
#  \begin{align*}
#   \partial_t H(Iu) &= \Re \int_{\mathbb{R}^2} \overline{Iu_t} (|Iu|^2 Iu -
#  \Delta Iu  - i I u_t)dx \\
#  & = \Re \int_{\mathbb{R}^2} \overline{Iu_t} ( |Iu|^2 Iu - I (|u|^2 u)) dx \neq 0.
#  \end{align*} 
#  *  The increment in modified energy involves a commutator,
#  $$H(Iu)(t) - H(Iu)(0) = \Re \int_0^t \int_{\mathbb{R}^2} \overline{Iu_t} ( |Iu|^2 Iu - I (|u|^2 u)) dx dt.
#  $$
#  *  Littlewood-Paley, Case-by-Case, **(Bi)linear Strichartz**, $X_{s,b}$....

# 
# # Remarks
# 
# *  The almost conservation property
# $$  \sup_{t \in [0, T_{lwp}]} \widetilde{E}[Iu(t)] \leq
# \widetilde{E}[Iu_0] + N^{-\alpha}$$
# leads to GWP for 
# $$
# s > s_\alpha = \frac{2}{2+\alpha}.
# $$
# *  The $I$-method is a **subcritical method**. 
# 
# *  The $I$-method **localizes the conserved density** in
#       frequency}}.  
# 
# *  There is a **multilinear corrections algorithm** for defining
#   other choices of 
#   $\widetilde{E}$ which yield a better AC property.

# # Multilinear Correction Terms

# 
# 
# ## Multilinear Correction Terms
# 
# * For $k \in {\mathbb{N}}$, define the **convolution hypersurface** 
# $$ \Sigma_k := \{ (\xi_1,\ldots,\xi_k) \in (\mathbb{R}^2)^k: \xi_1 + \ldots +
# \xi_k = 0 \} \subset (\mathbb{R}^2)^k.$$
#  * For $M: \Sigma_k \to \mathbb{C}$ and $u_1,\ldots,u_k$ nice, define
#    **$k$-linear functional** 
#  $$ \Lambda_k( M; u_1,\ldots,u_k ) :=
#  c_k ~\mathbb{R}e \int\limits_{\Sigma_k} M(\xi_1,\ldots,\xi_k) \widehat{u_1}(\xi_1) \ldots \widehat{u_k}(\xi_k).$$
# * For $k \in 2{\mathbb{N}}$ abbreviate
# $\Lambda_k (M; u) = \Lambda_k (M; u, \overline{u}, \ldots, \overline{u}).$
# * $\Lambda_k (M;u)$ invariant under interchange of even/odd arguments,
# $$
# M (\xi_1,\xi_2,\ldots,\xi_{k-1},\xi_k) \mapsto \overline{M}(\xi_2,\xi_1,\ldots,\xi_k,\xi_{k-1}).$$
# * We can define a symmetrization rule via group orbit.

# 
# ## Examples
# 
# 
# * 
# $$
# \int\limits_x u \overline{u} u \overline{u} dx = \int (\int e^{i x \cdot
#   \xi_1} \widehat{u} (\xi_1) d\xi_1) \ldots (\int e^{i x \cdot
#   \xi_4} \widehat{\overline{u}} (\xi_4) d\xi_4)  dx
# $$
# $$
# = \int_{\xi_1, \dots,  \xi_4} \left[\int_x e^{i x \cdot ( \xi_1 + \xi_2 +
#   \xi_3 + \xi_4)} dx\right]  
#   \widehat{u} (\xi_1) \widehat{\overline{u}} (\xi_2) \widehat{u} (\xi_3) \widehat{\overline{u}} (\xi_4) 
#   d \xi_{1, \ldots ,4}
# $$
# $$
# = \int\limits_{\Sigma_4} \widehat{u} (\xi_1) \widehat{\overline{u}} (\xi_2) 
# \widehat{u} (\xi_3) \widehat{\overline{u}} (\xi_4)
# = \Lambda_4 (1; u).
# $$
# 
# * 
# $$\Lambda_2 (-\xi_1 \cdot \xi_2; u) = \| \nabla u \|_{L^2}^2.$$
# 
# 
# 
# 
# Thus, $H[u] = \Lambda_2 ( - \xi_1 \cdot \xi_2; u) \pm \Lambda_4 (\frac{1}{2} ; u)$.

# 
# 
# ## Time Dependence of Multilinear Forms
# 
# Suppose $u$ nicely solves $NLS_3^+ (\mathbb{R}^2)$; $M$ is time
# independent, symmetric. 
# 
# ### How would you calculate
# 
# $$
# \partial_t \Lambda_k( M; u(t) )?
# $$
# 

# ## Time Differentiation Formula
# 
# 
# $$
# \partial_t \Lambda_k( M; u(t) ) =
# \Lambda_k( i M \alpha_k; u(t) ) - \Lambda_{k+2}( i k X(M); u(t) ) 
# $$
# $$
# =
# \Lambda_k( i M \alpha_k; u(t) ) - \Lambda_{k+2}( [i k X(M)]_{sym}; u(t)
# ).
# $$
# Here
# $$ \alpha_k(\xi_1,\ldots,\xi_k) := -|\xi_1|^2 + |\xi_2|^2 - \ldots - |\xi_{k-1}|^2 + |\xi_k|^2$$
# (so $\alpha_2 = 0$ on $\Sigma_2$) and 
# $$ X(M)(\xi_1,\ldots,\xi_{k+2}) := M( \xi_{123}, \xi_4, \ldots, \xi_{k+2}).$$
# We use the notation $\xi_{ab} := \xi_a + \xi_b$, $\xi_{abc} := \xi_a +
# \xi_b + \xi_c$, etc.  

# 
# 
# ## AC Quantities via Multilinear Corrections
# 
# 
# *  Abbreviate $m(\xi_j)$ as $m_j$. Define $\sigma_2$ s.t. $\| I \nabla u \|_{L^2}^2 = \Lambda_2 (\sigma_2; u):$ 
# $$\sigma_2(\xi_1,\xi_2) := -
#   \frac{1}{2} \xi_1 m_1 \cdot \xi_2 m_2 = \frac{1}{2} |\xi_1|^2 m_1^2$$
#  
# * With $\tilde \sigma_4$ (symmetric, time independent) {{to be determined}}, set
# $$
# {\widetilde{E}}  := \Lambda_2( \sigma_2 ; u ) + \Lambda_4( \tilde
# \sigma_4 ; u ).
# $$
# * Using the time differentiation formula, we calculate
# $$
# \partial_t \widetilde E = \Lambda_4 (  {{\left\{i \tilde
#       \sigma_4 \alpha_4 -     i 2[ X (\sigma_2)]_{sym} \right\}}} ; u) -
# \Lambda_6  ( [i 4 X(\tilde \sigma_4)]_{sym}; u).
# $$
# 
# 
# 
# We'd like to define $\tilde \sigma_4$ to cancel away the $\Lambda_4$
# contribution. 

# 
#  
#  ## Natural Choice of $\sigma_4$ Fails
#  
# Here is the natural choice:
#  $$ \tilde \sigma_4 =~ \frac{[2 i X(\sigma_2)]_{sym}}{i \alpha_4}.$$
# On $\Sigma_4$, we can reexpress $\alpha_4 = -|\xi_1|^2 + |\xi_2|^2
# -|\xi_3|^2 + |\xi_4|^2$ as
# $$
# \alpha_4 = -2 \xi_{12} \cdot \xi_{14} = -2 |\xi_{12}| |\xi_{14}| \cos
# \angle(\xi_{12},\xi_{14}), 
# $$
# and
# $$
# [2 i X(\sigma_2)]_{sym} = \frac{1}{4} ( - m_1^2 |\xi_1|^2 + m_2^2
# |\xi_2|^2 - m_3^2 |\xi_3|^2 + m_4^2 |\xi_4|^2 ).
# $$
# When all the $m_j = 1$ (so $\max_{j} |\xi_j | < N$), $\tilde \sigma_4$ is
# well-defined. However, $\alpha_4$ can also vanish when $\xi_{12}$ and
# $\xi_{14}$ are orthogonal. 
# 
# ### Small Divisor Problem
# 
# 
# 

#  
# ## Speculation on Integrable Systems?
# 
# For $NLS_3^+ (\mathbb{R})$,
# the resonant obstruction disappears. Thus, 
# $$ \widetilde E^1 = \Lambda_2 (\sigma_2) + \Lambda_4 (\tilde \sigma_4);$$
# $$ \partial_t \widetilde E^1 = - \Lambda_6 ( [i 4 X(\tilde \sigma_4)]_{sym}).$$
# We can then define, with $\tilde \sigma_6$ to be determined,
# $$\widetilde E^2 = \widetilde E^1 + \Lambda_6 (\tilde \sigma_6 );$$
# $$
# \partial_t \widetilde E^2 =  \Lambda_6 ( {{\{ i \tilde \sigma_6
#     \alpha_6 -  [i 4 X(\tilde \sigma_4)]_{sym}\} }}) + 
# \Lambda_{8}( [i 6 X(\tilde \sigma_6)]_{sym}).
# $$
# Let's define
# $$
# \tilde \sigma_6 = \frac{[i 4 X (\tilde \sigma_4)]_{sym}}{i \alpha_6}.
# $$

# 
# ## Speculation on Integrable Systems?
# 
# Thus, we formally obtain a continued-fraction-like algorithm.
# $$
# \tilde \sigma_6 = \frac{\left[i 4 X \left ( \frac{[2 i
#       X(\sigma_2)]_{sym}}{i \alpha_4}\right)\right]_{sym}}{i \alpha_6}, 
# $$
# $$
# \tilde \sigma_8 = \frac{\left[i 6 X \left( \frac{\left[i 4 X \left ( \frac{[2 i
#       X(\sigma_2)]_{sym}}{i \alpha_4}\right)\right]_{sym}}{i \alpha_6}
# \right)\right]_{sym}}{i \alpha_8}, \ldots.
# $$
# Each step gains two derivatives but costs two more factors.
# 
# **Speculation:** The multipliers $\tilde \sigma_6, \tilde \sigma_8,
# \ldots$ are well defined and lead to better AC properties. Same for
# other integrable systems.