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

# # Nonlinear Schrödinger as a Dynamical System
# 
# > #### [J. Colliander](http://colliand.com) ([UBC](http://www.math.ubc.ca))
# 
# #### [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=)
# 
# 

# ## 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 1
# 
# * Initial Value Problem for NLS
# * Conserved Quantities
# * Well-posedness Theory

# # Initial Value Problem for NLS

# # Initial Value Problem for NLS
# 
# $$
# \begin{equation*}
# \tag{$NLS^{\pm}_p (\Omega)$}
#  \left\{
#  \begin{matrix}
#  (i \partial_t  + \Delta) u = \pm |u|^{p-1} u \\
#  u(0,x) = u_0 (x), ~ x \in \Omega.
#  \end{matrix}
#  \right.
# \end{equation*}
# $$

# ## Studies explore interplay between:
# 
# * Dispersion
# * Nonlinearity

# Dilation invariance quantifies the balance between these effects.

# ## Dilation Invariance
# 
# One solution $u$ generates parametrized family $\{u^\lambda\}_{\lambda > 0}$ of solutions:
# 
# $$u:[0,T) \times \mathbb{R}^d_x \rightarrow \mathbb{C} ~{\mbox{solves}}~ NLS_p^{\pm}(\mathbb{R}^d)$$ 
# 
# $${\iff}$$
# 
# $$u^\lambda: [0,\lambda^2 T )\times \mathbb{R}^d_x \rightarrow \mathbb{C} ~{\mbox{solves}}~ NLS_p^{\pm}(\mathbb{R}^d)$$ 
# where
# $$
# u^\lambda (\tau, y) = \lambda^{-2/(p-1)} u( \lambda^{-2} \tau, \lambda^{-1} y).
# $$
# 

# Norms which are invariant under $u \longmapsto u_\lambda$ are **critical**. 

# ## Critical Regularity
# 

# 
# In the $L^2$-based Sobolev scale,
# $$
# \| D^s u^\lambda (t) \|_{L^2} = \lambda^{-\frac{2}{p-1} - s + \frac{d}{2}} \| D^s u (t)\|_{L^2}.
# $$
# 
# 
# 

# The **critical Sobolev index** for $NLS_p^{\pm}(\mathbb{R}^d)$ is
# $$
# s_c := \frac{d}{2} - \frac{2}{p-1}.
# $$

# 
# 

# 
# 
# | critical Sobolev index   | Regime                      | 
# |:---------------:|:-------------------------------:| 
# | $ s_c < 0$      | mass subcritical                |
# | $0 < s_c < 1$   | mass super/energy subcritical   |  
# | $s_c = 1$       | energy critical                 |  
# | $1 < s_c < \frac{d}{2}$| energy supercritical     |
# 
# 
# 
# 

# 
# ### The spatial domain $\Omega$?
# 
# 
# * Infinite measure space like $\mathbb{R}^d$ 
# * Finite measure space like $\mathbb{T}^d$

# ### The Choice of Sign?
#  
# * $+$ **defocusing**
# * $-$ **focusing**
# 
# Large data dynamics are completely different.

# > [numerical simulations of $NLS_5^+ (\mathbb{R}^5)$](https://wwejubwfy.s3.amazonaws.com/chirped-data-NLS5R5-simulation.pdf)

# # Conserved Quantities
# 
# 

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

# 
# ## Remarks
# 
# * Conserved quantities constrain the dynamics.
# * NLS defines a flow on a sphere in $L^2$.
# * Energy vividly reveals **focusing vs. defocusing** difference.
# * Mass is $L^2$; Momentum scales like $H^{1/2}$; Energy involves $H^1$.
# * Local conservation laws express **how** a quantity is
#   conserved: 
#   $\partial_t |u|^2= \nabla \cdot 2 \Im (\overline{u}
#   \nabla u)$. 

# # Variations on Conserved Quantities?

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

# * Energy/Compactness methods for building solutions.
# * Globalizing control to extend local-in-time solutions.

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

# * Bourgain's High/Low Frequency Decomposition
# * $I$-Method
# * Multilinear Correction Terms
# * Applications

# ## Monotone ## 
# $$\partial_t Q[u] > 0.$$

# * Virital identity $\implies$ blow-up.
# * Morawetz-type inequalities $\implies$ decay.

# # Well-posedness Theory

# # Local Well-posedness Theory
# 
# * Fixed point argument based on Contraction Mapping
# * Show the "Picard iterates" converge
# * Key innovation for the analysis: identify the right space!

# # Free Schrödinger Evolution

# $$
# \begin{equation*}
# \tag{$LS(\mathbb{R}^d)$}
#  \left\{
#  \begin{matrix}
#  (i \partial_t  + \Delta) u = 0 \\
#  u(0,x) = u_0 (x).
#  \end{matrix}
#  \right.
# \end{equation*}
# $$

# ## Explicit Solution Formula
# 
# $$ u_0 \longmapsto u(t,x) = e^{it \Delta} u_0$$

# ## Explicit Solution Formula

# 
# 
# * Fourier Multiplier Representation:
# $$
# e^{it \Delta} u_0 (x) = c_\pi\int_{\mathbb{R}^d} e^{i x \cdot \xi} e^{-i t |\xi|^2}
# \widehat{u_0} (\xi) d\xi.
# $$
# * Convolution Representation:
# $$
# e^{it \Delta} u_0 (x) = k_\pi \frac{1}{(it)^{d/2}} \int_{\mathbb{R}^d} e^{i
#   \frac{|x-y|^2}{4t}} u_0 (y) dy.
# $$

# ## Estimates for Schrödinger Propagator $e^{it \Delta} u_0$

# * Fourier Multiplier Representation $\implies$ Unitary in $H^s$:
# $$
# \| D_x^s e^{it \Delta} u_0 \|_{L^2_x} = \| D_x^s u_0 \|_{L^2_x}.
# $$

# * Convolution Representation $\implies$ Dispersive estimate:
# $$
# \| e^{it \Delta} u_0 \|_{L^\infty_x} \leq \frac{C}{t^{d/2}} \| u_0 \|_{L^1_x}.
# $$

# * Spacetime estimates? **Strichartz estimates** hold, for example,
# $$
# \| e^{it \Delta} u_0 \|_{L^4 ( \mathbb{R}_t \times \mathbb{R}^2_x)} \leq C 
# \| u_0  \|_{L^2 (\mathbb{R}^2_x)}.
#  $$

# # Schrödinger Evolution with Forcing

# $$
# \begin{equation*}
#  \left\{
#  \begin{matrix}
#  (i \partial_t  + \Delta) u = F \\
#  u(0,x) = u_0 (x).
#  \end{matrix}
#  \right.
# \end{equation*}
# $$

# ### Strichartz Estimates
# 
# $$ \| u \|_{L^q_t L^r_x (\mathbb{R}_t \times \mathbb{R}^d_x )}  \leq C \| u_0 \|_{L^2_x} 
# + \| F \|_{L^{Q'}_t L^{R'}_x(\mathbb{R}_t \times \mathbb{R}^d_x )}.
# $$
# ***
# 

# Admissibility for *independent* pairs $(q,r), (Q,R)$
# $$
# \frac{2}{q} + \frac{d}{r} = \frac{d}{2}, ~ q > 2.
# $$
# The $'$ denotes Hölder dual exponent.

# # Local-in-time theory for $NLS^{\pm}_3 (\mathbb{R}^2)$

# * $\forall ~u_0 \in L^2 (\mathbb{R}^2)~\exists ~T_{lwp} ( u_0 ) $ determined by 
# $$
# \| e^{it \Delta} u_0 \|_{L^4_{tx}  ([0,T_{lwp} ] \times \mathbb{R}^2)} <
# \frac{1}{100} ~{\mbox{such that}}~
# $$
# 

# $\exists$ unique $u \in C([0, T_{lwp} ]; L^2 ) \cap L^4_{tx} ([0,T_{lwp}] \times \mathbb{R}^2)$ solving $NLS_3^{+} (\mathbb{R}^2)$.

# 
# * $\forall ~ u_0 \in H^s (\mathbb{R}^2), s>0$, $T_{lwp} \thicksim \| u_0  \|_{H^s}^{-\frac{2}{s}}$ and regularity persists: $u \in C([0,T_{lwp}]; H^s (\mathbb{R}^2))$.

# * Define the **maximal forward existence time** $T^* (u_0)$ by 
# $$
# \| u \|_{L^4_{tx} ([0,T^* -\delta] \times \mathbb{R}^2) }< \infty
# $$
# for all $\delta > 0$ but diverges to $\infty$ as $\delta \searrow 0$.

# * $\exists ~$ **small data scattering threshold** $\mu_0 > 0$
# $$
# \| u_0 \|_{L^2} < \mu_0 \implies \|u \|_{L^4_{tx} (\mathbb{R} \times \mathbb{R}^2)}  < 2 \mu_0.
# $$

# # $H^1$ Global-in-time theory for $NLS^{+}_3 (\mathbb{R}^2)$

# 
# * $\forall ~ u_0 \in H^1 (\mathbb{R}^2), s>0$, $T_{lwp} \thicksim \| u_0  \|_{H^1}^{-2}$ and regularity persists: $u \in C([0,T_{lwp}]; H^1 (\mathbb{R}^2))$.

# * Conserved Quantities $\implies ~T_{lwp} > C(u_0)$
# $$
# {\mbox{Energy}}  = H[u(t)] = \frac{1}{2} \int_{\mathbb{R}^2} |\nabla u(t) |^2 dx +  \frac{2}{p+1} |u(t)|^{p+1}
# $$
# $$
# {\mbox{Mass}} = \| u \|_{L^2_x}^2 = \int_{\mathbb{R}^d} |u(t,x)|^2 dx
# $$

# * Globalize by iteration using $ ~T_{lwp} > C(u_0)$.
#