On discrete rarefaction waves in an NLS Toy Model for weak turbulence
Sebastian~Herr
We explore the rarefaction wave-like solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schr\"odinger equation on the torus.
Torus_Shock_v2.tex
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\title[Discrete Rarefaction Waves]{On discrete rarefaction waves in an NLS Toy Model for weak turbulence}


\author[S.~Herr]{Sebastian~Herr}

\address{Universit\"at Bielefeld, Fakult\"at f\"ur Mathematik, Postfach 10 01 31, 33501 Bielefeld, Germany}
\email{herr@math.uni-bielefeld.de}

\author[J.~L.~Marzuola]{Jeremy~L.~Marzuola}
\address{Mathematics Department, University of North Carolina - Chapel Hill, Chapel Hill, NC 27599, USA}
\email{marzuola@math.unc.edu}



\begin{document}

\begin{abstract}
  We explore the rarefaction wave-like solutions to a model Hamiltonian dynamical
  system recently derived to
  study frequency cascades in the cubic defocusing nonlinear
  Schr\"odinger equation on the torus.  
  \end{abstract}

\maketitle

%\tableofcontents

\section{Introduction}
\label{s:intro}

In a work of Colliander-Keel-Staffilani-Takaoka-Tao \cite{CKSTT}, the authors studied the $2d$ defocusing cubic toroidal
nonlinear Schr\"odinger equation,
\begin{equation}
  \label{e:dcnls}
  i u_t + \Delta u - |u|^2 u = 0, \ \ u(0,x) = u_0(x) \ \text{for} \ x \in \mathbb{T}^2,
\end{equation}
by developing their ``Toy Model System'' given by the equation
\begin{equation}
  \label{e:toy_model}
  -i\dt b_j (t) = -\abs{b_j(t)}^2 b_j(t) + 2 b_{j-1}^2 \overline{b_j}(t)
  + 2 b_{j+1}^2 \overline{b_j}(t),\quad \text{for $j=1\ldots N$}
\end{equation}
with ``boundary conditions''
\begin{equation}
\label{e:dirichletbc}
  b_0(t) = b_{N+1}(t) = 0.
\end{equation}
The $b_j$ approximate the mass associated with families of resonantly
interacting frequencies.  In \cite{CMOS1}, Section $5$, the authors derive a discrete Burgers style equation with a phase drag term (see \eqref{e:hydro1}, \eqref{e:hydro2} below) and study its numerical stability within \eqref{e:toy_model}.   The goal of this paper is to prove that a discrete rarefaction wave associated to the Burgers equation approximates the dynamics in \eqref{e:toy_model}. This shows that this mechanism transfers mass from low to high frequency nodes. On a short time scale, this is distinctly possible due to an explicit construction of a solution to a discrete Burgers equation recently posted in works of Ben-Naim et al \cite{B-NK1,B-NR1}.  In addition, while much of the global structure of the rarefaction wave-like solutions in \eqref{e:toy_model} remains challenging to describe fully, we present several computations that give insight into the longer time behavior of discrete rarefaction wave solutions as observed numerically in \cite{CMOS1} and are consistent with further mass transfer.

The main goal of developing \eqref{e:toy_model} in \cite{CKSTT} is the construction of a solution to \eqref{e:toy_model}
which transfers mass from low index $j$ to high $j$. In other words, the goal is to robustly construct frequency cascades to show that, as stated in \cite{CMOS1}, ``dispersive equations posed on tori have ``weakly
turbulent dynamics; while there may be no finite time singularity,
arbitrarily high index Sobolev norms can grow to be arbitrarily large, but
finite, in finite time.''  We note that here we are focusing on the dynamical system in \eqref{e:toy_model} and attempting to ascertain how robust the rarefaction wave structure is under the ``phase drift'' inherent to dispersive Schr\"odinger models and the Toy Model in particular.  For other works related to frequency cascades and the study of weak turbulence for NLS, we refer the reader to \cite{bourgain1995cauchy,bourgain1995aspects,bourgain1996growth,B04} as well as the interesting and recent works \cite{Carles:2012jv,Hani2,GK12,Ku1,Soh1,GT1,grebert2012beating,guardia2012growth,haus2012dynamics,faou2013weakly,hani2013modified}.  

The paper will proceed as follows.  In Section \ref{s:conlaw}, we
recall some discrete conservation laws related to the Toy Model to be
applied later.  Then, in Section \ref{s:discrar}, we recall the
modified discrete Burgers equation and corresponding rarefaction wave
approximation using the Madelung transformation from \cite{CMOS1}.  We
proceed in Section \ref{s:rarefaction} to discuss properties of
rarefaction wave solutions to a discrete Burgers equation, drawing
largely from an explicit solution in \cite{B-NK1,B-NR1}, and  study boundary effects in symmetric discrete Burgers equation. In Section \ref{sect:pert} we linearize about the explicit
rarefaction wave in \eqref{e:toy_model} and then study the robustness
of the discrete Burgers rarefaction wave rigorously  (and arguably
sharply) using a Gronwall
type argument.  Finally, we present flux computations related to
truncated conservation laws, discuss future work, open problems and an illustrative computation about the rarefaction wave linearization in discrete $L^2$ spaces, which we hope will provide for more robust control of mass transfer through rarefaction waves in the Toy Model in Section \ref{sect:rem}.

\section{Conserved Quantities}
\label{s:conlaw}


As they will be quite useful in our studies below, we recall here the results from Section 3 of \cite{CKSTT}, where the
Toy Model is studied as a Hamiltonian dynamical system. The
Hamiltonian is given by
\begin{equation}
  \label{e:hamiltonian}
  H [{\bf b}] = \sum_{j=1}^\infty \left( \frac14 | b_j |^4 - \Re ( \bar{b}_j^2 b_{j-1}^2) \right)
\end{equation}
and symplectic structure,
\begin{equation}
  \label{e:symplectic}
  i \frac{db_j}{dt} = \frac{\partial H [{\bf b}]}{\partial \bar b_j},
  \quad j = 1,\ldots, \infty.
\end{equation}
The model \eqref{e:toy_model} admits many of the symmetries of \eqref{e:dcnls},
including phase invariance, scaling, time translation and time
reversal.  However, many of these symmetries are redundant, and the
known only invariant, other than \eqref{e:hamiltonian} is the mass quantity,
\begin{equation}
\label{e:truncmass}
M [{\bf b} ]=  \sum_{j=1}^\infty |b_j|^2.
\end{equation}

\begin{rem}
While we are summing over all nodes above to observe conservation, as the equation with boundary conditions 
\eqref{e:dirichletbc} remains compactly supported on the same region, both $H$ and $M$ are still conserved when one sums only from $j = 1$ to $N$.  
\end{rem}


Using the structure of the equations, it can be seen that given $b (0) $
initially compactly supported on a finite number of nodes, the
solution $b(t)$ remains compactly supported for all time.  This easily
is seen from the structure of the equation.


Defining circles $\mathbb{T}_j$ for $j = 1,\dots,N$ as
\begin{equation*}
  \mathbb{T}_j = \{ {\bf b} = (b_1, \dots, b_N) \ |  \ |b|^2 = 1, \  |b_j| =1, \ b_k = 0 \ \text{for all} \ k \neq j \},
\end{equation*}
the authors in \cite{CKSTT} point out that the flow of \eqref{e:toy_model}, which is
referred to as $S(t) b_0$, leaves each ${\mathbb T}_j$ invariant.  In \cite{CMOS1}, it was observed that \eqref{e:toy_model} also has a natural probabilistic formulation and can be seen to have some basic recurrence properties.

\section{Discrete Rarefaction Waves, the Discrete Burgers Equation
  and Weak Turbulence}
\label{s:discrar}


In this section, we recall the main aspects of Section 5 of \cite{CMOS1}: First, we make the Madelung transformation given by
\begin{equation}
\label{e:mad}
b_j (t) = \sqrt{ \rho_j} e^{ i \phi_j (t) }.
\end{equation}
with \emph{out of phase} initial
interactions set by $\phi_j (0)= \phi_{j+1} (0) + \frac{\pi}{4}$.  Initially, the hydrodynamic
equations have a Burgers type structure
\begin{equation}
  \label{e:burgers}
  \left\{ \begin{array}{l}
      \dot\phi_j = 0  \\
      \dot\rho_j = -4 \rho_j \rho_{j-1}  - 4 \rho_j \rho_{j+1}  = -8 \rho_j \left( \frac{ \rho_{j+1} - \rho_{j-1} }{ 2} \right).
    \end{array} \right. 
\end{equation}
This system has beautiful discrete rarefaction waves propagating towards infinity and a backwards dispersive shock. We also refer the reader to \cite{HerrmannRademacher} for another example of discrete conservation law with a dispersive shock, cp. Subsection \ref{subsect:other}.  We call this the discrete Burgers equation since in the
continuum limit we would have
\begin{equation*}
  \rho_t = -8 \rho \nabla \rho,
\end{equation*}
which with initial data 
\begin{equation*}
\rho (0,x) = \left\{  \begin{array}{ll}
0 \ & \  x \leq 0, \\
1 \ & \  0 < x <  \infty.
\end{array}  \right.  
\end{equation*}
has the known rarefaction wave solution
\begin{equation*}
\rho (t,x) = \left\{  \begin{array}{ll}
0 \ & \  x < 0, \\
\frac{x}{8t} \ & \  0 < x <  8t, \\
1 \  & \ 8t < x.
\end{array}  \right.  
\end{equation*}

However, in our discrete system, there is drag in the phase coefficients that
does not allow us to permanently assume an out of phase framework:
\begin{align}
\label{e:hydro1} 
  \dot\phi_j & =  -\rho_j + 2 \rho_{j-1} \cos(2(\phi_{j-1}-\phi_j)) + 2 \rho_{j+1} \cos(2(\phi_{j+1}-\phi_j)),  \\
  \dot\rho_j & =  -4 \rho_j \rho_{j-1} \sin(2(\phi_{j-1}-\phi_j)) -4 \rho_j \rho_{j+1} \sin(2(\phi_{j+1}-\phi_j)). \label{e:hydro2}
\end{align}
Let us recall some numerical simulations from \cite{CMOS1} to motivate our analysis below.  There, it is numerically studied how solutions evolve, with the initial condition
\begin{equation}
\label{e:shock_ic}
b_j = \exp i \set{ (j-1)\pi/4}.
\end{equation}
In Figure \ref{f:shockshort_compare} we show the numerically computed time evolution of the Toy Model compared to that of a backward discrete Burgers equation.


 \begin{figure}
   \subfigure{\includegraphics[width=6cm,type=pdf,ext=.pdf,read=.pdf]{comp1_rho1_alt}}
   \subfigure{\includegraphics[width=6cm,type=pdf,ext=.pdf,read=.pdf]{comp2_rho1_alt}}
   \subfigure{\includegraphics[width=6cm,type=pdf,ext=.pdf,read=.pdf]{comp3_rho1_alt}}
   \subfigure{\includegraphics[width=6cm,type=pdf,ext=.pdf,read=.pdf]{comp4_rho1_alt}}
   \caption{A comparison of the Toy Model and discrete Burgers rarefaction waves.}
   \label{f:shockshort_compare}
 \end{figure}
 


\section{Analysis of the Rarefaction Wave}
\label{s:rarefaction}

In this section, we present some analytic results on the rarefaction wave.  In particular, we compare solutions to the discrete Toy Model in the hydrodynamic formulation to computations from an explicit solution to a discrete Burgers Equation, which behaves comparably to a continuous Burgers equation.

\subsection{Alternative Coordinates}\label{subsect:alt}

Since the drag term in the phase (the term $-\rho_j$ in \eqref{e:hydro1} and $-i |b_j|^2 b_j$ in \eqref{e:toy_model}) introduces errors in the Burgers evolution, we introduce the coordinate $\theta_j = \phi_j - \phi_{j-1}$.   In the new coordinates, we have
\begin{align}
\label{e:hydroalt1} 
  \dot\theta_j & =  -(\rho_j-\rho_{j-1}) - 2 ( \rho_j - \rho_{j-1}) \cos(2 \theta_j) \\
  &\hspace{1cm}+ 2 \rho_{j+1} \cos(2 \theta_{j+1})  - 2 \rho_{j-2} \cos(2 \theta_{j-1}) ,  \notag \\
  \dot\rho_j & =  4 \rho_j \rho_{j-1} \sin(2\theta_j) -4 \rho_j \rho_{j+1} \sin(2 \theta_{j+1}). \label{e:hydroalt2}
\end{align}

\subsection{Scaling Discrete Burgers}\label{subsect:scaling}

We wish to use an exact solution to a discrete Burgers equation.   The best treatment of which we have found in the works \cite{B-NK1,B-NR1}, where for the backward Discrete Burgers
\begin{equation}
  \label{e:dbackburgers}
      \dot{\tilde{\rho}}_j =  - \tilde{\rho}_j \left(  \tilde{\rho}_{j} - \tilde{\rho}_{j-1} \right),
\end{equation} 
an explicit solution is derived.  To do so, they introduce the transformation ${\tilde \rho}_j = \frac{q_j}{q_{j+1}}$,
and the problem is converted to the recursively solved linear system of ODE
\begin{equation*}
{\dot q}_j = q_{j-1}.
\end{equation*}
With initial data configured only for a rarefaction wave solution
\begin{equation}
\label{e:initraref}
\trho_j (0) = \left\{  \begin{array}{l}
0 \  \  j \leq 0, \\
1 \ \  1\leq j <  \infty,
\end{array}  \right.  
\end{equation} 
this has a solution of the form
\begin{equation*}
q_j(t) =  (1 +  t  + \dots + \frac{1}{(j-1)!} t^{j-1}).
\end{equation*}
Note, this solution does not have finite speed of propagation, in that every non-zero $q_j$ will change for all $t>0$, not just those to the left of a front as in the continuous shock solution.  However, the resulting solution in \eqref{e:dbackburgers} is
\begin{align*}
{\tilde \rho}_j (t) & =  \frac{ 1 + t  + \dots + \frac{1}{(j-1)!} t^{j-1} }{ 1 + t  + \dots + \frac{1}{(j)!} t^{j} }   =  1 - \frac{ \frac{1}{(j)!} t^{j} }{ 1 + t  + \dots + \frac{1}{(j)!} t^{j} } \\
& =  1 - \frac{1}{(j)!} t^{j} + h.o.t.
\end{align*}
for short $t$.
Then, to solve the Burgers equation,
\begin{align}
  \label{e:scaleddbackburgers}
      \dot{\tilde{\rho} }_j (\alpha,\beta, t)  =   -\alpha \tilde{\rho}_j \left(  \tilde{\rho}_{j} - \tilde{\rho}_{j-1} \right), \ 
      \tilde{\rho}_j (\alpha, \beta, 0)  =  \left\{  \begin{array}{l}
0 \  \  j \leq 0, \\
\beta \ \  0 < j <  \infty,
\end{array}  \right.  
\end{align}
we have 
\begin{equation*}
{\tilde \rho}_j  = \beta \frac{ 1 + \alpha \beta t  + \dots + \frac{1}{(j-1)!} (\alpha \beta t)^{j-1} }{ 1 + \alpha \beta t  + \dots + \frac{1}{(j)!} (\alpha \beta t)^{j} } .
\end{equation*}




Generally, in order to control both the dispersive shock\footnote{In discussions with Mark Hoefer, he has suggested that this backward propagating wave takes on more features of turbulence than of a dispersive shock front.  However, to investigate this involves connecting the oscillations in a robust way from the front of the backward propagating front in a more robust fashion.} created by a large jump near $j=N$ and the phase splitting mechanism at large amplitude for lattice sites near $j=1$, the parameter $\beta$ will be set by us to simply be $\epsilon/8$ to study rarefaction waves in the Toy Model, where the $8$ is a scaling parameter chosen for convenience.  Indeed, due to the scalings above and the nature of the Toy Model, we study the following "backwards" discrete Burgers equation,
\begin{equation}
  \label{e:dburgers}
      \dot{\tilde{\rho}}_j =  -8 \tilde{\rho}_j \left(\tilde{\rho}_{j} - \tilde{\rho}_{j-1} \right).
\end{equation}
The appearance of the factor $8$ instead of $4$ here has to do with the fact that we are comparing to the backward Burgers equation and not the symmetric version of Burgers that actually arises in the hydrodynamic formulation.  
We have via the remarkable explicit solution from \cite{B-NK1,B-NR1}, the rescaled solution
\begin{align}
\label{e:scaledburgers}
{\tilde \rho}_j =  \frac{\epsilon}{8} \frac{ 1 + (\epsilon t)  + \dots + \frac{1}{(j-1)!} (\epsilon t) ^{j-1} }{ 1 + (\epsilon t)   + \dots + \frac{1}{(j)!} (\epsilon t) ^{j} } =   \frac{\epsilon}{8} \frac{ \partial_t (1 + (\epsilon t)   + \dots + \frac{1}{(j)!} (\epsilon t) ^{j}) }{ 1 + (\epsilon t)   + \dots + \frac{1}{(j)!} (\epsilon t) ^{j} } . \notag
\end{align}

While we make use of this rescaling in order to get natural smallness in the phase drift term, we can always rescale the solution back to order $1$ by the same argument, see Remark \ref{rem:rescale} for a further discussion.


Now, as a leading order description of the behavior in the full Toy Model, we propose the following modified discrete Burgers equation for capturing the dynamics.  
\begin{equation}
  \label{e:modburgers}
  \left\{ \begin{array}{l}
      \dot {\tilde{\theta}}_j = -(\tilde{\rho}_j-\tilde{\rho}_{j-1})  \\
      \dot {\tilde{\rho}}_j = 8 \tilde{\rho}_j \tilde{\rho}_{j-1}  - 8 \tilde{\rho}_j \tilde{\rho}_{j}  = -8 \tilde{\rho}_j \left( \tilde{\rho}_{j} - \tilde{\rho}_{j-1} \right).
    \end{array} \right. 
\end{equation}
Since these completely decouple, we can use the exact solution for $\tilde \rho$ to solve explicitly for $\tilde \theta$.  

%Since these completely decouple, we observe that
%\begin{equation*}
%\ttheta_{j+1} (t)   \approx   \frac{\pi}{4} + \frac{(\epsilon t)^j}{j!} \left( 1 - \frac{(\epsilon t)}{j} \right) + h.o.t..
%\end{equation*}
%for times $(\epsilon t) < j$ 
%and 
%\begin{equation*}
%\ttheta_{j+1} (t)   \approx   \frac{\pi}{4} 
%\end{equation*}
%for $j < (\epsilon t)$.
%{\color{red}  Expand, move or simply cut this statement??}


We will show in our analysis of \eqref{e:toy_model} that errors arising around this approximation are small on the time scales we study.  It is however the errors in phase term that account primarily for the slight deviations from the pure rarefaction wave on the left and the approximation of discrete symmetric Burgers by discrete backwards Burgers on the right in Figure \ref{f:shockshort_compare}.

\subsection{The Shock vs. Rarefaction in the Toy Model}\label{subsect:shock}



Let us analyze the symmetric discrete Burgers equation in \eqref{e:burgers} with initial condition 
\begin{equation}
\label{e:initdatasymburgers}
\rho_j = 1 \ \text{for} \ 1 \leq j \leq 2 N, \ \ \rho_j = 0 \ \text{otherwise}.
\end{equation}  
Then, we observe that such an equation can be decomposed into a coupled system of equations for $s_j = \rho_{2j}$ and $r_{j} = \rho_{2j+1}$, which results in the system
\begin{equation*}
\dot s_j = -4 s_j (r_j - r_{j-1}) , \ \ 
\dot r_j = -4 r_j (s_{j+1} - s_{j}).
\end{equation*}
Now, if we look at the right-most points, we observe
\begin{equation*}
\dot s_N = 4 s_N (r_{N-1}) , \ \ \dot r_{N-1} = -4 r_{N-1} (s_N - s_{N-1}),
\end{equation*}
which given that $r_j, s_j > 0$ for $1 \leq j \leq N$ implies that $s_N$ is an increasing function.  As a result, this implies that $r_{N-1}$ is a decreasing function.  Propagating this down the line interactions, we observe that the symmetric Burgers causes a splitting from the right endpoint instead of a shock moving right, see Figure \ref{f:symdiscburg} for a numerical simulation of this effect.  Note that the leading order component on the left is still that of a rarefaction wave however.  It is indeed this wave front we believe acts as the envelope for the Toy Model rarefaction wave, however as we do not have explicit control on its evolution, we focus on the rarefaction wave coming from the appropriate backwards Burgers evolution as in \eqref{e:dbackburgers}.

 \begin{figure}
   \includegraphics[width=7cm,type=pdf,ext=.pdf,read=.pdf]{symdiscburgevol}
   \caption{The left moving split that generates the dispersive shock from the symmetric Burgers evolution of \eqref{e:initdatasymburgers}.  This been evolved to time $T=60.0$ on a lattice of size $N=1000$.}
   \label{f:symdiscburg}
 \end{figure}




\section{Pertubation theory}
\label{sect:pert}

\subsection{Linearization}\label{subsect:lin}

Let us now fix a lattice with $N$ nodes and explicitly study equations \eqref{e:hydroalt1} and \eqref{e:hydroalt2} with initial conditions given by
\begin{equation}
\label{e:pertid}
\theta_j (0)  =  \frac{\pi}{4}, \ \ \rho_j (0) = \frac{\epsilon}{8}
\end{equation}
for all $j = 1, \dots, N$ and $0$ otherwise. 

We wish to observe what sorts of error terms arise when we perturb around the dynamics in \eqref{e:modburgers} in the full Toy Model.  The difficulty here is that since we are not plugging in a solution to the full nonlinear problem, there will be linear terms we must bound through careful analysis of a large, linear dynamical system.  In particular, 
we can study the evolution of
\begin{align*}
\hat{\theta}_j   =  \theta_j - \tilde{\theta}_j, \ \hat{\rho}_j  =  \rho_j - \tilde{\rho_j}
\end{align*}
by linearizing about $\tilde{\theta}$, $\tilde{\rho}$ in \eqref{e:hydroalt1}, \eqref{e:hydroalt2}.  Before we proceed, for simplicity, let us define 
\begin{align}
\label{eqn:gammabd}
 - \tgamma_j & =  \int_0^t (\trho_j - \trho_{j-1}) ds \\
& \hspace{.5cm} =  \frac18  \log \left(  \frac{  1 + \epsilon t  + \dots + \frac{1}{(j)!} (\epsilon t)^{j}}{(1 + \epsilon t  + \dots + \frac{1}{(j-1)!} (\epsilon t)^{j-1}) }  \right)  \notag \\
& \hspace{1.0cm}=  \frac18  \log \left( 1 + \frac{ \frac{1}{(j)!} (\epsilon t)^{j}}{(1 + \epsilon t  + \dots + \frac{1}{(j-1)!} (\epsilon t)^{j-1}) }  \right) \notag \\
& \hspace{1.5cm} \leq  \frac18 \log (1+\frac{ \epsilon t}{j} )  \leq \frac{\epsilon t}{ 8 j} \ \ \text{if $\epsilon t \leq 1$},  \notag 
\end{align}
\begin{align}
\label{eqn:sigmabd}
 \tsigma_j & = (\trho_j - \trho_{j-1}) \\
\hspace{0.5cm} &=\frac{\epsilon}8  \frac{ \frac{1}{(j-1)!} (\epsilon t)^{j-1} }{ 1 + (\epsilon t)  + \dots + \frac{1}{(j-1)!} (\epsilon t)^{j-1} }  \times \notag \\
& \hspace{1.0cm} \left( 1 - \frac{(\epsilon t)}{j}  \frac{ 1 + (\epsilon t) + \dots + \frac{1}{(j-1)!} (\epsilon t)^{j-1}}{ 1 + (\epsilon t)  + \dots + \frac{1}{(j)!} (\epsilon t)^{j} }  \right) , \notag \\
& \hspace{1.5cm} \leq \frac{ \epsilon  }{8 (j-1)!} \ \ \text{if $\epsilon t \leq 1$}, \notag
\end{align}
which will be an important explicit, but generally small, component of our expansion.   Note, we will take $\tgamma = \min_j \tgamma_j$ and $\tsigma = \max_j \tsigma_j$ when convenient below. 


We observe by linearizing \eqref{e:hydroalt1}, that
\begin{align*}
\dot{\htheta}_j & =  - (\hrho_j - \hrho_{j-1}) ( 1 + 2  \cos ( \pi/2 + 2 ( \htheta_j + \tgamma_j))) \notag \\
& \hspace{.5cm} - 2 ( \trho_j - \trho_{j-1} ) \cos ( \pi/2 + 2 ( \htheta_j + \tgamma_j)) \notag \\
& \hspace{.75cm} + 2( \hrho_{j+1} + \trho_{j+1})  \cos ( \pi/2 + 2 ( \htheta_{j+1} + \tgamma_{j+1}))) \notag \\
& \hspace{1.0cm} - 2 ( \hrho_{j-2} + \trho_{j-2})  \cos ( \pi/2 + 2 ( \htheta_{j-1} + \tgamma_{j-1})))  \notag \\
& \hspace{1.25cm}= - (\hrho_j - \hrho_{j-1})  + 4 [(\trho_j - \trho_{j-1}) + (\hrho_j - \hrho_{j-1})] (\tgamma_j  + \htheta_j) \notag  \\
&\hspace{1.5cm}- 4 ( \trho_{j+1} + \hrho_{j+1}) (  \tgamma_{j+1} + \htheta_{j+1} )  + 4  ( \trho_{j-2} + \hrho_{j-2}) (  \tgamma_{j-1} + \htheta_{j-1} ) \notag  \\
& \hspace{2cm} + \widetilde{(h.o.t.)}_{1,j} ( \htheta + \tgamma, \hrho),  \notag  
\end{align*}
hence
\begin{align}
\dot{\htheta}_j& =  - (\hrho_j - \hrho_{j-1}) (1 - 4 \tgamma_j) + 4 (\trho_j - \trho_{j-1}) \htheta_j \label{eqn:thetadot}   \\
& \hspace{.5cm} - 4\tgamma_{j+1} \hrho_{j+1} -4 \trho_{j+1} \htheta_{j+1} \notag \\
& \hspace{.75cm} + 4\tgamma_{j-1} \hrho_{j-2} + 4 \trho_{j-2} \htheta_{j-1} \notag \\
& \hspace{1.0cm} + (f)_{1,j} (t) + (h.o.t.)_{1,j} ( \htheta + \tgamma, \hrho) ,  \notag
\end{align}
using the expansion
\begin{equation}
\cos (\pi/2 + x) = -x + \frac{x^3}{3!} - \dots
\end{equation} 
and defining
\begin{align*}
 (f)_{1,j} (t) & = 4 \tsigma_j \tgamma_j - 4( \trho_{j+1} \tgamma_{j+1} - \trho_{j-2} \tgamma_{j-1}) \\
 & = 4 \tsigma_j \tgamma_j - 4 \trho_{j+1} (\tgamma_{j+1} - \tgamma_{j-1}) \\
 & \hspace{1cm} - 4\tgamma_{j-1} ( \trho_{j+1}  - \trho_{j-2} ).
 \end{align*}
 Note here that $\widetilde{(h.o.t.)_1}$ and our final version of $(h.o.t.)_1$ are not the same as we have factored some terms into $f_{1,j}$ to conveniently keep track of terms with certain behaviors.

Next, linearizing \eqref{e:hydroalt2} we have
\begin{align}
\dot{\hrho}_j & =  -4 ( \hrho_j + \trho_j) (\hrho_{j+1} + \trho_{j+1})  \sin ( \pi/2 + 2 ( \htheta_{j+1} + \tgamma_{j+1}))) \notag \\
& \hspace{.5cm} + 4 ( \hrho_j + \trho_j) (\hrho_{j-1} + \trho_{j-1})  \sin ( \pi/2 + 2 ( \htheta_{j} + \tgamma_{j}))) \notag \\
& \hspace{.75cm} + 8 \tilde{\rho}_j \left( \tilde{\rho}_{j} - \tilde{\rho}_{j-1} \right) \notag \\
& \hspace{.25cm} =  -4 \hrho_j ( \trho_{j+1} - \trho_{j-1}) - 4 \trho_{j} \hrho_{j+1} + 4 \trho_j \hrho_{j-1}  \label{eqn:rhodot}  \\
& \hspace{.5cm}  + (f)_{2,j}+ \widetilde{(h.o.t.)}_{2,j}  ( \htheta + \tgamma, \hrho)  \notag
\end{align}
using the expansion
\begin{equation}
\sin (\pi/2 + x) = 1 - \frac{x^2}{2!} - \dots
\end{equation} 
and defining
\begin{align*}
 (f)_{2,j} (t) & =  -4 \trho_j (\tsigma_{j+1}  - \tsigma_{j}).
 \end{align*}
 This term will turn out to be the dominant contribution to our bootstrapping arguments below and accounts for the fact that we are not solving the symmetric Burgers equation in the model.   Again, here $\widetilde{(h.o.t.)_{2,j}}$ and our final version of $(h.o.t.)_{2,j}$ are not the same as we have factored some terms into $f_{2,j}$ to conveniently keep track of terms with certain behaviors.
 
Expanding the $\cos$ terms in the $\htheta+\tgamma$ term and defining
\[
\hsigma_j = \hrho_j - \hrho_{j-1},
\] 
we have
\begin{align*}
& (h.o.t.)_{1,j}  = -2 \hsigma_j [  \cos ( \pi/2 + 2 ( \htheta_j + \tgamma_j))  +  2 ( \htheta_j + \tgamma_j)] \\
& \hspace{.5cm} -2 \tsigma_j [  \cos ( \pi/2 + 2 ( \htheta_j + \tgamma_j))  + 2 ( \htheta_j + \tgamma_j)] \\
& \hspace{.2cm} + 2 (\hrho_{j+1} + \trho_{j+1}) [  \cos ( \pi/2 + 2 ( \htheta_{j+1} + \tgamma_{j+1}))  +  2 ( \htheta_{j+1} + \tgamma_{j+1})] \\
&  \hspace{.3cm}- 2 (\hrho_{j-2} + \trho_{j-2}) [  \cos ( \pi/2 + 2 ( \htheta_{j-1} + \tgamma_{j-1}))  +  2 ( \htheta_{j-1} + \tgamma_{j-1})]    \\
&= 4 (\hrho_j - \hrho_{j-1}) \htheta_j - 4 \hrho_{j+1} \htheta_{j+1} + 4 \hrho_{j-2} \htheta_{j-1} \\
& \hspace{.25cm} -2 ( \hsigma_j + \tsigma_j) \mathcal{O} (| \htheta_j + \tgamma_j|^3) + 2 (\hrho_{j+1} + \trho_{j+1}) \mathcal{O} (| \htheta_{j+1} + \tgamma_{j+1} |^3) \\ 
& \hspace{.5cm} - 2 (\hrho_{j-2} + \trho_{j-2}) \mathcal{O} (| \htheta_{j-1} + \tgamma_{j-1} |^3) 
\end{align*}
and	
\begin{align*}
(h.o.t.)_{2,j} & =   -4 ( \hrho_j + \trho_j) (\hrho_{j+1} + \trho_{j+1}) [ \sin ( \pi/2 + 2 ( \htheta_{j+1} + \tgamma_{j+1})) -1] \\
& \hspace{.2cm} + 4 ( \hrho_j + \trho_j) (\hrho_{j-1} + \trho_{j-1})  [\sin ( \pi/2 + 2 ( \htheta_{j} + \tgamma_{j})) -1]  \\
& =   -4 ( \hrho_j + \trho_j) (\hrho_{j+1} + \trho_{j+1})  \mathcal{O} (| \htheta_{j+1} + \tgamma_{j+1} |^2) \\
& \hspace{.5cm} + 4 ( \hrho_j + \trho_j) (\hrho_{j-1} + \trho_{j-1}) \mathcal{O} (| \htheta_j + \tgamma_j|^2).
\end{align*}
Note, morally $(h.o.t.)_1 \sim \mathcal{O} (|\hrho_j \htheta_j |) + \mathcal{O} (|\trho_j| | \htheta_j + \tgamma_j|^3) $, which turns out to be the most difficult of the terms to bound below.  This is consistent with the errors building up through the expected phase drift.


At this stage, we set out to explore in what sense a rarefaction wave-like solution from the backward Burgers equation approximates the solution to 
\begin{align*}
  \dot\theta_j & =  -(\rho_j-\rho_{j-1}) - 2 ( \rho_j - \rho_{j-1}) \cos(2 \theta_j) \\
  &\hspace{1cm}+ 2 \rho_{j+1} \cos(2 \theta_{j+1})  - 2 \rho_{j-2} \cos(2 \theta_{j-1}) ,  \notag \\
  \dot\rho_j & =  4 \rho_j \rho_{j-1} \sin(2\theta_j) -4 \rho_j \rho_{j+1} \sin(2 \theta_{j+1}). \label{e:hydroalt2}
\end{align*}
with initial data 
\begin{equation*}
\theta_j (0)  =  \frac{\pi}{4}, \ \ \rho_j (0) = \frac{\epsilon}{8}
\end{equation*}
for all $j = 1, \dots, N$ and $0$ otherwise. 

\subsection{A Gronwall Estimate}\label{subsect:gronw}
Multiplying \eqref{eqn:thetadot} by $\htheta_j$ and \eqref{eqn:rhodot} by $\hrho_j$, we obtain
\begin{align}
\label{eqn:diff-ineq1}
\frac{d}{dt} \| \htheta \|_{\ell^\infty}^2 & <  C\big( \| \trho \|_{\ell^\infty}  \| \htheta \|_{\ell^\infty}^2 + \| \hrho \|_{\ell^\infty}  \| \htheta \|_{\ell^\infty}    + \| (f)_{1} \|_{\ell^\infty} \|  \htheta \|_{\ell^\infty}  \\ \notag & \hspace{2cm}   + \| (h.o.t._1) \|_{\ell^\infty} \| \htheta \|_{\ell^\infty} \big), \\\label{eqn:diff-ineq2}
\frac{d}{dt}  \| \hrho \|_{\ell^\infty}^2 & <  C\big(\| \trho + \tgamma^2 \trho \|_{\ell^\infty} \| \hrho \|_{\ell^\infty}^2 + \| (f)_{2} \|_{\ell^\infty} \| \hrho \|_{\ell^\infty} \\
\notag  & \hspace{2cm} + \| (h.o.t._2) \|_{\ell^\infty} \|  \hrho \|_{\ell^\infty}\big).
\end{align}



The following lemma contains the crucial Gronwall estimates. It immediately follows from \eqref{eqn:diff-ineq1} and \eqref{eqn:diff-ineq2} , using $ab\leq \frac{T}{2} a^2+ \frac{1}{2T} b^2$ for all $T > 0$.
\begin{lem}
There exists $C>0$, such that for all $T > t >0$
\begin{align} 
\label{eqn:bootstrap1}
\| \htheta (t)\|_{\ell^\infty}^2 & \leq  C e^{\int_0^t \trho ds} \left( \int_0^t  T \| \hrho  \|_{\ell^\infty}^2 + T \|(f_1)\|_{\ell^\infty}^2 +  T \|(h.o.t.)_1 \|_{\ell^\infty}^2 ds \right) , \\
\label{eqn:bootstrap2}
\| \hrho (t)\|_{\ell^\infty}^2 & \leq C e^{\int_0^t \trho (1 + \tgamma^2) ds} \left( \int_0^t  T \|(f_2)\|_{\ell^\infty}^2 + T \|(h.o.t.)_2\|_{\ell^\infty}^2 ds \right) 
\end{align}
where $C \leq \exp(1/2)/2 < 1$.  
\end{lem}

\subsection{Main result}\label{subsect:main}
We proceed with a bootstrap argument to prove uniform bounds for $(\htheta, \hrho)$.  We will control the error terms with respect to the parameter $\epsilon$ for a grid of size $N$ up to a time $T = \epsilon^{-1}$. 

\begin{thm}\label{thm:main1alt}
There exists an $1 > \delta > 0$, such that for any $0 < \epsilon/8 \leq 1$, $N > 0$, given initial data \eqref{e:pertid} depending upon $\epsilon$ for equations \eqref{e:hydroalt1}-\eqref{e:hydroalt2}, the solution $(\theta,\rho)$ to \eqref{e:hydroalt1}, \eqref{e:hydroalt2} satisfies
$$\| \theta - \ttheta \|_{L^\infty \ell^\infty } \leq \frac{\delta}{8} , \ \  \|\rho- \trho \|_{L^\infty \ell^\infty} \leq   \frac{\delta \epsilon}{12} $$
for all $0 \leq t \leq T = \delta \epsilon^{-1} $.  Here, $(\ttheta (t), \trho (t))$ the explicit rarefaction wave solution to \eqref{e:modburgers} with initial data \eqref{e:pertid}. \end{thm}

\begin{proof}
Let us take a $0< \delta < 1$ to be chosen later.  We define $B_{r} (0)$ denote the closed ball of radius $r$ centered at $0$ in $L^\infty([0,T]; \ell^\infty)$, and hence take the bootstrap assumption to be that 
$$(\htheta,\hrho) \in B_{ \frac{\delta}{8}} (0)\times B_{ \frac{\delta \epsilon}{12}} (0)$$ 
for all time for a given $\alpha$ to be chosen later. In other words, we will assume the following bounds
\begin{equation}
\label{eqn:gronwallbootstrapalt}
\| ( \htheta )\|_{L^\infty_t \ell^\infty} \leq \frac{\delta}{8} , \ \ \| ( \hrho )\|_{L^\infty_t \ell^\infty} \leq \frac{\delta\epsilon}{12}.
\end{equation} 


Let us note here that similar to the computation in \eqref{eqn:gammabd}, we have
\begin{equation}
\label{eqn:expgronbdalt}
e^{\int_0^T \trho_j (s) ds}  \leq
\big(  1 + \epsilon T  + \dots + \frac{1}{(j)!} (\epsilon T)^{j}\big)^{\frac18}
\leq e^{\frac{ \epsilon T}{8}} \leq e^{\frac{\delta}8}
\end{equation}
for $T \leq \delta \epsilon^{-1}$.  
Due to the bootstrap assumption we have for $t \leq \delta \epsilon^{-1}$ the bound
$$\max_{j }|\htheta_j|+|\tgamma_j| \leq \delta + \frac{ \delta}{8} < 1,$$ 
hence we can write the explicit bounds 
\[
|(h.o.t.)_{1,j}|   \leq 12 \epsilon ( |\tgamma| + \delta) <  \frac{1}{2^5} \epsilon \delta^2, \ |(f)_{1,j} |  \leq 3 \epsilon |\tgamma| < \frac{1}{2} \epsilon \delta
\]
and
\[ 
 |(h.o.t.)_{2,j}|   \leq \frac{16}{2^{6}} \epsilon^2 ( |\tgamma|^2 + \delta^2) \leq \frac{1}{2^{8}} \epsilon^2 \delta^2 ,\ | (f)_{2,j} |  \leq \left( \frac{\epsilon}{4} \right)^2.
\]
Here, we have taken the coefficients $12$ and $16$ respectively for $|(h.o.t)_{1,j}|$, $|(h.o.t)_{2,j}|$ in order to bound all higher order terms by twice the worst bound on those of lowest order.  Careful control of such error terms will likely allow for somewhat sharper bounds, however these terms are much lower order compared to boundary effects, so we do not work carefully to optimize them.

In fact, we note that
\[ 
\| (f)_{2,j} \|_{L^\infty_t} \leq \frac{ \delta \epsilon^2 }{2^4}
\]
for all $1 < j <  N$ by taking into consideration the $\epsilon t$ factor that arises in the difference of $\sigma$'s from estimate \eqref{eqn:sigmabd} since the leading order term or order $\epsilon/8$ will cancel.  However, for $j = 1,N$, we compute explicitly from \eqref{eqn:sigmabd} that
\[
\| (f)_{2,1} \|_{L^\infty_t} , \  \| (f)_{2,N} \|_{L^\infty_t} \leq \frac{\epsilon^2}{2^4} (1 - \mathcal{O} (\delta)) \leq  \frac{\epsilon^2}{2^4}.
\]
Thus, we deduce that there exists $C > 0$ (can be fixed uniformly in $\epsilon$ for $\delta$ small) such that
\begin{align*}
|\text{r.h.s.\ of }\eqref{eqn:bootstrap1}|  & \leq   \frac{e^{\frac12 + \frac{\delta}8}}{2}   T^2 \epsilon^2 [  \frac{\delta^2}{2^4 3^2} + 2 \frac{ \delta}{2}^2   ]  \leq C \delta^4 < \delta^2 , \\
|\text{r.h.s.\ of }\eqref{eqn:bootstrap2}| & \leq  \frac{e^{\frac12 + \frac{\delta}8 ( 1 + \frac{\delta}{8}^2)}}{2}  T^2 \epsilon^2 [ \frac{ \epsilon^2}{2^8} + \frac{1}{2^{14}} \delta^4 \epsilon^2 ]  \\
&  \hspace{.5cm}  \leq \frac{\epsilon^2 \delta^2}{12^2} \left[ \frac{12^2}{2^8} + \delta^4 \frac{12^2}{2^{14}} \right]  < \frac{ \delta^2 \epsilon^2}{12^2}
\end{align*}
for $\delta <1$ sufficiently small, but chosen independently of $\epsilon$ and $N$.
As a result, we can close the bootstrapping argument independent of our choice of lattice size $N$ and any initial step size $\epsilon/8$.  
\end{proof}

\begin{rem}
\label{rem:rescale}

By nature of the construction, we now have that
\begin{equation*}
|b_j|^2 (t) = \rho_j (t) \leq \trho_j (t) + \left(\frac{\epsilon \delta}{12} \right).
\end{equation*}
Hence, near the end of our evolution the rarefaction wave at the left of the grid has size roughly
\begin{equation*}
\trho_1 (T) = \frac{\epsilon}{8} \frac{1}{1 + \epsilon T} \approx \frac{ \epsilon }{8}(1 - \delta),\end{equation*}
which is less than the initial amplitude $\epsilon/8$ even when compared to the error.  Hence, we observe that our method moves mass towards higher grid points.  Obviously we would like a much stronger proof of mass transfer by rarefaction wave dynamics, which remains an open problem relating to the global structure of the mass transfer in the full Toy Model.  
\end{rem}


\section{Additional Observations and Remarks}
\label{sect:rem}

\subsection{Flux Computation for Finite Approximations}
\label{sec:flux}

We have the Hamiltonian system
\begin{equation}
\left\{ \begin{array}{c}
-i \partial_t b_j  = -| b_j|^2 b_j + 2 b_{j-1}^2 \bar{b}_j + 2 b_{j+1}^2 \bar{b}_j, \\ 
i \partial_t \bar{b}_j  = -| \bar{b}_j|^2 \bar{b}_j + 2 \bar{b}_{j-1}^2 b_j + 2 \bar{b}_{j+1}^2 b_j,
\end{array} \right.
\end{equation}
which is only Hamiltonian with respect to the infinite sum unless we are certain that our initial data is compactly supported.  However, as suggested to us by Jonathan Mattingly \cite{matt-pc} based off of ideas in \cite{MSV-cascade}, let us take initial data supported on the infinite half lattice, yet restrict the Hamiltonian system to the first $N$ nodes and simply look at the flux in the energy at this sufficiently high node, where we now have
\begin{equation}
H_N = \sum_{j=1}^N \frac14 |b_j|^4 - \Re ( \bar{b}_j^2 b_{j-1}^2),
\end{equation}
which now is not perfectly conserved.  Indeed, we have 
%\begin{equation}
%\partial_t H_N = -2 |b_N|^2 \Im ( b_{N-1}^2 \bar{b}_{N+1}^2 - \frac12 b_N^2 b_{N+1}^2).
%\end{equation}
\begin{equation}
\partial_t H_N = 2 |b_N|^2 \Im ( 2 b_{N+1}^2 \bar{b}_{N-1}^2 -   b_{N+1}^2 \bar{b}_{N}^2).
\end{equation}

Moving from the exact formula to do some asymptotic analysis, if we assume roughly comparable amplitude (Note:  we believe is the case at say $j = \frac{N}{2}$ up to the time the rarefaction wave and dispersive shock meet) of the final three nodes gives
\begin{equation}
\partial_t H_N \approx  A_N^6 [4 \sin (2 \phi_{N+1} -2 \phi_{N-1}) - 2 \sin (2 \phi_{N+1} -2 \phi_{N}) ]
\end{equation}
taking $b_j= A_j e^{i \phi_j}$.
Hence, the Hamiltonian flux is seen to be positive (inward flow of energy) if 
\begin{equation}
[ \sin (2 \phi_{N+1} -2 \phi_{N-1}) - \frac12 \sin (2 \phi_{N+1} -2\phi_{N}) ] > 0.
\end{equation}
%which holds for instance if 
%$$0 < 2 \phi_{N+1} - 2\phi_{N-1} < \pi , \ \ -\pi < 2 \phi_{N+1} - 2 \phi_{N}  < 0,$$
%or as would be the case for $\phi_{j+1} = \frac{(j-1) \pi}{4}$ if $N$ is even.
In order to see outward flow of energy, the Hamiltonian flux is negative if 
\begin{equation}
[ \sin (2 \phi_{N+1} - 2\phi_{N-1}) - \frac12 \sin (2 \phi_{N+1} - 2 \phi_{N}) ] < 0.
\end{equation}
%which holds for instance if 
%$$-\pi < 2 \phi_{N+1} - \phi_{N-1} < 0 , \ \ 0 < 2 \phi_{N+1} - \phi_{N}  < \pi.$$
%or as would be the case for $\phi_{j+1} = \frac{(j-1) \pi}{4}$ if $N$ is odd.  
For $\phi_{j} = \frac{(j-1) \pi}{4}$, we observe 
\begin{equation}
[ \sin (2 \phi_{N+1} - 2\phi_{N-1}) - \frac12 \sin (2 \phi_{N+1} - 2 \phi_{N}) ] = -3/2 ,
\end{equation}
 which would actually result in an outward flow of energy, though as the Hamiltonian energy is not coercive, we do not gain much from this computation.

Alternatively, we look at the restricted mass flux,
\begin{equation}
M_N = \sum_{j=1}^N |b_j|^2,
\end{equation}
which now is not perfectly conserved.  Indeed, we have 
\begin{equation}
\partial_t M_N = -4  \Im ( b_{N+1}^2 \bar{b}_N^2).
\end{equation}
Making a similar asymptotic assumption at the endpoint, we have 
\begin{equation}
\partial_t M_N \approx  -4  |A_N|^4 \sin ( 2 (\phi_{N+1} - \phi_N)).
\end{equation}
Hence, we observe that the mass flux is {\it outgoing} for $\phi_{j} = \frac{(j-1) \pi}{4}$ since we then have 
\begin{equation}
\partial_t M_N \approx  -4  |A_N|^4 \sin ( \frac{\pi}{2}).
\end{equation}
%Later we will study rarefaction wave solutions in the full toy model and re-visit the mass flux computation using specific solutions.

\begin{rem}
\label{rem:massflux}
As pointed out by the anonymous referee, we can use the mass flux computation from Section \ref{sec:flux} to study the rarefaction wave solution from Theorem \ref{thm:main1alt}.  In such a case, we observe that for say the node $j=N/2$, which remains roughly fixed at $\rho_j (t) \sim \epsilon/8$ and $\theta_j (t) \sim \pi/4$, we have the total mass moved across this node of order 
$$ \epsilon^{-1} \epsilon^4 = \epsilon^3.$$
Initially, this looks like a rather small mass flux compared to the size of the solution overall.  However, we not first of all that in this setting $\epsilon$ need not necessarily be extremely small since the asymptotic methods are are done mostly on the side of the $\delta$ parameter. In addition, we note that numerically of course, the mid-point of the rarefaction wave solution remains stable much longer than the time scales we have controlled here.  Indeed, while the rarefaction wave moving left and the dispersive shock like solution moving right definitely change the structure of the backwards Burgers equation on a time scale of order $\epsilon^{-1}$, we have strong numerical evidence that away from the fronts of those waves the solution remains largely unchanged.  Hence, we expect that with greater global control over the dynamics, the mass flux computation can be shown to be much stronger than can be applied on the time scales in Theorem \ref{thm:main1alt}.
\end{rem}

\begin{rem}
\label{rem:rescale1}

As commented in \cite{CMOS1}, this analysis still leads to open questions about Sobolev norm growth in the full problem \eqref{e:dcnls} given the pointwise bounds on the error for $j \sim N/2$ on the same time scale, we observe that the flux computation in Section \ref{sec:flux} will continue moving mass towards high $j$ on this time scale.  In addition, computational checks of the constants suggest that the bootstrapping arguments in Theorem \ref{thm:main1alt} appear to go through for $\delta$ chosen even as large as $1/2$, meaning while we need our time interval to be $o(1)$, there should be parts of the argument that extend to time $1$.  Doing so in a rigorous fashion likely requires more analytic control on the global structure of the rarefaction wave-like solution both in discrete symmetric Burgers and in terms of the behavior in the Toy Model near the right boundary.  
\end{rem}




\subsection{An observation about $\| \cdot \|_{\ell^2}$ growth of $\hrho$, $\htheta$}\label{subsect:obs}

We present here an illustrative computation, which unfortunately at the moment we cannot apply in a perturbation theoretic argument as we would require stronger control of the behavior of solutions to \eqref{e:toy_model} at the endpoints of our finite region.  Let us  recall that
\begin{align*}
\dot{\hrho}_j  =  4\hrho_{j-1} \trho_j - 4 \hrho_j (\trho_{j+1} - \trho_{j-1} ) - 4\hrho_{j+1}\trho_j+ F_j,
\end{align*}
with \[F_j=2 \hrho_{j-1} \trho_{j} \tgamma_j^2  - 2\hrho_{j+1}\trho_{j} \tgamma_{j+1}^2 + (f)_{2,j} (t) + \mathcal{O} (|\hrho_j + \trho_j|^2 |\htheta_j +
  \tgamma_j|^2).\]
Combining terms from nearest neighbors we have
\begin{align*}
&\tfrac12  \partial_t {( \hat{\rho}_{j-1}^2)} + \tfrac12 \partial_t {(
  \hat{\rho}_{j}^2)} + \tfrac12 \partial_t { (\hat{\rho}_{j+1}^2 )}\\
& \hspace{.5cm}  = 4\hrho_{j-2}\hrho_{j-1}\trho_{j-1}-4\hrho_{j-1}^2(\trho_j-\trho_{j-2})-4\hrho_{j}\hrho_{j-1}\trho_{j-1}+F_{j-1}\hrho_{j-1}\\
&\hspace{1.0cm} +4\hrho_{j-1}\hrho_{j}\trho_{j}-4\hrho_{j}^2 (\trho_{j+1}-\trho_{j-1}) -4\hrho_{j+1}\hrho_{j}\trho_{j}+F_{j}\hrho_{j}\\
& \hspace{1.5cm} +4\hrho_{j}\hrho_{j+1}\trho_{j+1}-4\hrho_{j+1}^2 (\trho_{j+2}-\trho_{j}) -4\hrho_{j+2}\hrho_{j+1}\trho_{j+1}+F_{j+1}\hrho_{j+1}.
\end{align*}
By combining nearby terms, we observe that
\begin{align*}
&-4\hrho_{j-1}^2(\trho_j-\trho_{j-2})-4\hrho_{j}\hrho_{j-1}\trho_{j-1}+4\hrho_{j-1}\hrho_{j}\trho_{j}-4\hrho_{j}^2
  (\trho_{j+1}-\trho_{j-1}) \\
& \hspace{1cm} -4\hrho_{j+1}\hrho_{j}\trho_{j} +4\hrho_{j}\hrho_{j+1}\trho_{j+1}-4\hrho_{j+1}^2 (\trho_{j+2}-\trho_{j})\\
& \hspace{.5cm} = -4\hrho_{j-1}^2(\trho_j-\trho_{j-2}) +4 \hat{\rho}_j
\hat{\rho}_{j-1} ( \tilde{\rho}_{j} - \tilde{\rho}_{j-1})
- 4 \hat{\rho}_j^2( \tilde{\rho}_{j+1} - \tilde{\rho}_{j-1}) \\
& \hspace{1.75cm}+ 4\hat{\rho}_j \hat{\rho}_{j+1}( \tilde{\rho}_{j+1} - \tilde{\rho}_{j})
 -4\hrho_{j+1}^2 (\trho_{j+2}-\trho_{j})\\
& \hspace{1cm}  \leq   - 2 \hat{\rho}_{j-1}^2 ( \tilde{\rho}_{j} - \tilde{\rho}_{j-2})
 -2 \hat{\rho}_j^2 ( \tilde{\rho}_{j+1} - \tilde{\rho}_{j-1}) - 2 \hat{\rho}_{j+1}^2 ( \tilde{\rho}_{j+2} - \tilde{\rho}_{j}) \\
 & \hspace{2cm} \leq 0
\end{align*}
by Cauchy-Schwarz. Summation yields
\[
\tfrac12 \partial_t \sum_{j=1}^{N} \hat{\rho}_{j}^2 \leq 
2 \hrho_N^2 ( \trho_N + \trho_{N-1} )
+\sum_{j=1}^{N} F_{j}\hrho_{j} 
\]
since the $\hrho$ terms are compactly supported on the interval $j = 1, \dots N$.  Hence, we observe that on the time scale of evolution, the leading order terms here involve only the right endpoints and are by construction positive, which fits with the conservation of mass in \eqref{e:toy_model}.

\subsection{Other Discrete Conservation Laws}\label{subsect:other}
In \cite{HerrmannRademacher}, the authors study Fermi-Pasta-Ulam Systems of the form
\begin{equation*}
\dot{r}_j = v_{j+1} - v_j, \\
\dot{v}_j = \phi' (r_j) - \phi' ( r_{j-1}),
\end{equation*}
where $\phi (r)  = e^{1-r} - (1-r)$.  Continuous limits $(r_j, v_j) (t) = (r,v) (\epsilon t, \epsilon j)$ of such models also satisfy the Burgers equation
\begin{equation*}
\partial_t r - \partial_x v = 0, \quad \partial_t v - \partial_x \phi' (r) = 0.
\end{equation*}
Rarefaction waves and dispersive shocks are observed and studied in depth numerically and some analysis is done on conservative shock formation.   In future work, we hope similar analysis can be done to that for \eqref{e:toy_model}. 


\subsection*{Acknowledgements}
The first author was supported by the German Research Foundation, CRC 701.
The second author was supported by a combination of an IBM Junior
  Faculty Development Award through the University of North Carolina, NSF Grant 1312874, plus Guest Lectureships through Universit\"at Bielefeld Summer 2012 and Karlsruhe Institute of Technology in Summer 2013.  
  
  Thanks especially to Gideon Simpson for many helpful conversations about this topic and for allowing us to modify his code to create the pictures seen here.  The authors also wish to thank Jianfeng Lu, Hiro Oh and Jonathan Mattingly for interesting discussions, but particularly Mattingly for suggesting the flux computation displayed in Section \ref{sec:flux}.  In addition, the authors thank the anonymous referee for pointing out many places the exposition could be improved or clarified and in particular for such a careful reading of the article.
\bibliographystyle{abbrv}

\bibliography{ToyModel2}



\end{document}


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