# Approximation of (some) Random FPUT Lattices by KdV Equations

Joshua A. McGinnis  University of Pennsylvania, Philadelphia  and  J. Douglas
Wright  Drexel University, Philadelphia

###### Abstract.

We consider a Fermi-Pasta-Ulam-Tsingou lattice with randomly varying
coefficients. We discover a relatively simple condition which when placed on
the nature of the randomness allows us to prove that small amplitude/long
wavelength solutions are almost surely rigorously approximated by solutions of
Korteweg-de Vries equations for very long times. The key ideas combine energy
estimates with homogenization theory and the technical proof requires a novel
application of autoregressive processes.

##  1\. Introduction

Consider a variable mass Fermi-Pasta-Ulam-Tsingou (FPUT) lattice111We write
the equations as a first order system as opposed to its possibly more familiar
second order form
$m(j)\ddot{x}(j)={\mathcal{V}}^{\prime}(x(j+1)-x(j))-{\mathcal{V}}^{\prime}(x(j)-x(j-1)).$
The change of variables leading from this to (1.1) is $q(j)=x(j+1)-x(j)$ and
$p(j)=\dot{x}(j)$.:

(1.1) $\dot{q}(j,t)=\delta^{+}p(j,t)\quad\text{and}\quad
m(j)\dot{p}(j,t)=\delta^{-}[{\mathcal{V}}^{\prime}(q)](j,t).$

Here $t\in{\bf{R}}$, $j\in{\bf{Z}}$ and the unknowns $q(j,t)$ (the relative
displacement) and $p(j,t)$ (the velocity) are real-valued. The mass
coefficients $m(j)$ are strictly positive and

(1.2) ${\mathcal{V}}(q):={1\over 2}q^{2}+{1\over 3}q^{3}$

is the spring potential222This choice of the spring potential—which is an
instance of the “$\alpha$-potential” from [9]—is made mainly for simplicity.
We could allow more complicated potentials and, so long as we had
${\mathcal{V}}^{\prime}(0)>0$ and ${\mathcal{V}}^{\prime\prime}(0)\neq 0$,
only minor changes to our results would occur.. Lastly

$\delta^{+}f(j)=f(j+1)-f(j)\quad\text{and}\quad\delta^{-}f(j)=f(j)-f(j-1)$

are the right and left finite-difference operators.

Models of this sort are ubiquitous in applications. A partial list: molecular
dynamics, lamination, nondestructive testing, vehicular traffic, granular
media, metamaterials, chemistry/biochemistry, and power generation [21]. The
system (1.1) also plays a major role as a paradigm for the mathematical
analysis of wave propagation—especially solitary waves—in nonlinear dispersive
settings and it is the system’s famous connection to the Korteweg-de Vries
(KdV) equation wherein our interest lies. Here are several important
mathematical results about that connection:

  * •

When $m(j)$ is constant, long-wavelength (say like $1/\epsilon$, where
$0<\epsilon\ll 1$), small-amplitude (order of $\epsilon^{2}$) solutions are
well-approximated over long time scales (order of $1/\epsilon^{3}$) by
solutions of KdV equations. The relative $\ell^{2}$-error made by the
approximation in this case is ${\mathcal{O}}(\epsilon)$. See [26] for the
earliest formal derivation and [23] for the first rigorous result.

  * •

The same sort of result holds when $m(j)$ is $N$-periodic (that is,
$m(j+N)=m(j)$ for all $j$). Indeed, the spring potentials may also be taken to
be $N-$periodic (for instance, replace ${\mathcal{V}}(q)$ with
${\mathcal{V}}_{j}(q)=\kappa(j)q^{2}/2+\beta(j)q^{3}/3$ where $\kappa(j)$ and
$\beta(j)$ are $N-$periodic). See [2, 10].

While there have been a few derivations of KdV from random versions of the
FPUT lattice previously (specifically [14, 25]), all have been purely formal
with no rigorous quantitative results. Even conjectures for the size of the
error have been absent. We have been working for some time to remedy this. In
our article [20] we discovered that if the $m(j)$ are independent identically
distributed (i.i.d.) random variables then the accuracy of long-wavelength
approximations is substantially diminished and, consequently, only shorter
time scales and the linear problem (that is when
${\mathcal{V}}^{\prime}(q)=q$) are within reach. Precisely, we showed333We
also allow for heterogeneity in ${\mathcal{V}}$ as well as in $m$ in [20]; our
results apply to the case where ${\mathcal{V}}^{\prime}(q)$ replaced by
${\mathcal{V}}^{\prime}_{j}(q)=\kappa(j)q$ with $\kappa(j)$ another collection
of i.i.d. random variables. that long-wavelength solutions (again like
$1/\epsilon)$ converge almost surely and strongly, but rather slowly, to
solutions of a wave equation on time scales on the order of $1/\epsilon$: the
relative $\ell^{2}$-error made by the approximation is almost surely
${\mathcal{O}}(\sqrt{\epsilon\ln|\ln(\epsilon)|})$. Numerics indicate that our
error estimate is close to sharp. In [19], McGinnis proved a similar result
for a 2D lattice.

Furthermore, formal and numerical studies of random FPUT and other similar
random lattice problems report that the waves in such systems experience a
notable deterioration of their amplitude as time evolves (see, for instance,
[11, 15, 16, 18]). We have carried out our own simulations of the nonlinear
problem (1.1) with i.i.d. random variables as coefficients in the long-
wavelength/small amplitude regime. These simulations demonstrate that for time
scales longer than $1/\epsilon$, solutions of (1.1) attenuate substantially;
KdV-like dynamics (namely, resolution into solitons) is not observed. We
include the results of our simulations below in Section 7, Figure 1. In short,
we do not believe that when the coefficients are i.i.d. random variables a KdV
approximation is appropriate or possible.

However, there are more sorts of randomness than simply taking the
coefficients to be i.i.d.. In this paper we consider the random case, but we
restrict the randomness in such a way that we can prove a fully rigorous KdV
approximation. We believe that this is the first example of such a result
involving randomness and nonlinear dispersive systems, though there are
several earlier results which carefully derive—but do not fully justify—KdV as
an effective equation for the evolution of long water waves over randomly
varying topography [22, 4, 6, 5].

Here is our assumption on the the masses:

######  Hypothesis 1.1.

The masses are given by

(1.3) $m(j)=1+\delta^{+}\delta^{-}\zeta(j)$

where $\zeta(j)$, $j\in{\bf{Z}}$, are i.i.d. random variables with zero mean,
variance $\sigma^{2}$ and support contained in $(-1/4,1/4)$.

We refer to (1.3) as a transparency condition and we call (1.1) subject to
Hypothesis 1.1 the transparent random mass FPUT lattice. The use of
“transparent” here is due to an observation from simulations: if the masses
meet this condition then waves propagate relatively cleanly through the
lattice without too much “back scattering” or “internal reflection.” Our idea
for making this choice was inspired by the derivation of KdV as an effective
equation for water waves over a random bottom in [22] where the topography is
given as a perfect spatial derivative. The condition on the support of
$\zeta(j)$ is there to ensure that $m(j)$ are strictly positive (for if
$|\zeta(j)|<1/4$ then the triangle inequality tells us $m(j)>0$). It also
guarantees that $\sigma^{2}<\infty$.

Our main result in a nutshell: for suitable initial conditions, solutions of
the transparent random mass FPUT lattice almost surely satisfy

(1.4)
$\begin{split}q(j,t)&=\epsilon^{2}\left[A(\epsilon(j-t),\epsilon^{3}t)+B(\epsilon(j+t),\epsilon^{3}t)\right]+{\mathcal{O}}_{\ell_{2}}(\epsilon^{2}\sqrt{|\ln(\epsilon)|})\\\
p(j,t)&=\epsilon^{2}\left[-A(\epsilon(j-t),\epsilon^{3}t)+B(\epsilon(j+t),\epsilon^{3}t)\right]+{\mathcal{O}}_{\ell_{2}}(\epsilon^{2}\sqrt{|\ln(\epsilon)|})\end{split}$

for $|t|\leq T_{0}/\epsilon^{3}$, where $A$ and $B$ solve the KdV equations

$2\partial_{T}A+\left({1\over
12}+2\sigma^{2}\right)\partial_{w}^{3}A+\partial_{w}A^{2}=0\quad\text{and}\quad
2\partial_{T}B-\left({1\over
12}+2\sigma^{2}\right)\partial_{l}^{3}B-\partial_{l}B^{2}=0.$

The fully technical version of our result appears in Theorem 6.1 below.

######  Remark 1.2.

To the uninitiated, it may look like the size of the error exceeds the size of
the approximation. However, the long-wave scaling of the spatial coordinate
gives
$\|\epsilon^{2}A(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}={\mathcal{O}}(\epsilon^{3/2})$
which indicates a relative $\ell^{2}$-error of
${\mathcal{O}}\left(\sqrt{\epsilon|\ln(\epsilon)|}\right)$.

Our paper is organized as follows. Section 2 spells out some notation and
other ground rules. Section 3 proves a general approximation theorem for
(1.1). While motivated by KdV approximations, the result applies more broadly.
Section 4 contains the derivation of KdV from (1.1) under Hypothesis 1.1; this
is the heart of the paper. Section 5 contains a multitude of estimates which
set the stage for the application of the general approximation theorem. It is
in this section where probability plays a major role and where the technical
guts of our result live. Section 6 ties everything together with the statement
and proof of our main result, the technical version of (1.4). Then we present
the result of supporting numerics in Section 7. We close out with a big list
of open questions in Section 8.

Acknowledgements: The authors would like to thank Amanda French, C. Eugene
Wayne and Atilla Yilmaz for helpful conversations related to this project.
Also, JDW would like to recognize the National Science Foundation who
supported this research with grant DMS-2006172.

##  2\. Preliminaries

###  2.1. Function/sequence spaces

For a doubly infinite sequence $f:{\bf{Z}}\to{\bf{R}}$ we put, as per usual,
$\|f\|_{\ell^{2}}:=\sqrt{\sum_{j\in{\bf{Z}}}f^{2}(j)}$ and
$\|f\|_{\ell^{\infty}}:=\sup_{j\in{\bf{Z}}}|f(j)|$. Of course $\ell^{2}$ and
$\ell^{\infty}$ are the sets of all sequences where the associated norms are
finite. If we write $\|f,g\|_{\ell^{2}}$ we mean
$\|f\|_{\ell^{2}}+\|g\|_{\ell^{2}}$, the norm on $\ell^{2}\times\ell^{2}$. The
analogous convention applies to $\|f,g\|_{\ell^{\infty}}$. For functions
$F:{\bf{R}}\to{\bf{R}}$ and non-negative integers $n$ and $r$ we put

$\|F\|_{H^{n}(r)}:=\sqrt{\int_{\bf{R}}(1+X^{2})^{r}F^{2}(X)dX+\int_{\bf{R}}(1+X^{2})^{r}(\partial_{X}^{n}F)^{2}(X)dX}$

and $H^{n}(r)$ is the closure of the set of all smooth functions with respect
to this norm. We define $H^{n}:=H^{n}(0)$, $L^{2}(r):=H^{0}(r)$ and
$L^{2}:=H^{0}(0)$. Next,
$\|F\|_{W^{n,\infty}}:=\sup_{X\in{\bf{R}}}|F(X)|+|\partial_{X}^{n}F(X)|$ and
$W^{n,\infty}$ is the associated function space. By $L^{\infty}$ we mean
$W^{0,\infty}$. All of the spaces listed above are Banach spaces.

###  2.2. Probability

All probabilistic components in the paper descend through the random variables
$\zeta(j)$. Associated probabilities are represented by ${\mathbb{P}}$ and
expectations by ${\mathbb{E}}$.

###  2.3. ${\mathcal{O}}$, $o$ and $C$ notation

We use the following version of Landau’s “big ${\mathcal{O}}$/little $o$”
notation. Given two real-valued functions, $f(\epsilon)$ and $g(\epsilon)$, we
say $f(\epsilon)={\mathcal{O}}(g(\epsilon))$ if for some $C_{\star}>0$ and
$\epsilon_{\star}>0$, $|f(\epsilon)|\leq C_{\star}|g(\epsilon)|$ for
$\epsilon\in(0,\epsilon_{\star})$. We say $f(\epsilon)=o(g(\epsilon))$ if
$\lim_{\epsilon\to 0^{+}}|f(\epsilon)/g(\epsilon)|=0$. If $Y$ is a Banach
space and we have a family of elements $u_{\epsilon}\in Y$, we write
$u_{\epsilon}={\mathcal{O}}_{Y}(g(\epsilon))$ if
$\|u_{\epsilon}\|_{Y}={\mathcal{O}}(g(\epsilon))$ by the earlier definition.
If we write $f(\epsilon)=g(\epsilon)+{\mathcal{O}}(h(\epsilon))$ (or similar)
we mean $f(\epsilon)-g(\epsilon)={\mathcal{O}}(h(\epsilon))$. We at times
default to “big $C$” notation: if we simply write $f(\epsilon)\leq
Cg(\epsilon)$ and omit qualifiers then we mean
$f(\epsilon)={\mathcal{O}}(g(\epsilon))$.

Some quantities will depend on the random variables $\zeta(j)$. For such
quantities, if we write $f(\epsilon)={\mathcal{O}}(g(\epsilon))$ we mean this
in an almost sure sense. Specifically, there exist constants $C_{\star}>0$
(almost surely finite) and $\epsilon_{\star}$ (almost surely positive) such
that $|f(\epsilon)|\leq C_{\star}|g(\epsilon)|$ for
$\epsilon\in(0,\epsilon_{\star})$. The values of $C_{\star}$ and
$\epsilon_{\star}$ may depend upon the realization of the $\zeta(j)$. The same
“almost sure” point of view holds for ${\mathcal{O}}_{Y}$ and $o$ too.

To be clear: we always mean ${\mathcal{O}}$, $o$ and their ilk rigorously, and
we always mean them in the almost sure sense.

##  3\. Approximation in general

We begin by proving a general approximation theorem using the strategy
described in Section 5.3 of [10] (itself inspired by [23]). Suppose that for
$\epsilon\in(0,1)$ we have some functions $\widetilde{q}_{\epsilon}(j,t)$ and
$\widetilde{p}_{\epsilon}(j,t)$ (the approximators) that we expect are good
approximations to solutions of (1.1) when $\epsilon$ is small. By this we mean
that we know that $\widetilde{q}_{\epsilon}(j,t)$ and
$\widetilde{p}_{\epsilon}(j,t)$ nearly solve (1.1) in the sense that the
residuals

(3.1)
$\operatorname{Res}_{1}:=\delta^{+}\widetilde{p}_{\epsilon}-\partial_{t}\widetilde{q}_{\epsilon}\quad\text{and}\quad\operatorname{Res}_{2}:={1\over
m}\delta^{-}\left[{\mathcal{V}}^{\prime}(\widetilde{q}_{\epsilon})\right]-\partial_{t}\widetilde{p}_{\epsilon}$

are small relative to $\epsilon$. To validate the approximation over the time
scale $|t|\leq T_{0}/\epsilon^{3}$ we need information about

(3.2) $\begin{split}\alpha_{1}(\epsilon)&:=\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\widetilde{q}_{\epsilon},\widetilde{p}_{\epsilon}\|_{\ell^{2}},\\\
\beta_{1}(\epsilon)&:=\inf_{|t|\leq
T_{0}/\epsilon^{3}}\|\widetilde{q}_{\epsilon},\widetilde{p}_{\epsilon}\|_{\ell^{2}},\\\
\alpha_{2}(\epsilon)&:=\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\partial_{t}\widetilde{q}_{\epsilon}\|_{\ell^{\infty}}\quad\text{and}\quad\\\
\alpha_{3}(\epsilon)&:=\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\operatorname{Res}_{1}\|_{\ell^{2}}+\|\operatorname{Res}_{2}\|_{\ell^{2}}.\end{split}$

In particular, we assume:

(3.3)
$\alpha_{1}(\epsilon)=o(1),\quad\alpha_{2}(\epsilon)={\mathcal{O}}(\epsilon^{3})\quad\text{and}\quad\alpha_{3}(\epsilon)=o(\beta_{1}(\epsilon)\epsilon^{3}).$

Our goal is to show that if we have approximators with these features then the
true solution of (1.1) whose initial conditions are consistent with those of
the approximators remains close over the long time scale. The result we prove
here is specialized to FPUT lattices where the spring potentials are
homogeneous and of the form (1.2), but requires only the following non-
degeneracy condition on the masses:

(3.4)
$\inf_{j\in{\bf{Z}}}m(j)>0\quad\text{and}\quad\sup_{j\in{\bf{Z}}}m(j)<\infty.$

The condition on the support of $\zeta(j)$ in Hypothesis 1.1 implies the
above, though we do not require all of that hypothesis in this section.

Here is the result:

######  Theorem 3.1.

Suppose that the mass coefficients satisfy (3.4), the approximators
$\widetilde{q}_{\epsilon}(j,t)$ and $\widetilde{p}_{\epsilon}(j,t)$ meet (3.3)
and the initial conditions for (1.1) satisfy

$\|q(0)-\widetilde{q}_{\epsilon}(0),p(0)-\widetilde{p}_{\epsilon}(0)\|_{\ell^{2}}={\mathcal{O}}\left({\alpha_{3}(\epsilon)\over\epsilon^{3}}\right).$

Then the solution $(q(t),p(t))$ of (1.1) satisfies the absolute error estimate

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|q(t)-\widetilde{q}_{\epsilon}(t),p(t)-\widetilde{p}_{\epsilon}(t)\|_{\ell^{2}}={\mathcal{O}}\left({\alpha_{3}(\epsilon)\over\epsilon^{3}}\right)$

as well as the relative error estimate

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}{\|q(t)-\widetilde{q}_{\epsilon}(t),p(t)-\widetilde{p}_{\epsilon}(t)\|_{\ell^{2}}\over\|q(t),p(t)\|_{\ell^{2}}}={\mathcal{O}}\left({\alpha_{3}(\epsilon)\over\beta_{1}(\epsilon)\epsilon^{3}}\right).$

###### Proof.

We introduce the errors:

$\eta:=q-\widetilde{q}_{\epsilon}\quad\text{and}\quad\xi:=p-\widetilde{p}_{\epsilon}$

where $q(j,t)$ and $p(j,t)$ solve (1.1). Time differentiation of these
expressions together with (1.1) and some algebra get us:

(3.5)
$\dot{\eta}=\delta^{+}\xi+\operatorname{Res}_{1}\quad\text{and}\quad\dot{\xi}={1\over
m}\delta^{-}({\mathcal{W}}^{\prime}(\eta,\widetilde{q}_{\epsilon}))+\operatorname{Res}_{2}.$

In the above

${\mathcal{W}}(a,b):={\mathcal{V}}(b+a)-{\mathcal{V}}(b)-{\mathcal{V}}^{\prime}(b)a={1\over
2}(1+2b)a^{2}+{1\over 3}a^{3}$

so that

${\mathcal{W}}^{\prime}(a,b):=\partial_{a}{\mathcal{W}}(a,b)={\mathcal{V}}^{\prime}(b+a)-{\mathcal{V}}^{\prime}(b)=(1+2b)a+a^{2}.$

Now we define the energy functional:

${\mathcal{H}}(u,v;t):=\sum_{j\in{\bf{Z}}}\left({1\over
2}m_{j}v(j)^{2}+{\mathcal{W}}(u(j),\widetilde{q}_{\epsilon}(j,t))\right).$

In the above $(u,v)=(u(j),v(j))$ is in $\ell^{2}\times\ell^{2}$. Under our
assumptions, the square root of this quantity is equivalent to the
$\ell^{2}\times\ell^{2}$ norm in the following sense: there exist
$\epsilon_{*}\in(0,1)$ and $C_{*}>1$ such that

(3.6) $\begin{split}&0<\epsilon<\epsilon_{*},\ \|u\|_{\ell^{2}}\leq 1,\
|t|\leq T_{0}/\epsilon^{3}\\\
\implies&C_{*}^{-1}\|u,v\|_{\ell^{2}}\leq\sqrt{{\mathcal{H}}(u,v;t)}\leq
C_{*}\|u,v\|_{\ell^{2}}.\end{split}$

Here are the details. First of all, simple estimation gives

${1\over
2}\inf_{j\in{\bf{Z}}}m(j)\|v\|_{\ell^{2}}^{2}\leq\sum_{j\in{\bf{Z}}}{1\over
2}m_{j}v(j)^{2}\leq{1\over 2}\sup_{j\in{\bf{Z}}}m(j)\|v\|_{\ell^{2}}^{2}$

and thus (3.4) tells us that $\sqrt{\sum_{j\in{\bf{Z}}}{1\over
2}m_{j}v(j)^{2}}$ is equivalent to $\|v\|_{\ell^{2}}$. This gives the “$v$”
part of (3.6).

For the “$u$” part, recall that $\|f\|_{\ell^{\infty}}\leq\|f\|_{\ell^{2}}$
and so the assumption $\alpha_{1}(\epsilon)=o(1)$ implies that
$\|\widetilde{q}_{\epsilon}\|_{\ell^{\infty}}=o(1)$ as well. Thus for
$\epsilon$ sufficiently small we have
$\|\widetilde{q}_{\epsilon}\|_{\ell^{\infty}}\leq 1/30$. This, in conjunction
with the assumption that $\|u\|_{\ell^{2}}\leq 1$, gives us

$\left|\sum_{j\in Z}\left(\widetilde{q}_{\epsilon}(j,t)+{1\over
3}u(j)\right)u(j)^{2}\right|\leq{11\over 30}\|u\|_{\ell^{2}}^{2}.$

In turn the triangle inequality gives:

${2\over 15}\|u\|_{\ell^{2}}^{2}\leq\sum_{j\in
Z}\begin{array}[t]{@{}c@{}}\underbrace{{1\over
2}(1+2\widetilde{q}_{\epsilon}(j,t)))u^{2}(j)+{1\over 3}u^{3}(j)}\\\
{\mathcal{W}}(u(j),\widetilde{q}_{\epsilon}(j,t))\end{array}\leq{13\over
15}\|u\|_{\ell^{2}}^{2}.$

So we have (3.6).

For the next step, we suppose that $\eta(j,t)$ and $\xi(j,t)$ solve (3.5) and
put $H(t):={\mathcal{H}}(\eta(t),\xi(t);t)$. Differentiation of $H(t)$ with
respect to $t$ gives:

$\dot{H}(t)=\sum_{j\in{\bf{Z}}}\left(m_{j}\xi(j,t)\dot{\xi}(j,t)+{\mathcal{W}}^{\prime}(\eta(j,t),\widetilde{q}_{\epsilon}(j,t))\dot{\eta}(j,t)+{\partial_{b}{\mathcal{W}}}(\eta(j,t),\widetilde{q}_{\epsilon}(j,t))\partial_{t}\widetilde{q}_{\epsilon}(j,t)\right).$

Using (3.5) (and suppressing some dependencies) results in:

$\dot{H}=\sum_{j\in{\bf{Z}}}\left(\xi(\delta^{-}({\mathcal{W}}^{\prime}(\eta,\widetilde{q}_{\epsilon}))+\operatorname{Res}_{2})+{\mathcal{W}}^{\prime}(\eta,\widetilde{q}_{\epsilon})(\delta^{+}\xi+\operatorname{Res}_{1})+{\partial_{b}{\mathcal{W}}}(\eta,\widetilde{q}_{\epsilon})\partial_{t}\widetilde{q}_{\epsilon}\right).$

We sum by parts and terms cancel:

$\dot{H}=\sum_{j\in{\bf{Z}}}\left(\xi\operatorname{Res}_{2}+{\mathcal{W}}^{\prime}(\eta,\widetilde{q}_{\epsilon})\operatorname{Res}_{1}+{\partial_{b}{\mathcal{W}}}(\eta,\widetilde{q}_{\epsilon})\partial_{t}\widetilde{q}_{\epsilon}\right).$

Subsequently, Cauchy-Schwarz and the like get us:

$\dot{H}\leq\|\xi\|_{\ell^{2}}\|\operatorname{Res}_{2}\|_{\ell^{2}}+\|{\mathcal{W}}^{\prime}(\eta,\widetilde{q}_{\epsilon})\|_{\ell^{2}}\|\operatorname{Res}_{1}\|_{\ell^{2}}+\|{\partial_{b}{\mathcal{W}}}(\eta,\widetilde{q}_{\epsilon})\|_{\ell^{1}}\|\partial_{t}\widetilde{q}_{\epsilon}\|_{\ell^{\infty}}.$

One easily computes that
${\partial_{b}{\mathcal{W}}}(\eta,\widetilde{q}_{\epsilon})=\eta^{2}.$ In
which case we conclude, using the earlier formula for ${\mathcal{W}}^{\prime}$
and routine estimates, that

$\dot{H}\leq\|\xi\|_{\ell^{2}}\|\operatorname{Res}_{2}\|_{\ell^{2}}+\left(\left(1+2\|\widetilde{q}_{\epsilon}\|_{\ell^{\infty}}\right)\|\eta\|_{\ell^{2}}+\|\eta\|_{\ell^{2}}^{2}\right)\|\operatorname{Res}_{1}\|_{\ell^{2}}+\|\eta\|_{\ell^{2}}^{2}\|\partial_{t}\widetilde{q}_{\epsilon}\|_{\ell^{\infty}}.$

Next we use (3.3) to get:

$\dot{H}\leq
2\alpha_{3}(\epsilon)\left(\|\eta\|_{\ell^{2}}+\|\xi\|_{\ell^{2}}\right)+2\alpha_{2}(\epsilon)\|\eta\|_{\ell^{2}}^{2}.$

Then we use (3.6):

$\dot{H}\leq
2C^{2}_{*}\left(\alpha_{3}(\epsilon)H^{1/2}+\alpha_{2}(\epsilon)H\right).$

Since $\dot{H}={2}H^{1/2}{d\over dt}H^{1/2}$ the above can be recast as

${d\over dt}{H^{1/2}}\leq
C_{*}^{2}\left(\alpha_{3}(\epsilon)+\alpha_{2}(\epsilon)H^{1/2}\right).$

We have assumed $\alpha_{2}(\epsilon)={\mathcal{O}}(\epsilon^{3})$ so the
above implies

${d\over dt}{H^{1/2}}\leq
C_{*}^{2}\left(\alpha_{3}(\epsilon)+C_{2}\epsilon^{3}H^{1/2}\right)$

for a constant $C_{2}>0$.

An application of Grönwall’s inequality gets us:

$H^{1/2}(t)\leq
C_{2}^{-1}\left(e^{C_{*}^{2}C_{2}\epsilon^{3}t}-1\right){\alpha_{3}(\epsilon)\over\epsilon^{3}}+e^{C_{*}^{2}C_{2}\epsilon^{3}t}{H^{1/2}(0)}.$

Using (3.6) again:

$\|\eta(t),\xi(t)\|_{\ell^{2}}\leq
C_{*}^{2}C_{2}^{-1}\left(e^{C_{*}^{2}C_{2}\epsilon^{3}t}-1\right){\alpha_{3}(\epsilon)\over\epsilon^{3}}+C_{*}^{2}e^{C_{*}^{2}C_{2}\epsilon^{3}t}\|\eta(0),\xi(0)\|_{\ell^{2}}.$

We take the supremum of this over $|t|\leq T_{0}/\epsilon^{3}$ and get

$\sup_{|t|\leq T_{0}/\epsilon^{3}}\|\eta(t),\xi(t)\|_{\ell^{2}}\leq
C_{\star}\left({\alpha_{3}(\epsilon)\over\epsilon^{3}}+\|\eta(0),\xi(0)\|_{\ell^{2}}\right).$

The constant $C_{\star}>0$ is independent of $\epsilon$.

In conclusion, if we assume that

$\|\eta(0),\xi(0)\|_{\ell^{2}}={\mathcal{O}}\left({\alpha_{3}(\epsilon)\over\epsilon^{3}}\right)$

then we have shown

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\eta(t),\xi(t)\|_{\ell^{2}}={\mathcal{O}}\left({\alpha_{3}(\epsilon)\over\epsilon^{3}}\right).$

This is the absolute error estimate. As for the relative error a standard
reverse triangle inequality argument shows that

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}{\|\eta(t),\xi(t)\|_{\ell^{2}}\over\|q(t),p(t)\|_{\ell^{2}}}={\mathcal{O}}\left({\alpha_{3}(\epsilon)\over\beta_{1}(\epsilon)\epsilon^{3}}\right).$

∎

##  4\. Derivation of the effective equations

Now that we have Theorem 3.1, we can move on to deriving the KdV equations
from (1.1). The procedure for the derivation is a multiple scales expansion,
inspired by [3]. We assume the following form of our approximators:

(4.1)
$\begin{split}\widetilde{q}_{\epsilon}(j,t)&=\sum_{n=0}^{3}\epsilon^{n+2}Q_{n}(j,\epsilon
j,\epsilon
t,\epsilon^{3}t)\quad\text{and}\quad\widetilde{p}_{\epsilon}(j,t)=\sum_{n=0}^{3}\epsilon^{n+2}P_{n}(j,\epsilon
j,\epsilon t,\epsilon^{3}t)\end{split}$

where the $Q_{n}=Q_{n}(j,X,\tau,T)$ and $P_{n}=P_{n}(j,X,\tau,T)$ are maps

${\bf{Z}}\times{\bf{R}}\times{\bf{R}}\times{\bf{R}}\to{\bf{R}}.$

Of course we are viewing $\epsilon$ as being small. Given that we put
$X=\epsilon j$ in $\widetilde{q}_{\epsilon}$ and $\widetilde{p}_{\epsilon}$,
we think of $X$ as being the long-wave length scale and $j$ being the
microscopic length scale.

For expansions of the sort we are carrying out, it pays to be organized at the
outset. First we define the following operators for functions $U=U(j,X)$:

$\begin{split}S^{\pm}_{j}U(j,X):=U(j\pm
1,X)\quad\text{and}\quad\delta_{j}^{\pm}U(j,X):=\pm\left(U(j\pm
1,X)-U(j,X)\right).\end{split}$

These are partial shifts and partial finite-differences with respect to $j$.
Next, for $\epsilon>0$ put

$\begin{split}D^{+}U(j,X):=\pm(U(j\pm 1,X\pm\epsilon)-U(j,X)).\end{split}$

If $u(j)=U(j,\epsilon j)$ then $\delta^{\pm}u(j)=D^{\pm}U(j,\epsilon j).$ That
is to say, $D^{\pm}$ are the total finite-difference operators.

Expanding the right-hand sides of $D^{\pm}U(j,X)$ in (formal) Taylor series
with respect to $\epsilon$ gives $\displaystyle
D^{\pm}U(j,X)=\delta_{j}^{\pm}U(j,X)+\sum_{n=1}^{\infty}\epsilon^{n}{(\pm
1)^{n+1}\over n!}S_{j}^{\pm}\partial_{X}^{n}U(j,X).$ Truncating the sum at
$n=M$ would give a formal error on the order of $\epsilon^{M+1}$ and so we
define

$\epsilon^{M+1}E^{\pm}_{M}:=D^{\pm}-\delta_{j}^{\pm}-\sum_{n=1}^{M}\epsilon^{n}{(\pm
1)^{n+1}\over n!}S_{j}^{\pm}\partial_{X}^{n}.$

These operators give the exact error made by such a truncation. Note that for
$M=0$ we just ignore the sum, i.e. $\epsilon
E_{0}^{\pm}:=D^{\pm}-\delta_{j}^{\pm}$.

If we plug (4.1) into the residuals (3.1) and carry out some substantial
algebra we find that

$\begin{split}\operatorname{Res}_{1}&=\epsilon^{2}Z_{12}+\epsilon^{3}Z_{13}+\epsilon^{4}Z_{14}+\epsilon^{5}Z_{15}+\epsilon^{6}W_{1}\quad\text{and}\quad\\\
\operatorname{Res}_{2}&={1\over
m}\left(\epsilon^{2}Z_{22}+\epsilon^{3}Z_{23}+\epsilon^{4}Z_{24}+\epsilon^{5}Z_{25}+\epsilon^{6}W_{2}\right)\end{split}$

where

$\begin{split}Z_{12}:=&\delta_{j}^{+}P_{0}\\\ Z_{22}:=&\delta_{j}^{-}Q_{0}\\\
Z_{13}:=&\delta_{j}^{+}P_{1}+{{S}_{j}^{+}}\partial_{X}P_{0}-\partial_{\tau}Q_{0}\\\
Z_{23}:=&\delta_{j}^{-}Q_{1}+{{S}_{j}^{-}}\partial_{X}Q_{0}-m\partial_{\tau}P_{0}\\\
Z_{14}:=&\delta_{j}^{+}P_{2}+S_{j}^{+}\partial_{X}P_{1}+{1\over
2}S_{j}^{+}\partial_{X}^{2}P_{0}-\partial_{\tau}Q_{1}\\\
Z_{24}:=&\delta_{j}^{-}Q_{2}+S_{j}^{-}\partial_{X}Q_{1}-{1\over
2}S_{j}^{-}\partial_{X}^{2}Q_{0}+\delta_{j}^{-}Q_{0}^{2}-m\partial_{\tau}P_{1}\\\
Z_{15}:=&\delta_{j}^{+}P_{3}+S_{j}^{+}\partial_{X}P_{2}+{1\over
2}S_{j}^{+}\partial_{X}^{2}P_{1}+{1\over 6}S_{j}^{+}\partial_{X}^{3}P_{0}\\\
&-\partial_{\tau}Q_{2}-\partial_{T}Q_{0}\\\
Z_{25}:=&\delta_{j}^{-}Q_{3}+S_{j}^{-}\partial_{X}Q_{2}-{1\over
2}S_{j}^{-}\partial_{X}^{2}Q_{1}+{1\over 6}S_{j}^{-}\partial_{X}^{3}Q_{0}\\\
&+2\delta_{j}^{-}(Q_{0}Q_{1})+\partial_{X}Q_{0}^{2}-m\partial_{\tau}P_{2}-m\partial_{T}P_{0}\\\
\end{split}$

and

(4.2)
$\begin{split}W_{1}:=&\sum_{n=0}^{3}E_{3-n}^{+}P_{n}-\partial_{\tau}Q_{3}-\sum_{n=1}^{3}\epsilon^{n-1}\partial_{T}Q_{n}\\\
W_{2}:=&\sum_{n=0}^{3}E_{3-n}^{-}Q_{n}+E_{1}^{-}Q_{0}^{2}+2E_{0}^{-}(Q_{0}Q_{1})\\\
&+D^{-}\left(2Q_{0}Q_{2}+\left(Q_{1}+\epsilon
Q_{2}+\epsilon^{2}Q_{3}\right)^{2}\right)-m\partial_{\tau}P_{3}-m\sum_{n=1}^{3}\epsilon^{n-1}\partial_{T}P_{n}.\end{split}$

The usual way to proceed is to select the $Q_{0},P_{0},\dots,Q_{3},P_{3}$ so
that each $Z_{1k}=Z_{2k}=0$. In this case we would have
$\operatorname{Res}_{1}=\epsilon^{6}W_{1}$ and
$\displaystyle\operatorname{Res}_{2}={1\over m}\epsilon^{6}W_{2}$ which we can
then estimate using the formulas for the $Q_{n}$ and $P_{n}$. This strategy
works perfectly well in the homogeneous and periodic problems as all the terms
are rigorously the size they formally appear to be, modulo an annoying factor
of $\epsilon^{-1/2}$ caused by the long-wave scaling. But it fails in the
random problem; the randomness leads to terms which are much larger than they
appear.

Our modified strategy is to solve

$Z_{12}=Z_{22}=Z_{13}=Z_{23}=Z_{14}=Z_{24}=0$

(which will largely determine $Q_{0},P_{0},\dots,Q_{2},P_{2}$) and then to do
“something else” for $Z_{15}$ and $Z_{25}$. At the end of this, we find that
$\operatorname{Res}_{1}=\epsilon^{5}Z_{15}+\epsilon^{6}W_{1}$ and
$\displaystyle\operatorname{Res}_{2}={1\over
m}\left(\epsilon^{5}Z_{25}+\epsilon^{6}W_{2}\right).$ In Section 5 we show
that these are ${\mathcal{O}}_{\ell^{2}}(\epsilon^{5}\sqrt{|\ln(\epsilon)|})$.
This is enough to apply Theorem 3.1 and get the error estimates shown in
(1.4).

###  4.1. A tutorial on solving $Z_{1k}=Z_{2k}=0$

Each pair of equations $Z_{1k}=Z_{2k}=0$ will have the form

(4.3)
$\begin{split}\delta_{j}^{+}P_{k-2}&=\bar{F}_{0}(X,\tau,T)+\sum_{n=1}^{N}f_{n}(j)\bar{F}_{n}(X,\tau,T)\\\
\delta_{j}^{-}Q_{k-2}&=\bar{G}_{0}(X,\tau,T)+\sum_{n=1}^{N}g_{n}(j)\bar{G}_{n}(X,\tau,T).\end{split}$

The sequences $f_{n}(j)$ and $g_{n}(j)$ are mean-zero random variables which
come, in one way or another, from $\zeta(j)$; they depend only on the
microscale coordinate. The $\bar{F}_{n}$ and $\bar{G}_{n}$ functions do not
depend on the microscale coordinate at all. They will be made up of pieces of
the various $P_{n}$ and $Q_{n}$ where $n<k-2$. In this way (4.3) allows us to
figure out $P_{k-2}$ and $Q_{k-2}$ from the earlier functions.

We decompose (4.3) into a “long-wave” part (those pieces that do not depend on
the microscale coordinate $j$ at all) and a “microscale” part (those that do).
The long-wave part just consists of the terms $\bar{F}_{0}$ and $\bar{G}_{0}$
and so we set

(4.4) $\bar{F}_{0}=0\quad\text{and}\quad\bar{G}_{0}=0.$

This is a sort of solvability condition that will wind up giving us the long-
wave dynamics; how it all plays out will be seen when we get in the weeds
below.

The microscale part is what is left over:

(4.5)
$\begin{split}\delta_{j}^{+}P_{k-2}=\sum_{l=1}^{N}f_{n}(j)\bar{F}_{n}(X,\tau,T)\quad\text{and}\quad\delta_{j}^{-}Q_{k-2}=\sum_{l=1}^{N}g_{n}(j)\bar{G}_{n}(X,\tau,T).\end{split}$

We can just write down a solution for this:

(4.6)
$\begin{split}P_{k-2}(j,X,\tau,T)&=\bar{P}_{k-2}(X,\tau,T)+\sum_{l=1}^{N}\chi_{n}(j)\bar{F}_{n}(X,\tau,T)\quad\text{and}\quad\\\
Q_{k-2}(j,X,\tau,T)&=\bar{Q}_{k-2}(X,\tau,T)+\sum_{l=1}^{N}\kappa_{n}(j)\bar{G}_{n}(X,\tau,T)\end{split}$

where we select $\chi_{n}$ and $\kappa_{n}$ so that

$\delta^{+}\chi_{n}=f_{n}\quad\text{and}\quad\delta^{-}\kappa_{n}=g_{n}.$

Solving these equations for $\chi_{n}$ and $\kappa_{n}$ from $f_{n}$ and
$g_{n}$ is one of the key steps in the whole procedure and as we shall show
the transparency condition makes this a relatively easy affair…at least at
first. The functions $\bar{P}_{k-2}(X,\tau,T)$ and $\bar{Q}_{k-2}(X,\tau,T)$
are “constants of integration”; in most cases we determine these from (4.4) at
a later point in the derivation.

Now we get into actually solving the equations.

###  4.2. $Z_{12}=Z_{22}=0$

These read $\delta_{j}^{+}P_{0}=0$ and $\delta_{j}^{-}Q_{0}=0$ which tells us
that

(4.7) $Q_{0}(j,X,\tau,T)=\bar{Q}_{0}(X,\tau,T)\quad\text{and}\quad
P_{0}(j,X,\tau,T)=\bar{P}_{0}(X,\tau,T).$

######  Remark 4.1.

In this section any function with a “bar” on top will not depend on $j$. We
make this convention so that we do not need to perpetually clutter up our
algebra with functional dependencies. For the same reason it is helpful to
keep in mind that $m$ and $\zeta$ depend only on $j$ and not on the other
variables.

###  4.3. $Z_{13}=Z_{23}=0$

Using (4.7) these equations become

$\delta_{j}^{+}P_{1}=\partial_{\tau}\bar{Q}_{0}-\partial_{X}\bar{P}_{0}\quad\text{and}\quad\delta_{j}^{-}Q_{1}=m\partial_{\tau}\bar{P}_{0}-\partial_{X}\bar{Q}_{0}.$

Using the transparency condition (1.3) converts these to

$\delta_{j}^{+}P_{1}=\partial_{\tau}\bar{Q}_{0}-\partial_{X}\bar{P}_{0}\quad\text{and}\quad\delta_{j}^{-}Q_{1}=\partial_{\tau}\bar{P}_{0}-\partial_{X}\bar{Q}_{0}+\delta^{+}\delta^{-}\zeta\partial_{\tau}\bar{P}_{0}.$

Following the steps from the tutorial in Section 4.1 we see that the long-wave
part (4.4) of these equations is

(4.8)
$\begin{split}\partial_{\tau}\bar{Q}_{0}-\partial_{X}\bar{P}_{0}=0\quad\text{and}\quad\partial_{\tau}\bar{P}_{0}-\partial_{X}\bar{Q}_{0}=0.\end{split}$

This is the wave equation wearing a fake mustache and glasses and we readily
solve it:

(4.9)
$\bar{Q}_{0}=A(X-\tau,T)+B(X+\tau,T)\quad\text{and}\quad\bar{P}_{0}=-A(X-\tau,T)+B(X+\tau,T).$

######  Remark 4.2.

We use the convention that

$w=X-\tau\quad\text{and}\quad l=X+\tau$

so that $A=A(w,T)$ and $B=B(l,T)$. Note that $A$ and $B$ do not depend on $j$.
It is these functions that will ultimately solve KdV equations.

After (4.8) we are left with the microscale part

$\delta_{j}^{+}P_{1}=0\quad\text{and}\quad\delta_{j}^{-}Q_{1}=\delta^{+}\delta^{-}\zeta\partial_{\tau}\bar{P}_{0}.$

The solution formula (4.6) gives
$Q_{1}=\bar{Q}_{1}+\chi\partial_{\tau}\bar{P}_{0}$ where we want
$\delta^{-}\chi=\delta^{+}\delta^{-}\zeta$. Finding $\chi$ is easily done as
we can simply cancel a $\delta^{-}$ from both sides and put
$\chi=\delta^{+}\zeta$. This is so simple because of the transparency
condition (1.3) and this is one of the reasons we have assumed it. Likewise
(4.6) says that we should put $P_{1}=\bar{P}_{1}$, but it will turn out that
$\bar{P}_{1}$ will be zero so we just enforce that now. In short we have

(4.10)
$\begin{split}Q_{1}=\bar{Q}_{1}+\delta_{j}^{+}\zeta\partial_{\tau}\bar{P}_{0}\quad\text{and}\quad
P_{1}=0.\end{split}$

Note that $\delta_{j}^{+}\zeta$ is bounded in $j$ because of the compact
support assumption in Hypothesis 1.1.

######  Remark 4.3.

What if we had not made the transparency assumption but instead assumed that
$m(j)=1+z(j)$ where $z(j)$ are i.i.d. mean zero random variables? The long-
wave part is the same as above but now the microscale part is
$\delta_{j}^{-}Q_{1}=z\partial_{\tau}\bar{P}_{0}.$ To use the solution formula
(4.6) we would want to find $\chi$ so that $\delta^{-}\chi=z$, or rather

$\chi(j)=\chi(j-1)+z(j).$

This equation tells us that $\chi(j)$ is a random walk with steps given by
$z(j)$ and as such we expect $\chi$ to grow like $\sqrt{j}$.

To see why this is an issue, notice that $Q_{1}$ would include the term
$\chi(j)A_{w}(X-\tau,T),$ which then would show up in the approximator (4.1)
as $\epsilon^{3}\chi(j)A_{w}(\epsilon(j-t),\epsilon^{3}t).$ The term $A_{w}$
is propagating to the right with roughly unit speed and thus when $t\sim
1/\epsilon^{3}$ will be located at $j\sim 1/\epsilon^{3}$. In turn this
indicates $\chi(j)\sim\epsilon^{-3/2}$ towards the end of the approximation
time interval. Hence the term $\epsilon^{3}\chi A_{w}$ would be substantially
larger than it appears: the techniques from [20] show that almost surely

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\epsilon^{3}\chi(\cdot)A_{w}(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}\leq
C\epsilon\sqrt{\ln|\ln(\epsilon)|}.$

Were $\chi={\mathcal{O}}_{\ell^{\infty}}(1)$ the right-hand side of the
preceding estimate would be $C\epsilon^{5/2}$ (see Lemma 5.6 below). And so we
find that the “$\epsilon^{3}$ term” in the approximator is more than an order
of magnitude larger than it should be, bigger in fact that the leading order
term in the approximation. Disaster!

The lesson learned: if a term in our approximation involves a random walk it
will ultimately be at least $\epsilon^{-3/2}$ larger than it formally appears
to be. We call this difficulty a random walk disaster.

###  4.4. $Z_{14}=Z_{24}=0$

The relations (1.3), (4.7), (4.8), (4.10) and a little algebra convert these
equations to

$\begin{split}\delta_{j}^{+}P_{2}&=\partial_{\tau}\bar{Q}_{1}-{1\over
2}\partial_{X\tau}^{2}\bar{Q}_{0}+\delta_{j}^{+}\zeta\partial_{X}^{2}\bar{P}_{0}\quad\text{and}\quad\delta_{j}^{-}Q_{2}=-\partial_{X}\bar{Q}_{1}+{1\over
2}\partial_{X}^{2}\bar{Q}_{0}-\delta_{j}^{-}\zeta\partial^{2}_{X}\bar{Q}_{0}.\end{split}$

The long-wave part of this is

$\begin{split}0&=\partial_{\tau}\bar{Q}_{1}-{1\over
2}\partial_{X\tau}^{2}\bar{Q}_{0}\quad\text{and}\quad
0=-\partial_{X}\bar{Q}_{1}+{1\over 2}\partial_{X}^{2}\bar{Q}_{0}\end{split}$

which can be solved by putting

(4.11) $\bar{Q}_{1}={1\over 2}\partial_{X}\bar{Q}_{0}={1\over
2}\left(\partial_{w}A+\partial_{l}B\right).$

This leaves the microscale part

(4.12)
$\begin{split}\delta_{j}^{+}P_{2}&=\delta_{j}^{+}\zeta\partial_{X}^{2}\bar{P}_{0}\quad\text{and}\quad\delta_{j}^{-}Q_{2}=-\delta_{j}^{-}\zeta\partial^{2}_{X}\bar{Q}_{0}.\end{split}$

Which as per (4.6) we solve by taking

(4.13)
$\begin{split}P_{2}&=\bar{P}_{2}+\zeta\partial_{X}^{2}\bar{P}_{0}\quad\text{and}\quad
Q_{2}=\bar{Q}_{2}-\zeta\partial_{X}^{2}\bar{Q}_{0}.\end{split}$

Once again, the transparency condition (1.3) made finding this solution a
simple matter of cancelation; it is the reason why the transparency condition
has two finite-differences on $\zeta$. If we had put only one finite-
difference in (1.3) then another random walk disaster as described in Remark
4.3 would occur when we solve (4.12).

###  4.5. Something else for $Z_{15}$ and $Z_{25}$

The relations (1.3), (4.7), (4.8), (4.10), (4.11), (4.13) and quite a bit of
algebra get us:

$\begin{split}Z_{15}=&-\partial_{T}\bar{Q}_{0}-\partial_{\tau}\bar{Q}_{2}+\partial_{X}\bar{P}_{2}+{1\over
6}\partial_{X}^{3}\bar{P}_{0}\\\
&+\delta_{j}^{+}P_{3}+\left(\zeta+S_{j}^{+}\zeta\right)\partial_{X}^{3}\bar{P}_{0}\\\
Z_{25}=&-\partial_{T}\bar{P}_{0}-\partial_{\tau}\bar{P}_{2}+\partial_{X}\bar{Q}_{2}-\left({1\over
12}-2\sigma^{2}\right)\partial_{X}^{3}\bar{Q}_{0}+\partial_{X}(\bar{Q}_{0}^{2})\\\
&+\delta_{j}^{-}Q_{3}-\left(\zeta+S_{j}^{-}\zeta+{1\over
2}\delta_{j}^{-}\zeta+\zeta{\delta_{j}^{+}\delta_{j}^{-}\zeta}+2\sigma^{2}\right)\partial_{X}^{3}\bar{Q}_{0}\\\
&+\delta_{j}^{-}\delta_{j}^{+}\zeta\partial_{X}(\bar{Q}^{2}_{0})-{\delta_{j}^{+}\delta_{j}^{-}\zeta}\left(\partial_{T}\bar{P}_{0}+\partial_{\tau}\bar{P}_{2}\right).\end{split}$

Recall that $\sigma^{2}$ is the variance of $\zeta(j)$. We need $Z_{15}$ and
$Z_{25}$ to be small relative to $\epsilon$, but zeroing them out completely
happens to be too restrictive; we will need to modify the microscale part of
the decomposition described in Section 4.1. But before we get there we deal
with the long-wave part.

####  4.5.1. Kill the long-wave part with KdV equations

As per normal, we zero out the long-wave parts of $Z_{15}$ and $Z_{25}$. We
have conveniently arranged all such terms in the first line on the right of
the preceding formulas for $Z_{15}$ and $Z_{25}$ and so we put

(4.14)
$\begin{split}0&=-\partial_{T}\bar{Q}_{0}-\partial_{\tau}\bar{Q}_{2}+\partial_{X}\bar{P}_{2}+{1\over
6}\partial_{X}^{3}\bar{P}_{0}\\\
0&=-\partial_{T}\bar{P}_{0}-\partial_{\tau}\bar{P}_{2}+\partial_{X}\bar{Q}_{2}-\left({1\over
12}-2\sigma^{2}\right)\partial_{X}^{3}\bar{Q}_{0}+\partial_{X}(\bar{Q}_{0}^{2}).\end{split}$

Within (4.14) lurk the KdV equations; here is how we coax them into the
daylight. Let

$\begin{split}\bar{Q}_{2}(X,\tau,T)&=A_{2}(X-\tau,X+\tau,T)+B_{2}(X-\tau,X+\tau,T)\\\
\quad\text{and}\quad\bar{P}_{2}(X,\tau,T)&=-A_{2}(X-\tau,X+\tau,T)+B_{2}(X-\tau,X+\tau,T)\end{split}$

and use (4.9) in (4.14) to get

$\begin{split}0=&\partial_{T}A+\partial_{T}B+2\partial_{l}A_{2}-2\partial_{w}B_{2}-{1\over
6}(-\partial_{w}^{3}A+\partial_{l}^{3}B)\\\
0=&-\partial_{T}A+\partial_{T}B-2\partial_{l}A_{2}-2\partial_{w}B_{2}+\left({1\over
12}-2\sigma^{2}\right)(\partial_{w}^{3}A+\partial_{l}^{3}B)\\\
&-\left(\partial_{w}A^{2}+2\partial_{w}AB+2A\partial_{l}B+\partial_{l}B^{2}\right).\end{split}$

Subtracting these gives:

(4.15) $\begin{split}0=&2\partial_{T}A+\left({1\over
12}+2\sigma^{2}\right)\partial_{w}^{3}A+\partial_{w}A^{2}\\\
+&4\partial_{l}A_{2}-\left({1\over
4}-2\sigma^{2}\right)\partial_{l}^{3}B+\left(2\partial_{w}AB+2A\partial_{l}B+\partial_{l}B^{2}\right).\end{split}$

If we let ${\mathcal{B}}$ be an $l$-antiderivative of $B$ (specifically
${\mathcal{B}}(l,T):=\int_{0}^{l}B(y,T)dy$) and set

(4.16) $A_{2}={1\over 4}\left[\left({1\over
4}-2\sigma^{2}\right)\partial_{l}^{2}B-\left(2\partial_{w}A{\mathcal{B}}+2AB+B^{2}\right)\right]$

many terms in (4.15) die. What survives is

(4.17) $\begin{split}0=2\partial_{T}A+\left({1\over
12}+2\sigma^{2}\right)\partial_{w}^{3}A+\partial_{w}A^{2}.\end{split}$

This is a KdV equation! A parallel argument (after adding instead subtracting
equations a few steps above) shows we should take

(4.18) $B_{2}={1\over 4}\left[\left({1\over
4}-2\sigma^{2}\right)\partial_{w}^{2}A-\left(A^{2}+2AB+2{\mathcal{A}}\partial_{l}B\right)\right]\\\
$

with ${\mathcal{A}}$ a $w$-antiderivative of $A$ (specifically
${\mathcal{A}}(w,T):=\int_{0}^{w}A(y,T)dy$). In which case we get that $B$
solves another KdV equation:

(4.19) $\begin{split}0=2\partial_{T}B-\left({1\over
12}+2\sigma^{2}\right)\partial_{l}^{3}B-\partial_{l}B^{2}.\end{split}$

To summarize: taking $A$, $B$, $A_{2}$ and $B_{2}$ as we have just described
means that (4.14) is satisfied.

####  4.5.2. Handle with the microscopic part using autoregressive processes.

The next step in dealing with $Z_{15}$ and $Z_{25}$ is to control the
microscopic parts that are left over after (4.14):

(4.20)
$\begin{split}Z_{15}=&\delta_{j}^{+}P_{3}+\left(\zeta+S_{j}^{+}\zeta\right)\partial_{X}^{3}\bar{P}_{0}\\\
Z_{25}=&\delta_{j}^{-}Q_{3}-\left(\zeta+S_{j}^{-}\zeta+{1\over
2}\delta_{j}^{-}\zeta+\zeta{\delta_{j}^{+}\delta_{j}^{-}\zeta}+2\sigma^{2}\right)\partial_{X}^{3}\bar{Q}_{0}\\\
&+\delta_{j}^{-}\delta_{j}^{+}\zeta\partial_{X}(\bar{Q}^{2}_{0})-{\delta_{j}^{+}\delta_{j}^{-}\zeta}\left(\partial_{T}\bar{P}_{0}+\partial_{\tau}\bar{P}_{2}\right).\end{split}$

Many, but not all, of these terms in $Z_{25}$ can be eliminated with the same
cancelation tricks that worked earlier. To see this, we let

$\begin{split}P_{3}&=\gamma_{1}\partial_{X}^{3}\bar{P}_{0}\quad\text{and}\quad\\\
Q_{3}&=\left(\gamma_{2}+{1\over
2}\zeta\right)\partial_{X}^{3}\bar{Q}_{0}-\delta_{j}^{+}\zeta\partial_{X}(\bar{Q}^{2}_{0})+{\delta_{j}^{+}\zeta}\left(\partial_{T}\bar{P}_{0}+\partial_{\tau}\bar{P}_{2}\right)\end{split}$

where for the moment we leave $\gamma_{1}=\gamma_{1}(j)$ and
$\gamma_{2}=\gamma_{2}(j)$ unspecified. Substituting the above into (4.20)
gives

(4.21)
$\begin{split}Z_{15}=&\left[\delta_{j}^{+}\gamma_{1}+\zeta+S_{j}^{+}\zeta\right]\partial_{X}^{3}\bar{P}_{0}\\\
Z_{25}=&\left[\delta_{j}^{-}\gamma_{2}-\left(\zeta+S_{j}^{-}\zeta+\zeta{\delta_{j}^{+}\delta_{j}^{-}\zeta}+2\sigma^{2}\right)\right]\partial_{X}^{3}\bar{Q}_{0}.\end{split}$

If we followed the strategy from the tutorial in Section 4.1, we would put
$\delta_{j}^{+}\gamma_{1}=-\zeta-S_{j}^{+}\zeta$ and
$\delta^{-}\gamma_{2}=\zeta+S^{-}\zeta+\zeta\delta^{+}\delta^{-}\zeta+2\sigma^{2}$
and get $Z_{15}=Z_{25}=0$. Since the $\zeta(j)$ are i.i.d. random variables we
would then find that $\gamma_{1}(j)$ and $\gamma_{2}(j)$ are random walks,
which leads us to another disaster as described in Remark 4.3 (this time in
the residual terms). Why not just stack another finite-difference on $\zeta$
in the transparency condition? This would help in $Z_{15}$ but would not be
useful in handling the parts stemming from
$\zeta{\delta_{j}^{+}\delta_{j}^{-}\zeta}$ in $Z_{25}$.

To avoid these problematic random walks we take $\gamma_{1}$ and $\gamma_{2}$
to solve

(4.22)
$\begin{split}\delta^{+}\gamma_{1}&=-\epsilon\operatorname{sgn}(j)\gamma_{1}-\left(\zeta+S^{+}\zeta\right)\quad\text{and}\quad\\\
\delta^{-}\gamma_{2}&=-\epsilon\operatorname{sgn}(j)\gamma_{2}+\left(\zeta+S^{-}\zeta+\zeta\delta^{+}\delta^{-}\zeta+2\sigma^{2}\right).\end{split}$

In which case we find that (4.21) becomes

$Z_{15}=-\epsilon\operatorname{sgn}(j)\gamma_{1}\partial_{X}^{3}\bar{P}_{0}\quad\text{and}\quad
Z_{25}=-\epsilon\operatorname{sgn}(j)\gamma_{2}\partial_{X}^{3}\bar{Q}_{0}.$

The extra factors of $\epsilon$ on the right-hand sides here means that our
choices of $P_{3}$ and $Q_{3}$ are formally as good as putting
$Z_{15}=Z_{25}=0$. Estimates for $P_{3}$ and $Q_{3}$ (and consequently
$Z_{15}$, $Z_{25}$ and the residuals) ultimately require us to understand
$\gamma_{1}$ and $\gamma_{2}$. The equations in (4.22) are examples of
autoregressive processes [12]. These are dissipative cousins of random walks
and with classical probabilistic methods we will show that they roughly cost
us a factor of $\epsilon^{-1/2}$ (see Lemma 5.10 below) instead of the
$\epsilon^{-3/2}$ we get from using random walks. This is big but not too big
for our estimates to handle.

###  4.6. Summing up.

At this point we have completely determined all the functions
$P_{0},\dots,Q_{3}$ in the approximation. As it can be challenging to sort
through it all, we close out this section by summarizing the derivation.

######  Definition 4.4.

Suppose $A(w,T)$ and $B(l,T)$ solve the KdV equations (4.17) and (4.19) and
$\gamma_{1}(j)$ and $\gamma_{2}(j)$ solve the autoregressive processes (4.22).
Take $A_{2}(w,l,T)$ and $B_{2}(w,l,T)$ as in (4.16) and (4.18). Define
$Q_{k}(j,X,\tau,T)$ and $P_{k}(j,X,\tau,T)$ via

$Q_{0}=A+B$, | $P_{0}=-A+B$
---|---
$Q_{1}={1\over 2}\partial_{X}Q_{0}+\delta_{j}^{+}\zeta\partial_{\tau}P_{0}$, | $P_{1}=0$
$Q_{2}=A_{2}+B_{2}-\zeta\partial_{X}^{2}Q_{0}$ | $P_{2}=-A_{2}+B_{2}+\zeta\partial_{X}^{2}P_{0}$
$\begin{aligned} Q_{3}=&\left(\gamma_{2}+{1\over 2}\zeta\right)\partial_{X}^{3}Q_{0}-\delta_{j}^{+}\zeta\partial_{X}(Q^{2}_{0})\\\ +&{\delta_{j}^{+}\zeta}\left(\partial_{T}P_{0}-\partial_{\tau}A_{2}+\partial_{\tau}B_{2}\right)\end{aligned}$ | $\begin{aligned} P_{3}=&\gamma_{1}\partial_{X}^{3}P_{0}\\\ &\phantom{boo}\end{aligned}$

where it is understood that $w=X-\tau$ and $l=X+\tau$. Then we call
$\widetilde{q}_{\epsilon}(j,t)$ and $\widetilde{p}_{\epsilon}(j,t)$, as
defined in (4.1), the extended KdV approximators.

In this section we have proven:

######  Lemma 4.5.

The extended KdV approximators have

$\operatorname{Res}_{1}=\epsilon^{6}\left(-\operatorname{sgn}(j)\gamma_{1}\partial_{X}^{3}P_{0}+W_{1}\right)\quad\text{and}\quad\displaystyle\operatorname{Res}_{2}={\epsilon^{6}\over
m}\left(-\operatorname{sgn}(j)\gamma_{2}\partial_{X}^{3}Q_{0}+W_{2}\right)$

with $W_{1}$ and $W_{2}$ given at (4.2).

We move on to proving many estimates related to the extended KdV
approximators.

##  5\. Estimates on the approximators and residuals

To streamline some of the forthcoming statements we put forth the following
convention:

######  Definition 5.1.

We say $A$ and $B$ are good solutions of KdV on $[-T_{0},T_{0}]$ if they
satisfy (4.17) and (4.19) along with the estimate

$0<\sup_{|T|\leq
T_{0}}\|A(\cdot,T)\|_{H^{7}(1)}+\|B(\cdot,T)\|_{H^{7}(1)}<\infty.$

######  Remark 5.2.

The existence of good solutions of KdV on intervals of arbitrary length is by
now classical (see [24]). The lower bound is just to guarantee that the
approximation is not trivial.

In this section we prove:

######  Proposition 5.3.

Assume Hypothesis 1.1. Let $\widetilde{q}_{\epsilon}(j,t)$ and
$\widetilde{p}_{\epsilon}(j,t)$ be the extended KdV approximators as in
Definition 4.4 where we further assume that that $A$ and $B$ are good
solutions of KdV on $[-T_{0},T_{0}]$. Then almost surely the quantities
defined at (3.2) satisfy

$\alpha_{1}(\epsilon)={\mathcal{O}}(\epsilon^{3/2}),\
\alpha_{2}(\epsilon)={\mathcal{O}}(\epsilon^{3}),\
\alpha_{3}(\epsilon)={\mathcal{O}}(\epsilon^{5}\sqrt{|\ln(\epsilon)|})\
{\text{ and }}\ \beta^{-1}_{1}(\epsilon)={\mathcal{O}}(\epsilon^{-3/2}).$

Estimates on terms which do not involve $\gamma_{1}$ or $\gamma_{2}$ can be
handled using well-understood techniques found in previous works, whereas the
rest require new ideas. All dependence on $\gamma_{1}$ and $\gamma_{2}$ enters
through $P_{3}$ and $Q_{3}$, the latter of which has some terms without them.
And so we put

(5.1) $Q_{3\gamma}:=\gamma_{2}\partial_{X}^{3}Q_{0}\quad\text{and}\quad
Q_{30}:=Q_{3}-\gamma_{2}\partial_{X}^{3}Q_{0}.$

To be clear, $Q_{30}$ has no instances of a $\gamma$ within.

Similarly if in the formulas for $W_{1}$ and $W_{2}$ we eliminate any term
with a $\gamma$ in it we get:

$\begin{split}W_{10}:=&\sum_{n=0}^{2}E_{3-n}^{+}P_{n}-\sum_{n=1}^{2}\epsilon^{n-1}\partial_{T}Q_{n}-\epsilon^{2}\partial_{T}Q_{30}\\\
W_{20}:=&\sum_{n=0}^{2}E_{3-n}^{-}Q_{n}+E_{0}^{-}Q_{30}+E_{1}^{-}Q_{0}^{2}+2E_{0}^{-}(Q_{0}Q_{1})\\\
&+D^{-}\left(2Q_{0}Q_{2}+\left(Q_{1}+\epsilon
Q_{2}+\epsilon^{2}Q_{30}\right)^{2}\right)-m\sum_{n=1}^{2}\epsilon^{n-1}\partial_{T}P_{n}.\end{split}$

Thus the terms with a $\gamma$ are:

(5.2) $W_{1\gamma}:=W_{1}-W_{10}\quad\text{and}\quad
W_{2\gamma}:=W_{2}-W_{20}.$

###  5.1. Terms without $\gamma_{1}$ and $\gamma_{2}$

In this part we prove:

######  Lemma 5.4.

Assume Hypothesis 1.1. Let $\widetilde{q}_{\epsilon}(j,t)$ and
$\widetilde{p}_{\epsilon}(j,t)$ be the extended KdV approximators as in
Definition 4.4 where we further assume that that $A$ and $B$ are good
solutions of KdV on $[-T_{0},T_{0}]$. Then

$\sum_{n=0}^{2}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\left(\|P_{n}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|Q_{n}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)={\mathcal{O}}(\epsilon^{-1/2}),$
$\sup_{|t|\leq T_{0}/\epsilon^{3}}\|Q_{30}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}={\mathcal{O}}(\epsilon^{-1/2}),$

and

$\sup_{|t|\leq T_{0}/\epsilon^{3}}\left(\|W_{10}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|W_{20}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)={\mathcal{O}}(\epsilon^{-1/2}).$

######  Remark 5.5.

Note that in Hypothesis 1.1 we assumed that $|\zeta(j)|<1/4$ for all $j$. A
consequence of this is that none of the estimates in Lemma 5.4 depend on the
realization of the $\zeta(j)$. That is to say there is no probability needed
to understand this lemma.

###### Proof.

The proof is similar to that of Proposition 4.2 of [10], though there are a
few small, but substantive, differences. The main tool we need is:

######  Lemma 5.6.

Let $M\geq 0$ be an integer. Suppose that $f(j)\in\ell^{\infty}$ and $F(X)\in
H^{M+1}$. If $u_{\epsilon}(j):=f(j)F(\epsilon j)$ then

$\|u\|_{\ell^{\infty}}\leq\|f\|_{\ell^{\infty}}\|F\|_{L^{\infty}},$
$\|u\|_{\ell^{2}}\leq C\epsilon^{-1/2}\|f\|_{\ell^{\infty}}\|F\|_{H^{1}},$
$\|E_{M}^{\pm}u\|_{\ell^{2}}\leq
C\epsilon^{-1/2}\|f\|_{\ell^{\infty}}\|F\|_{H^{M+1}}$

and

$\|D^{\pm}u\|_{\ell^{2}}\leq
C\epsilon^{-1/2}\|f\|_{\ell^{\infty}}\|F\|_{H^{1}}.$

The constants $C>0$ depend only on $M$.

###### Proof.

Lemma 4.3 of [10] is nearly identical to this, but has the requirement that
$f(j)$ be $N$-periodic. Still we can piggyback the proof of our result on that
one. The first estimate is all but obvious. For the second we have the easy
estimate
$\|u\|_{\ell^{2}}\leq\|f\|_{\ell^{\infty}}\|F_{\epsilon}\|_{\ell^{2}}$ where
$F_{\epsilon}(j):=F(\epsilon j)$, $j\in{\bf{Z}}$. But then the second estimate
of Lemma 4.3 of [10] applies and shows $\|F_{\epsilon}\|_{\ell^{2}}\leq
C\epsilon^{-1/2}\|F\|_{H^{1}}$. For the third, a direct computation shows that
$\displaystyle E_{M}^{+}u(j)=f(j+1)(E_{M}^{+}F_{\epsilon})(j)$ which implies
$\|E_{M}^{+}u\|_{\ell^{2}}\leq\|f\|_{\ell^{\infty}}\|E_{M}^{+}F_{\epsilon}\|_{\ell^{2}}$.
The third estimate from Lemma 4.3 of [10] implies that
$\|E_{M}^{+}F_{\epsilon}\|_{\ell^{2}}\leq C\epsilon^{-1/2}\|F\|_{H^{M+1}}$. To
get an estimate for $E_{M}^{-}$ is similar. The final estimate, for
$D^{\pm}u$, follows from, the definition of $D^{\pm}$, the triangle inequality
and the second estimate in this lemma.

∎

We also need the following, to control the antiderivatives in $A_{2}$ and
$B_{2}$:

######  Lemma 5.7.

Suppose that $F(X)\in L^{2}(1)$ then $\mathcal{F}(X):=\int_{0}^{X}F(y)dy$ is
in $L^{\infty}$ and
$\|\mathcal{F}\|_{L^{\infty}}\leq\sqrt{\pi}\|F\|_{L^{2}(1)}$.

###### Proof.

We use Cauchy-Schwarz and the fact that $\int_{\bf{R}}(1+y^{2})^{-1}=\pi$. To
wit:

$\begin{split}\left|\mathcal{F}(X)\right|\leq&\int_{0}^{X}(1+y^{2})^{-1/2}(1+y^{2})^{1/2}|F(y)|dy\\\
\leq&\sqrt{\int_{\bf{R}}(1+y^{2})^{-1}dy}\sqrt{\int_{\bf{R}}(1+y^{2})|F(y)|^{2}dy}=\sqrt{\pi}\|F\|_{L^{2}(1)}.\end{split}$

Taking the supremum over $X$ seals the deal. ∎

Armed with Lemmas 5.6 and 5.7 we can get into proving the estimates in the
Lemma 5.4. There are many terms and handling each would inflate this paper
like a bounce house. So we do not do that. Instead we show how to estimate a
“prototype” term which captures the nuances. That term is

$g=E_{0}^{-}\left(\delta_{j}^{+}\zeta{\mathcal{A}}\partial_{l}^{2}B\right)$

which some digging will show appears in $\operatorname{Res}_{2}$. Using the
estimate for $E_{0}^{-}$ from Lemma 5.6 we have

$\|g\|_{\ell^{2}}\leq
C\epsilon^{-1/2}\|\delta^{+}_{j}\zeta\|_{\ell^{\infty}}\|{\mathcal{A}}\partial_{l}^{2}B\|_{H^{1}}.$

By the triangle inequality and the definition of $\delta_{j}^{+}$ we have
$\|\delta^{+}_{j}\zeta\|_{\ell^{\infty}}\leq 2\|\zeta\|_{\ell^{\infty}}$ and
the supposition that the support of $\zeta(j)$ is in $(-1/4,1/4)$ ultimately
gives $\|\delta^{+}_{j}\zeta\|_{\ell^{\infty}}\leq 1/2$. Also classical
Sobolev-Hölder inequalities tell us that
$\|{\mathcal{A}}\partial_{l}^{2}B\|_{H^{1}}\leq\|{\mathcal{A}}\|_{W^{1,\infty}}\|\partial_{l}^{2}B\|_{H^{1}}\leq\|{\mathcal{A}}\|_{W^{1,\infty}}\|B\|_{H^{3}}$.

Since
$\|{\mathcal{A}}\|_{W^{1,\infty}}=\|{\mathcal{A}}\|_{L^{\infty}}+\|{\mathcal{A}}_{w}\|_{L^{\infty}}$
and ${\mathcal{A}}$ is an antiderivative of $A$ we can use Lemma 5.7 to
conclude that $\|{\mathcal{A}}\|_{L^{\infty}}\leq\sqrt{\pi}\|A\|_{L^{2}(1)}.$
Likewise, Sobolev’s inequality tells us that
$\|{\mathcal{A}}_{w}\|_{L^{\infty}}=\|A\|_{L^{\infty}}\leq C\|A\|_{H^{1}}$. So
all together we have

$\|g\|_{\ell^{2}}\leq
C\epsilon^{-1/2}\left(\|A\|_{L^{2}(1)}+\|A\|_{H^{1}}\right)\|B\|_{H^{3}}.$

Since $A$ and $B$ are assumed to be good solutions of KdV on $[-T_{0},T_{0}]$
we get $\sup_{|t|\leq T_{0}/\epsilon^{3}}\|g\|_{\ell^{2}}\leq
C\epsilon^{-1/2},$ which is the targeted estimate.

All the other terms are handled using the same sorts of steps used above. We
close the proof with a comment on the regularity needed. The most smoothness
required for $A$ and $B$ comes from the terms in $\partial_{T}Q_{30}$. As in
[23, 2, 10], one finds that $\partial_{w}^{6}A$ and $\partial_{l}^{6}B$ make
an appearance and so, to deploy estimates like in Lemma 5.6, we need $A$ and
$B$ to be in $H^{7}$. ∎

###  5.2. The autoregressive part

Now we need to put bounds on terms where $\gamma_{1}$ and $\gamma_{2}$ appear.
The first question: how big are these sequences? The equations in (4.22) which
these satisfy are examples of autoregressive models, specifically AR(1)
processes [12]. We have the following almost sure estimate for solutions of
such processes:

######  Lemma 5.8.

Suppose that $z(n)$, $n\geq 0$, are i.i.d. random variables with zero mean and
compact support. Fix $\theta\in(-1,1)$ and let

(5.3) $\chi(n):=\sum_{k=0}^{n-1}\theta^{k}z(n-k).$

Then there exists a constant $C>0$ so that

$\sup_{n>0}{|\chi(n)|\over\sqrt{\ln(e+n)}}\leq C\sqrt{1\over{1-\theta^{2}}}.$

The constant $C$ depends on the realization of $z(n)$ but does not depend on
$\theta$; it is almost surely finite.

###### Proof.

The result is a consequence of of Hoeffding’s inequality, whose proof can be
found in [13]:

######  Theorem 5.9.

Let $w(0),\dots,w(n-1)$ be mean-zero, independent random variables with
$-b_{k}\leq w(k)\leq b_{k}$ almost surely and
$\chi(n)=\displaystyle\sum_{k=0}^{n-1}w(k).$ Then for any $\mu\geq 0$

${\mathbb{P}}(|\chi(n)|\geq\mu)\leq 2\exp\left(-{\mu^{2}\over
2\sum_{k=0}^{n-1}b_{k}^{2}}\right).$

We apply this to (5.3); let $w_{n}(k):=\theta^{k}z(n-k)$. Since
${\mathbb{E}}[z(j)]=0$ we have ${\mathbb{E}}[w_{n}(k)]=0$ for all choices of
$n$ and $k$. Since the $z(j)$ are independent it follows that, for fixed $n$,
the $w_{n}(k)$ are independent with respect to $k$. The support of $z(j)$ is
compact so there is $a\geq 0$ for which the support lies in $[-a,a]$. Then the
support of $\theta^{k}z(n-k)$ is in $[-a\theta^{k},a\theta^{k}]$. Thus
$w_{n}(0),\dots,w_{n}(n-1)$ pass the hypotheses of Theorem 5.9 with
$b_{k}=a\theta^{k}$ and we have:

${\mathbb{P}}[|\chi(n)|\geq\mu]\leq 2\exp\left(-{\mu^{2}\over
2a^{2}\sum_{k=0}^{n-1}\theta^{2k}}\right)=2\exp\left(-{\mu^{2}(1-\theta^{2})\over
2a^{2}(1-\theta^{2n})}\right).$

Now let $\displaystyle\mu(n):=\sqrt{\ln(e+n){4a^{2}(1-\theta^{2n})\over
1-\theta^{2}}}$ so that

${\mathbb{P}}[|\chi(n)|\geq\mu(n)]\leq 2\exp\left(-{\mu^{2}(1-\theta^{2})\over
2a^{2}(1-\theta^{2n})}\right)=2\exp\left(-2\ln(e+n)\right)={2\over(e+n)^{2}}.$

Since $\sum_{n\geq 0}{2/(e+n)^{2}}$ is finite, the Borel-Cantelli Lemma [8]
tells us that, almost surely, $|\chi(n)|\geq\mu(n)$ happens for at most
finitely many $n$. For a given realization of $z(n)$ let $N_{\omega}$ be the
largest value of $n$ at which $|\chi(n)|\geq\mu(n)$ and put
$c_{\omega}:=\max_{1\leq n\leq N_{\omega}}|\chi(n)|/\mu(n).$ Thus we have

$|\chi(n)|\leq 2ac_{\omega}\sqrt{\ln(e+n){1-\theta^{2n}\over
1-\theta^{2}}}\leq 2ac_{\omega}\sqrt{{\ln(e+n)\over 1-\theta^{2}}}$

for all $n$. Putting $C=2ac_{\omega}$ completes the proof.

∎

With Lemma 5.8 we can prove

######  Lemma 5.10.

Take Hypothesis 1.1 as given. Suppose that $\gamma_{1}(j)$ and $\gamma_{2}(j)$
solve (4.22) and $\gamma_{1}(0)=\gamma_{2}(0)=0$. Then there exists a constant
$C>0$ such that for all $\epsilon\in(0,1)$ we have

$\sup_{j\in{\bf{Z}}}{|\gamma_{1}(j)|+|\gamma_{2}(j)|\over\sqrt{\ln(e+|j|)}}\leq
C\epsilon^{-1/2}.$

The constant $C$ depends on the realization of the $\zeta(j)$ but is almost
surely finite.

###### Proof.

We prove the estimate for $\gamma_{2}$ as the one for $\gamma_{1}$ is similar
but easier. Taking $j>0$ in the second equation in (4.22) gives

$\gamma_{2}(j)-\gamma_{2}(j-1)=-\epsilon\gamma_{2}(j)+\left(\zeta(j)+S^{-}\zeta(j)+\zeta(j)\delta^{+}\delta^{-}\zeta(j)+2\sigma^{2}\right)$

or rather

$\gamma_{2}(j)={1\over 1+\epsilon}\gamma_{2}(j-1)+{1\over
1+\epsilon}\left(\zeta(j)+S^{-}\zeta(j)+\zeta(j)\delta^{+}\delta^{-}\zeta(j)+2\sigma^{2}\right).$

If we take $\gamma_{2}(0)=0$ then the we can find $\gamma_{2}(j)$ (for $j>0$)
from the above by iteration. In particular we have

$\begin{split}\gamma_{2}(j)=&{1\over
1+\epsilon}\sum_{k=0}^{j-1}\theta_{\epsilon}^{k}\zeta(j-k)+{1\over
1+\epsilon}\sum_{k=0}^{j-1}\theta_{\epsilon}^{k}S^{-1}\zeta(j-k)\\\ +&{1\over
1+\epsilon}\sum_{k=0}^{j-1}\theta_{\epsilon}^{k}\left(\zeta(j-k)\delta^{+}\delta^{-}\zeta(j-k)+2\sigma^{2}\right)\\\
=:&{1\over
1+\epsilon}\left(\gamma_{21}(j)+\gamma_{22}(j)+\gamma_{23}(j)\right)\end{split}$

where we have put $\theta_{\epsilon}:={1/(1+\epsilon)}.$ To be clear
$\gamma_{21}(j)$, $\gamma_{22}(j)$ and $\gamma_{23}(j)$ correspond to the
three sums in the order of their appearance.

The random variables $\zeta(j)$ meet the hypotheses of Lemma 5.8 and so we can
apply the results to $\gamma_{21}(j)$ and $\gamma_{22}(j)$ forthwith to get:

$\sup_{j\in{\bf{Z}}}{|\gamma_{21}(j)|\over\sqrt{\ln(e+|j|)}}\leq C\sqrt{1\over
1-\theta_{\epsilon}^{2}}\quad\text{and}\quad\sup_{j\in{\bf{Z}}}{|\gamma_{22}(j)|\over\sqrt{\ln(e+|j|)}}\leq
C\sqrt{1\over 1-\theta_{\epsilon}^{2}}$

for some $C>0$ which is almost surely finite. An easy calculation shows that
${1/(1-\theta_{\epsilon}^{2})}={(1+\epsilon^{2})/(2\epsilon+\epsilon^{2})}<{1/\epsilon}$
when $\epsilon\in(0,1)$. Thus we have

$\sup_{j\geq 0}{|\gamma_{21}(j)|+|\gamma_{22}(j)|\over\sqrt{\ln(e+|j|)}}\leq
C\epsilon^{-1/2}.$

Dealing with $\gamma_{23}(j)$ is a bit more complicated because the summands
are not independent. We have
$\gamma_{23}(j)=\sum_{k=0}^{j-1}\theta_{\epsilon}^{k}v(j-k)$ where

$v(j)=\zeta(j)\zeta(j+1)+\zeta(j)\zeta(j-1)-2\zeta(j)^{2}+2\sigma^{2}.$

From this we see that $v(j)$ and $v(j+1)$ are dependent. As are $v(j)$ and
$v(j+2)$, since $\zeta(j+1)$ appears in both. But $v(j+3)$ and $v(j)$ have no
terms in common and it follows that they are independent. Thus
$\left\\{v(3l)\right\\}_{l\in{\bf{Z}}}$ is an i.i.d. collection of random
variables. As are $\left\\{v(3l+1)\right\\}_{l\in{\bf{Z}}}$ and
$\left\\{v(3l+2)\right\\}_{l\in{\bf{Z}}}.$ We break up $\gamma_{23}(k)$
accordingly:

$\gamma_{23}(j)=\sum_{\begin{subarray}{c}0\leq k\leq j-1\\\
k=0\textrm{mod}3\end{subarray}}\theta_{\epsilon}^{k}v(j-k)+\sum_{\begin{subarray}{c}0\leq
k\leq j-1\\\
k=1\textrm{mod}3\end{subarray}}\theta_{\epsilon}^{k}v(j-k)+\sum_{\begin{subarray}{c}0\leq
k\leq j-1\\\ k=2\textrm{mod}3\end{subarray}}\theta_{\epsilon}^{k}v(j-k).$

Each of the three sums passes the hypotheses of Lemma 5.8, though there are
some small subtleties. We estimate the first as the others are all but the
same. Put $k=3l$ to find

$\begin{split}\sum_{\begin{subarray}{c}0\leq k\leq j-1\\\
k=0\textrm{mod}3\end{subarray}}\theta_{\epsilon}^{k}v(j-k)&=\sum_{l=0}^{\lfloor(j-1)/3\rfloor}\theta_{\epsilon}^{3l}v(j-3l).\end{split}$

Then we have from Lemma 5.8:

$\sum_{l=0}^{\lfloor(j-1)/3\rfloor}\theta_{\epsilon}^{3l}v(j-3l)\leq
C\sqrt{\ln(e+\lfloor(j-1)/3\rfloor)}{\sqrt{1\over 1-\theta_{\epsilon}^{6}}}.$

As $\theta_{\epsilon}^{2}>0$ we find
$1/(1-\theta_{\epsilon}^{6})=1/(1-\theta_{\epsilon}^{2})(1+\theta_{\epsilon}^{2}+\theta_{\epsilon}^{4})\leq
1/(1-\theta_{\epsilon}^{2})<1/\epsilon$. This, along with the fact that $\ln$
is an increasing function, gives

$\sum_{l=0}^{\lfloor(j-1)/3\rfloor}\theta_{\epsilon}^{3l}v(j-3l)\leq
C\sqrt{\ln(e+|j|)}\epsilon^{-1/2}$

which in turn leads to the estimate we are after.

We need estimates for $\gamma_{2}(j)$ when $j<0$ too. If we take $j<0$ in the
second equation of (4.22) we get

$\gamma_{2}(j)-\gamma_{2}(j-1)=\epsilon\gamma_{2}(j)+\left(\zeta(j)+S^{-}\zeta(j)+\zeta(j)\delta^{+}\delta^{-}\zeta(j)+2\sigma^{2}\right).$

We rearrange this:

$\gamma_{2}(j-1)=(1-\epsilon)\gamma_{2}(j)-\left(\zeta(j)+S^{-}\zeta(j)+\zeta(j)\delta^{+}\delta^{-}\zeta(j)+2\sigma^{2}\right).$

As we have taken $\gamma_{2}(0)=0$ the above formula gives us $\gamma_{2}(-1)$
and, more generally, $\gamma_{2}(j)$, $j<0$, by iteration. For $j=-l<0$ we
obtain:

$\begin{split}\gamma_{2}(-l)=-&\sum_{k=0}^{l-1}\vartheta_{\epsilon}^{k}\zeta(-l+k+1)-\sum_{k=0}^{l-1}\vartheta_{\epsilon}^{k}S^{-1}\zeta(-l+k+1)\\\
-&\sum_{k=0}^{l-1}\vartheta_{\epsilon}^{k}\left(\zeta(-l+k+1)\delta^{+}\delta^{-}\zeta(-l+k+1)+2\sigma^{2}\right)\end{split}$

where $\vartheta_{\epsilon}:=1-\epsilon$. The first two sums pass the
hypotheses of Lemma 5.8 and since
${1/(1-\vartheta_{\epsilon}^{2})}={1/(2\epsilon-\epsilon^{2})}<{1/\epsilon}$
when $\epsilon\in(0,1)$ we can bound both as we did for $\gamma_{21}$ and
$\gamma_{22}$ earlier. And the same skullduggery about independence that
worked for $\gamma_{23}$ works for the third sum. All together we get

$\sup_{j\leq 0}{|\gamma_{2}(j)|\over\sqrt{\ln(e+|j|)}}\leq C\epsilon^{-1/2}.$

That completes the proof of Lemma 5.10. ∎

Next we prove the main workhorse lemma for controlling $\gamma$ terms in our
approximation:

######  Lemma 5.11.

If $F=F(w,T)$ has $\sup_{|T|\leq T_{0}}\|F(\cdot,T)\|_{H^{1}(1)}<\infty$ then

(5.4) $\sup_{|t|\leq T_{0}/\epsilon^{3}}\|\gamma_{k}(\cdot)F(\epsilon(\cdot\pm
t),\epsilon^{3}t)\|_{\ell^{2}}\leq
C\epsilon^{-1}\sqrt{|\ln(\epsilon)|}\sup_{|T|\leq
T_{0}}\|F(\cdot,T)\|_{H^{1}(1)}$

for $k=1,2$.

If in addition $\sup_{|T|\leq T_{0}}\|F(\cdot,T)\|_{H^{2}(1)}<\infty$ then

(5.5) $\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|E_{0}^{\pm}\left(\gamma_{k}(\cdot)F(\epsilon(\cdot\pm
t),\epsilon^{3}t)\right)\|_{\ell^{2}}\leq
C\epsilon^{-1}\sqrt{|\ln(\epsilon)|}\sup_{|T|\leq
T_{0}}\|F(\cdot,T)\|_{H^{2}(1)}$

for $k=1,2$. (The choices for $+$ or $-$ in $E_{0}^{\pm}$ and
$F(\epsilon(\cdot\pm t),\epsilon^{3}t)$ are not linked.)

The constant $C>0$ is almost surely finite.

###### Proof.

First we tackle (5.4). We handle $k=1$ and the “$-$” sign. The other cases are
no different. First

$\begin{split}\|\gamma_{1}(\cdot)F(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}^{2}=\sum_{j\in{\bf{Z}}}\gamma_{1}^{2}(j)F(\epsilon(j-t),\epsilon^{3}t)^{2}.\end{split}$

Using the estimate from Lemma 5.10 gets

$\begin{split}\|\gamma_{1}(\cdot)F(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}^{2}\leq&C\epsilon^{-1}\sum_{j\in{\bf{Z}}}{\ln(e+|j|)}F(\epsilon(j-t),\epsilon^{3}t)^{2}.\end{split}$

A simple estimate leads us to

$\begin{split}\|\gamma_{1}(\cdot)F(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}^{2}.\leq&C\epsilon^{-1}\sup_{j\in{\bf{Z}}}{\ln(e+|j|)\over
1+(\epsilon(j-t))^{2}}\sum_{j\in{\bf{Z}}}(1+(\epsilon(j-t))^{2})F(\epsilon(j-t),\epsilon^{3}t)^{2}.\end{split}$

We can apply the second estimate in Lemma 5.6 to the sum (with
$u=(1+(\epsilon(j-t))^{2})F(\epsilon(j-t),\epsilon^{3}t)^{2}$) and find

$\begin{split}\|\gamma_{1}(\cdot)F(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}^{2}\leq&C\epsilon^{-2}\sup_{j\in{\bf{Z}}}{\ln(e+|j|)\over
1+(\epsilon(j-t))^{2}}\|\sqrt{1+(\cdot)^{2}}F(\cdot,\epsilon^{3}t)\|_{H^{1}}^{2}\\\
\leq&C\epsilon^{-2}\sup_{j\in{\bf{Z}}}{\ln(e+|j|)\over
1+(\epsilon(j-t))^{2}}\|F(\cdot,\epsilon^{3}t)\|_{H^{1}(1)}^{2}\\\
\end{split}$

Thus

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\gamma_{1}(\cdot)F(\epsilon(\cdot-t),\epsilon^{3}t)\|_{\ell^{2}}\leq
C\epsilon^{-1}\sqrt{\sup_{|t|\leq
T_{0}/\epsilon^{3}}\sup_{j\in{\bf{Z}}}{\ln(e+|j|)\over
1+(\epsilon(j-t))^{2}}}\sup_{|T|\leq T_{0}}\|F(\cdot,T)\|_{H^{1}(1)}.$

From this we see that the proof of (5.4) will be complete once we show that
there is $C=C(T_{0})>0$ such that $0<\epsilon<1$ implies

${\sup_{|t|\leq T_{0}/\epsilon^{3}}\sup_{j\in{\bf{Z}}}{\ln(e+|j|)\over
1+(\epsilon(j-t))^{2}}}\leq C|\ln(\epsilon)|.$

The proof is mainly elementary Calculus, but that does not mean it is obvious.
Here are the details. Let $f_{\epsilon}(y,t):=\displaystyle{\ln(e+|y|)\over
1+(\epsilon(y-t))^{2}}$. We show $\sup_{|t|\leq
T_{0}/\epsilon^{3}}\sup_{y\in{\bf{R}}}f_{\epsilon}(y,t)\leq C|\ln(\epsilon)|$.
Since $f_{\epsilon}(y,t)=f_{\epsilon}(-y,-t)$ we have $\sup_{|t|\leq
T_{0}/\epsilon^{3}}\sup_{y\in{\bf{R}}}f_{\epsilon}(y,t)=\sup_{0\leq t\leq
T_{0}/\epsilon^{3}}\sup_{y\in{\bf{R}}}f_{\epsilon}(y,t)$. If $t\geq 0$ and
$y\geq 0$ then $|y-t|\leq|-y-t|$ which implies $f_{\epsilon}(-y,t)\leq
f_{\epsilon}(y,t)$. Thus $\sup_{|t|\leq
T_{0}/\epsilon^{3}}\sup_{y\in{\bf{R}}}f_{\epsilon}(y,t)=\sup_{0\leq t\leq
T_{0}/\epsilon^{3}}\sup_{y\geq 0}f_{\epsilon}(y,t)$.

Next we argue that $f_{\epsilon}(y,t)$ achieves its supremum at a point in
$(0,\infty)$. Clearly $f_{\epsilon}(y,t)$ is non-negative and
$f_{\epsilon}(y,t)\to 0$ as $y\to\infty$. It is easy enough to show that
$\lim_{y\to 0^{+}}\partial_{y}f_{\epsilon}(y,t)>0$ when $t\geq 0$. Since
$f_{\epsilon}$ is smooth (except at $y=0$), these considerations imply the
existence of $y_{\epsilon}(t)\in(0,\infty)$ for which
$f_{\epsilon}(y_{\epsilon}(t),t)=\sup_{y\in{\bf{R}}}f_{\epsilon}(y,t)$ and
$\partial_{y}f_{\epsilon}(y_{\epsilon}(t),t)=0$.

So we search for solutions of $\partial_{y}f_{\epsilon}(y,t)=0$ with $y\geq
0$. We claim that for $t\geq 0$ and $\epsilon\in(0,1)$ that

(5.6) $\partial_{y}f_{\epsilon}(y,t)=0\text{ and }y\geq 0\implies t\leq y\leq
t+{1\over\epsilon}.$

Given the claim, $t\leq y_{\epsilon}(t)\leq t+\epsilon^{-1}$ follows and as
such:

$f_{\epsilon}(y_{\epsilon}(t),t)={\ln(e+y_{\epsilon}(t))\over
1+(\epsilon(y_{\epsilon}(t)-t))^{2}}\leq\ln(e+t+\epsilon^{-1}).$

In turn we have $\sup_{0\leq t\leq T_{0}/\epsilon^{3}}\sup_{y\geq
0}f_{\epsilon}(y,t)\leq\ln(e+T_{0}\epsilon^{-3}+\epsilon^{-1})\leq
C|\ln(\epsilon)|$ for a constant depending only on $T_{0}$.

So we will be done if we establish the claim. Routine computations show that
$\partial_{y}f_{\epsilon}(y,t)=0$ if and only if

(5.7) ${1+\epsilon^{2}(y-t)^{2}\over 2\epsilon^{2}(y-t)}=(e+y)\ln(e+y).$

Note that if $0\leq y<t$ then the left-hand side of (5.7) is negative whereas
the right-hand side is positive. So there can be no solutions with $y<t$ and
this implies the left-hand inequality in (5.6).

For the right-hand inequality, let us assume that $y-t>\epsilon^{-1}$ and
$t\geq 0$. This gives

${1+\epsilon^{2}(y-t)^{2}\over 2\epsilon^{2}(y-t)}={1\over
2\epsilon^{2}(y-t)}+{y-t\over 2}<{1\over 2\epsilon}+{y-t\over 2}.$

Next, since $t\geq 0$ then $y-t\leq y$ which implies
$\displaystyle{1+\epsilon^{2}(y-t)^{2}\over 2\epsilon^{2}(y-t)}<{1\over
2\epsilon}+{y\over 2}.$ Since $y-t>\epsilon^{-1}$ and $t\geq 0$ we have
$y>\epsilon^{-1}$. Thus $\displaystyle{1+\epsilon^{2}(y-t)^{2}\over
2\epsilon^{2}(y-t)}<y.$ Also we clearly have $y<(e+y)\ln(e+y)$ and so all told

${1+\epsilon^{2}(y-t)^{2}\over 2\epsilon^{2}(y-t)}<(e+y)\ln(e+y)$

when $y-t>\epsilon^{-1}$ and $t\geq 0$. This precludes
$\partial_{y}f_{\epsilon}(y,t)=0$ and the right inequality in (5.6) follows.
Thuse we are done with the proof of (5.4).

The estimate (5.5) follows from (5.4) with a few tricks. First we have by
direct calculation
$E_{0}^{+}\left(\gamma_{1}F\right)=\gamma_{1}(j+1)(E_{0}^{+}F).$ Second, if we
let ${\mathcal{I}}_{\epsilon}G(X)=\epsilon^{-1}\int_{X}^{X+\epsilon}G(Y)dY$
then the Fundamental Theorem of Calculus and the definition of $E_{0}^{+}$
tell us $E_{0}^{+}F={\mathcal{I}}_{\epsilon}\partial_{w}F$. One can show (see
the argument that leads to equation (3.4) in [20]) that
$\|{\mathcal{I}}_{\epsilon}G\|_{H^{n}(r)}\leq C\|G\|_{H^{n}(r)}$. Putting it
all together we get

$\begin{split}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|E_{0}^{\pm}\left(\gamma_{k}(\cdot)F(\epsilon(\cdot\pm
t),\epsilon^{3}t)\right)\|_{\ell^{2}}&=\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\gamma_{k}(\cdot+1){\mathcal{I}}_{\epsilon}\partial_{w}F(\epsilon(\cdot\pm
t),\epsilon^{3}t)\|_{\ell^{2}}\\\ &\leq
C\epsilon^{-1}\sqrt{|\ln(\epsilon)|}\sup_{|T|\leq
T_{0}}\|{\mathcal{I}}_{\epsilon}\partial_{w}F(\cdot,T)\|_{H^{1}(1)}\\\ &\leq
C\epsilon^{-1}\sqrt{|\ln(\epsilon)|}\sup_{|T|\leq
T_{0}}\|F(\cdot,T)\|_{H^{2}(1)}.\end{split}$

∎

Now we can control all the $\gamma$ dependent terms.

######  Lemma 5.12.

Assume Hypothesis 1.1. Let $\widetilde{q}_{\epsilon}(j,t)$ and
$\widetilde{p}_{\epsilon}(j,t)$ be the extended KdV approximators as in
Definition 4.4 where we further assume that that $A$ and $B$ are good
solutions of KdV on $[-T_{0},T_{0}]$. Then almost surely

$\sup_{|t|\leq T_{0}\epsilon^{-3}}\left(\|P_{3}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|Q_{3\gamma}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)={\mathcal{O}}(\epsilon^{-1}\sqrt{|\ln(\epsilon)|})$

and

$\begin{split}\sup_{|t|\leq
T_{0}\epsilon^{-3}}\left(\|W_{1\gamma}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|W_{2\gamma}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)={\mathcal{O}}(\epsilon^{-1}\sqrt{|\ln(\epsilon)|}).\end{split}$

###### Proof.

The estimates for $P_{3}$ and $Q_{3\gamma}$ are immediate from Lemma 5.11 and
their definitions. A direct calculation shows that

$W_{1\gamma}=E_{0}^{+}P_{3}-\partial_{\tau}Q_{3\gamma}-\epsilon^{2}\partial_{T}Q_{3\gamma}.$

Each of these can be estimated with Lemma 5.11 as well. Another calculation
gives

$\begin{split}W_{2\gamma}=&E^{-}_{0}Q_{3\gamma}-m\partial_{\tau}P_{3}-m\epsilon^{2}\partial_{T}P_{3}\\\
&+2\epsilon^{2}D^{-}(Q_{3\gamma}Q_{1})+2\epsilon^{3}D^{-}(Q_{3\gamma}Q_{2})+2\epsilon^{4}D^{-}(Q_{3\gamma}Q_{30})+\epsilon^{4}D^{-}Q_{3\gamma}^{2}.\end{split}$

The first line of the above we estimate with Lemma 5.11. The ones in the
second line all hinge on estimating terms of the form
$D^{-}(Q_{3\gamma}Q_{l})$ for different choices of $l$. The definition of
$D^{-}$ and the triangle inequality give
$\|D^{-}(Q_{3\gamma}Q_{l})(\cdot,\epsilon\cdot)\|_{\ell^{2}}\leq
2\|Q_{3\gamma}(\cdot,\epsilon\cdot)Q_{l}(\cdot,\epsilon\cdot)\|_{\ell^{2}}$
Then we use the fact that
$\|fg\|_{\ell^{2}}\leq\|f\|_{\ell^{2}}\|g\|_{\ell^{2}}$ to get
$\|D^{-}(Q_{3\gamma}Q_{l})(\cdot,\epsilon\cdot)\|_{\ell^{2}}\leq
2\|Q_{3\gamma}(\cdot,\epsilon\cdot)\|_{\ell^{2}}\|Q_{l}(\cdot,\epsilon\cdot)\|_{\ell^{2}}.$
At this point the remainder of the estimates follow from earlier estimates on
the component $Q_{k}$ and bookkeeping.

∎

###  5.3. Finishing up

We are now in position to prove Proposition 5.3.

###### Proof.

We begin with $\alpha_{1}(\epsilon)$. From its definition, (4.1), (5.1) and
the triangle inequality we have

$\begin{split}\alpha_{1}(\epsilon)\leq&\sum_{n=0}^{2}\epsilon^{n+2}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\left(\|Q_{n}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|P_{n}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)\\\ +&\epsilon^{5}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|Q_{30}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\epsilon^{5}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\left(\|Q_{3\gamma}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|P_{3}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)\end{split}$

Then Lemmas 5.4 and 5.12 gives us
$\alpha_{1}(\epsilon)={\mathcal{O}}(\epsilon^{3/2}).$

For $\alpha_{3}(\epsilon)$, we use its definition, Lemma 4.5, (5.2) and the
triangle inequality to obtain

$\begin{split}\alpha_{3}(\epsilon)\leq&\epsilon^{6}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\left(\|\gamma_{1}(\cdot)\partial_{X}^{3}P_{0}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|\gamma_{2}(\cdot)\partial_{X}^{3}Q_{0}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)\\\ +&\epsilon^{6}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\left(\|W_{10}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|W_{20}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right)\\\ +&\epsilon^{6}\sup_{|t|\leq
T_{0}/\epsilon^{3}}\left(\|W_{1\gamma}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}+\|W_{2\gamma}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{2}}\right).\end{split}$

Then Lemmas 5.4 , 5.11 and 5.12 give
$\alpha_{3}(\epsilon)={\mathcal{O}}(\epsilon^{5}\sqrt{|\ln(\epsilon)|}).$

To prove the estimate for $\alpha_{2}(\epsilon)$, from (4.1) we have

$\partial_{t}\widetilde{q}_{\epsilon}=\epsilon^{3}\partial_{\tau}Q_{0}+\begin{array}[t]{@{}c@{}}\underbrace{\epsilon^{5}\partial_{T}Q_{0}+\epsilon^{4}\sum_{k=1}^{3}\left(\epsilon^{k}\partial_{\tau}Q_{k}+\epsilon^{k+2}\partial_{T}Q_{k}\right)}\\\
\epsilon^{4}\widetilde{h}_{\epsilon}\end{array}.$

Since $\|f\|_{\ell^{\infty}}\leq\|f\|_{\ell^{2}}$ we have
$\|\widetilde{h}_{\epsilon}\|_{\ell^{\infty}}\leq\|\widetilde{h}_{\epsilon}\|_{\ell^{2}}.$
All terms appearing in $\widetilde{h}_{\epsilon}$ have been estimated in one
place or another previously and each is
${\mathcal{O}}_{\ell^{2}}(\epsilon^{-1/2})$ at worst so that we get
$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\epsilon^{4}\widetilde{h}_{\epsilon}\|_{\ell^{\infty}}={\mathcal{O}}(\epsilon^{7/2})$.
On the other hand using the first estimate in Lemma 5.6 shows that

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|\epsilon^{3}\partial_{\tau}Q_{0}(\cdot,\epsilon\cdot,\epsilon
t,\epsilon^{3}t)\|_{\ell^{\infty}}\leq\sup_{|t|\leq
T_{0}/\epsilon^{3}}\epsilon^{3}\left(\|A_{w}(\cdot,\epsilon^{3}t)\|_{W^{1,\infty}}+\|B_{l}(\cdot,\epsilon^{3}t)\|_{W^{1,\infty}}\right)\leq
C\epsilon^{3}.$

So all told we have $\alpha_{2}(\epsilon)={\mathcal{O}}(\epsilon^{3})$.

Next, if we let

(5.8) ${\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon}(j,t):=\sum_{k=1}^{3}\epsilon^{k+2}Q_{k}(j,\epsilon
j,\epsilon
t,\epsilon^{3}t)\quad\text{and}\quad{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\displaystyle
p$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\textstyle
p$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.52188pt}$}}}}\cr\hbox{$\scriptstyle
p$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.51563pt}$}}}}\cr\hbox{$\scriptscriptstyle
p$}}}}}_{\epsilon}(,t):=\sum_{k=1}^{3}\epsilon^{k+2}P_{k}(j,\epsilon
j,\epsilon t,\epsilon^{3}t)$

then the estimates from Lemmas 5.4 and 5.12 lead to

(5.9) $\sup_{|t|\leq T_{0}/\epsilon^{3}}\|{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon},{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\displaystyle
p$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\textstyle
p$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.52188pt}$}}}}\cr\hbox{$\scriptstyle
p$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.51563pt}$}}}}\cr\hbox{$\scriptscriptstyle
p$}}}}}_{\epsilon}\|_{\ell^{2}}={\mathcal{O}}(\epsilon^{5/2}).$

So long $A$ and $B$ are not both identically zero it is easy using the
conservation laws of KdV to find that $\inf_{|T|\leq
T_{0}}\|A(\cdot,T)\|_{H^{7}(1)}+\|B(\cdot,T)\|_{H^{7}(1)}\geq b>0$, for some
$b$. This leads, by the triangle inequality, to
$\beta_{1}(\epsilon)=\inf_{|t|\leq
T_{0}/\epsilon^{3}}\|\widetilde{q}_{\epsilon},\widetilde{p}_{\epsilon}\|_{\ell^{2}}\geq
C\epsilon^{3/2}.$ This completes the proof. ∎

##  6\. The main event

Now we can state and prove our main theorem in full detail.

######  Theorem 6.1.

Let $m(j)$ be a realization of the mass coefficients subject to Hypothesis
1.1. Fix $T_{0}>0$ and $\Phi,\Psi\in H^{7}(1)$. Let $(q(j,t),p(j,t))$ be the
solution of the transparent random mass FPUT lattice (1.1) with initial data

$q(j,0)=\epsilon^{2}\Phi(\epsilon j)\quad\text{and}\quad
p(j,0)=\epsilon^{2}\Psi(\epsilon j).$

Let $A(w,T)$ and $B(l,t)$ be the solutions of the KdV equations (4.17) and
(4.19) with initial data

$A(w,0)={1\over 2}\Phi(w)-{1\over 2}\Psi(w)\quad\text{and}\quad B(l,0)={1\over
2}\Phi(l)+{1\over 2}\Psi(l).$

Then there exits $\epsilon_{\star}=\epsilon_{\star}(m,T_{0},\Phi,\Psi)$
(almost surely positive) and $C_{\star}=C_{\star}(m,T_{0},\Phi,\Psi)>0$
(almost surely finite) such that, for all $\epsilon\in(0,\epsilon_{\star})$,
we have the absolute $\ell^{2}$-error estimates

$\begin{split}&\sup_{|t|\leq
T_{0}/\epsilon^{3}}{\left\|q(\cdot,t)-\epsilon^{2}\left[A(\epsilon(\cdot-t),\epsilon^{3}t)+B(\epsilon(\cdot+t),\epsilon^{3}t)\right]\right\|_{\ell^{2}}}\leq
C_{\star}\epsilon^{2}\sqrt{|\ln(\epsilon)|}\quad\text{and}\quad\\\
&\sup_{|t|\leq
T_{0}/\epsilon^{3}}{\left\|p(\cdot,t)-\epsilon^{2}\left[-A(\epsilon(\cdot-t),\epsilon^{3}t)+B(\epsilon(\cdot+t),\epsilon^{3}t)\right]\right\|_{\ell^{2}}}\leq
C_{\star}\epsilon^{2}\sqrt{|\ln(\epsilon)|}.\end{split}$

If at least one of $\Phi$ or $\Psi$ is non-zero then the associated relative
$\ell^{2}$-error estimates are
${\mathcal{O}}(\sqrt{\epsilon|\ln(\epsilon)|})$.

###### Proof.

Take $A$ and $B$ with the initial data as in the statement and form
$\widetilde{q}_{\epsilon}$ and $\widetilde{p}_{\epsilon}$ as in Definition
4.4. Then we have the estimates for $\alpha_{1}(\epsilon)$,
$\alpha_{2}(\epsilon)$, $\alpha_{3}(\epsilon)$ and $\beta_{1}(\epsilon)$ as in
Proposition 5.3, which is to say we have met condition (3.3) from the
statement of Theorem 3.1.

Note that

$q(j,t)-\widetilde{q}_{\epsilon}(j,t)=q(j,t)-\epsilon^{2}[A(\epsilon(j-t),\epsilon^{3}t)+B(\epsilon(j+t),\epsilon^{3}t)]-{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon}(j,t)$

where ${\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon}$ is given in (5.8). In (5.9) we showed that
${\mathchoice{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon}={\mathcal{O}}_{\ell^{2}}(\epsilon^{5/2})$ for $|t|\leq
T_{0}/\epsilon^{3}$. The initial conditions for $p$, $q$, $A$ and $B$ are
arranged so that

$q(j,0)-\widetilde{q}_{\epsilon}(j,0)=-{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon}(j,0)$

And so we have
$\|q(0)-\widetilde{q}_{\epsilon}(0)\|_{\ell^{2}}=\|{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\displaystyle
q$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=4.8229pt}$}}}}\cr\hbox{$\textstyle
q$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.37604pt}$}}}}\cr\hbox{$\scriptstyle
q$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.41144pt}$}}}}\cr\hbox{$\scriptscriptstyle
q$}}}}}_{\epsilon}(0)\|_{\ell^{2}}={\mathcal{O}}(\epsilon^{5/2})$. Similarly
we have

$p(j,t)-\widetilde{p}_{\epsilon}(j,t)=p(j,t)-\epsilon^{2}[-A(\epsilon(j-t),\epsilon^{3}t)+B(\epsilon(j+t),\epsilon^{3}t)]-{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\displaystyle
p$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\textstyle
p$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.52188pt}$}}}}\cr\hbox{$\scriptstyle
p$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.51563pt}$}}}}\cr\hbox{$\scriptscriptstyle
p$}}}}}_{\epsilon}(j,t)$

and
$\|p(0)-\widetilde{p}_{\epsilon}(0)\|_{\ell^{2}}=\|{\mathchoice{{\ooalign{\hbox{\raise
6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\displaystyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\displaystyle
p$}}}}{{\ooalign{\hbox{\raise 6.25pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.25pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=4.30554pt\vrule
height=0.0pt,width=5.03125pt}$}}}}\cr\hbox{$\textstyle
p$}}}}{{\ooalign{\hbox{\raise 5.81944pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.81944pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=3.01389pt\vrule
height=0.0pt,width=3.52188pt}$}}}}\cr\hbox{$\scriptstyle
p$}}}}{{\ooalign{\hbox{\raise 5.53241pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.53241pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=2.15277pt\vrule
height=0.0pt,width=2.51563pt}$}}}}\cr\hbox{$\scriptscriptstyle
p$}}}}}_{\epsilon}(0)\|_{\ell^{2}}={\mathcal{O}}(\epsilon^{5/2})$.

Since
$\alpha_{3}(\epsilon)/\epsilon^{3}={\mathcal{O}}(\epsilon^{2}\sqrt{|\ln(\epsilon)|})$
and $\epsilon^{5/2}=o(\epsilon^{2}\sqrt{|\ln(\epsilon)|})$, these estimates
imply that we meet the condition on the initial data in the statement of
Theorem 3.1. Thus we have the conclusion

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}\|q(t)-\widetilde{q}_{\epsilon}(t),p(t)-\widetilde{p}_{\epsilon}(t)\|_{\ell^{2}}={\mathcal{O}}\left(\epsilon^{2}\sqrt{|\ln(\epsilon)|}\right).$

Then the triangle inequality plus (5.8) give

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}{\left\|q(\cdot,t)-\epsilon^{2}\left[A(\epsilon(\cdot-t),\epsilon^{3}t)+B(\epsilon(\cdot+t),\epsilon^{3}t)\right]\right\|_{\ell^{2}}}={\mathcal{O}}(\epsilon^{2}\sqrt{|\ln(\epsilon)|})$

and

$\sup_{|t|\leq
T_{0}/\epsilon^{3}}{\left\|p(\cdot,t)-\epsilon^{2}\left[-A(\epsilon(\cdot-t),\epsilon^{3}t)+B(\epsilon(\cdot+t),\epsilon^{3}t)\right]\right\|_{\ell^{2}}}={\mathcal{O}}(\epsilon^{2}\sqrt{|\ln(\epsilon)|}).$

This is the absolute error estimate in the theorem. The relative error
estimate follows from the estimate on $\beta_{1}(\epsilon)$. ∎

##  7\. Numerics

In this section we report the outcomes of a variety of numerical simulations
of solutions of (1.1). In all cases our methodology is to truncate (1.1) to
$|j|\leq M$ where $M\gg 1$ and enforce periodic boundary conditions ($M$ is
always taken to be so incredibly vast that the solutions are never large
anywhere near the edges of the computational domain). The resulting system is
a large finite-dimensional ODE which we solve with a standard RK4 algorithm.
This is essentially the same method as used in [10, 20]. The calculations were
performed in MATLAB.

###  7.1. Amplitude attenuation

The first experiment simulates (1.1) with a number of choices for $m(j)$.
These are:

  * •

$m(j)=1$ for all $j$, that is, they are constant.

  * •

$m(j)=1+(-1)^{j}/4$, i.e. $2-$periodic.

  * •

$m(j)$ meet the transparency condition (1.3) where $\zeta(j)$ are drawn from
the uniform distribution on $[-1/8,1/8]$.

  * •

$m(j)$ are i.i.d. random variables, drawn uniformly from $[1/2,3/2]$.

For all these cases, we choose as initial conditions:

(7.1) $q(j,0)=3\epsilon^{2}\operatorname{sech}^{2}(\sqrt{6}\epsilon
j)\quad\text{and}\quad
p(j,0)=-3\epsilon^{2}\operatorname{sech}^{2}(\sqrt{6}\epsilon j).$

We take $\epsilon=1/2,1/4,1/8$ and $1/16$ and simulate from $t=0$ out to
$t=3/\epsilon^{3}$.

Famously, solutions of KdV equations with smooth and localized initial data
will, over time, resolve into the sum of separated solitary waves of fixed
amplitude [7]. Thus, if the solution of the FPUT lattice is well-approximated
by a KdV equation we expect the $\ell^{\infty}$-norm to at least roughly
stabilize over long time periods. And so in Figure 1 we plot
$\|q(\cdot,T/\epsilon^{3}),p(\cdot,T/\epsilon^{3})\|_{\ell^{\infty}}/\epsilon^{2}$
vs $T$, for $0\leq T\leq 3$. (The scaling here is to be consistent with the
long wave scaling so that we may compare various choices of $\epsilon$ on the
same plot.) We see exactly this stabilization in the plots for the constant,
$2$-periodic and transparent cases. Furthermore, the stabilization becomes
more pronounced as $\epsilon$ decreases, which is consistent with the rigorous
KdV approximation theorems here and in [23, 2, 10]. But when the masses are
taken to be i.i.d., there is an obvious, pronounced decay of the amplitude;
this attenuation (up to the scaling) becomes stronger as $\epsilon$ decreases.
This is why we said in the Introduction that a KdV approximation for the
i.i.d. problem is not appropriate.

(a) $m(j)$ are constant

(b) $m(j)$ are $2$-periodic

(c) $m(j)$ are transparent

(d) $m(j)$ are i.i.d.

Figure 1. Scaled $\ell^{\infty}$-amplitude vs scaled time for solutions of
(1.1) with long-wave data as in (7.1).

###  7.2. Numerical computation of optimal error bound

In the second experiment we aim to corroborate the conclusions of our main
result, Theorem 6.1. We simulate (1.1) with $m(j)$ subject to the transparency
condition (1.3) where $\zeta(j)$ are drawn from the uniform distribution on
$[-1/8,1/8]$. In this case, $\sigma^{2}=1/192$. We choose the initial data so
that $B(l,T)$ is zero and $A(w,T)$ is an exact solitary wave solution of
(4.17), namely $3\operatorname{sech}^{2}\left({\sqrt{6\over
1+24\sigma^{2}}(w-T)}\right)$. That is to say, we take

(7.2) $q(j,0)=3\epsilon^{2}\operatorname{sech}^{2}\left({\sqrt{6\over
1+24\sigma^{2}}\epsilon j}\right)\quad\text{and}\quad
p(j,0)=-3\epsilon^{2}\operatorname{sech}^{2}\left({\sqrt{6\over
1+24\sigma^{2}}\epsilon j}\right).$

Figure 2. Ten realizations of absolute $\ell^{2}$-error vs $\epsilon$, loglog
plot, for solutions of (1.1) with “solitary wave-like” data as in (7.2).

We simulate for $\epsilon=2^{-l/2}$ where $l=2,\dots,10$ and run the
simulations from $t=0$ to $t=3/\epsilon^{3}$. (When $\epsilon=1/32$ this takes
a very long time!) To be clear, we fix a realization and then vary $\epsilon$
as stated with the same realization used throughout. Then we compute the
overall absolute error, specifically:

$\begin{split}E_{\epsilon}:=&\sup_{0\leq t\leq
T_{0}/\epsilon^{3}}\left\|q(\cdot,t)-3\epsilon^{2}\operatorname{sech}^{2}\left({\sqrt{6\over
1+24\sigma^{2}}\left(\epsilon(\cdot-t-\epsilon^{2}t)\right)}\right)\right\|_{\ell^{2}}\\\
+&\sup_{0\leq t\leq
T_{0}/\epsilon^{3}}\left\|p(\cdot,t)+3\epsilon^{2}\operatorname{sech}^{2}\left({\sqrt{6\over
1+24\sigma^{2}}\left(\epsilon(\cdot-t-\epsilon^{2}t)\right)}\right)\right\|_{\ell^{2}}.\end{split}$

Then we repeat for another realization (ten different realizations all
together). If we plot $E_{\epsilon}$ vs $\epsilon$ on a loglog plot, Theorem 2
tells us the best fit line to the data should have slope somewhere around $2$,
or larger. The results are shown in Figure 2; all ten realizations on the same
graph. The line of best fit has slope exceeding $2.5$ in each case; they are
$2.5715$, $2.5982$, $2.5677$, $2.5445$, $2.5778$, $2.5948$, $2.5510$,
$2.5760$, $2.5499$, $2.5563$.

This numerically computed slope is over $.5$ larger than what we expect from
our rigorous estimate. That is to say, the numerics indicate that the
approximation of the transparent mass FPUT lattice by KdV is a fair bit better
than what our results from Theorem 6.1 show. We repeated the same experiment
for many different realizations and the numerically computed slope was always
near to $2.5$. Thus we conjecture that the absolute $\ell^{2}$-error is at
worst ${\mathcal{O}}(\epsilon^{5/2})$, which is the same size as the error for
the constant and periodic problems. Of course we do not know how to prove such
a thing at this time.

##  8\. What’s next.

Our results are the first piece of much larger program aimed at bringing
stochastic homogenization to nonlinear dispersive problems. Here are a number
of open problems, some of which should be relatively straightforward given our
results here and others of which will require substantial new technical ideas.

  1. (1)

Prove a result analogous to Theorem 6.1 but in expectation instead of in the
almost sure sense. In our work on the linear i.i.d. lattice [20] we proved
approximation results in both senses and the strategy for the expectation
result is almost surely transferable.

  2. (2)

Study (1.1) but allow spatial heterogeneity in the spring potentials as well.
We expect an analogous transparency condition can be used to achieve a result
similar to the one here.

  3. (3)

Confirm (or reject!) the conjecture that the sharp order of the KdV
approximation error for the transparent mass FPUT lattice is smaller than
${\mathcal{O}}(\epsilon^{5/2}$). The error estimate we prove here is due
entirely to our use of the autoregressive processes in the extended
approximation. But perhaps a yet more clever option exists to handle the terms
which we encountered at $Z_{15}$ and $Z_{25}$.

  4. (4)

If one can get the conjectured sharp error estimate, it opens the door to
replacing the transparency condition (1.3) with

$m(j)=1+\delta^{-}\zeta(j).$

Here there is only one finite-difference on $\zeta(j)$ (which are still
i.i.d.) instead of two. Some preliminary simulations indicate this condition,
which we call the translucent random mass FPUT lattice, should have a valid
KdV approximation.

  5. (5)

The transparency condition is hardly natural or obvious. This raises the
question: are there conditions that one can place on $m(j)$ that are less
rigid than the transparency (or translucency) conditions? Note that the
transparency condition implies that the $m(j)$ are correlated with one
another; perhaps strictures placed upon correlation lengths in $m(j)$ can be
used to get results similar to our results here.

  6. (6)

How can one rigorously capture the amplitude attenuation seen in the numerics
for the i.i.d. random mass problem in Figure 1? Clearly KdV is the wrong
approach, but perhaps there is some other modulation equation that can capture
the dynamics. The articles [15, 16] suggest the use of nonlinear diffusion
equations. On the other hand, an analysis of a spectral problem associated
with linear random lattices by [1] indicates a connection to Anderson
localization and the authors of [17] utilize Boltzmann models in a high-
dimensional linear version of (1.1).

  7. (7)

Can our approach be carried over to the problem of long water waves over
random bathymetry? Our transparency condition is inspired by a similar
condition on the bathymetry proposed in [22]. Can a continuous version of an
autoregressive process (that is, an Ornstein-Uhlenbeck process) be used to
control the residuals in that problem and prove a rigorous approximation?

  8. (8)

How about nonlinear wave equations with random coefficients? Nonlinear
Schrödinger equations (discrete or continuous)? Any of the multitude of
equations named for Joseph Valentin Boussinesq? Or problems in higher spatial
dimensions?

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