# On the open Toda chain with external forcing

Percy Deift, Luen-Chau Li, Herbert Spohn,
Carlos Tomei, Thomas Trogdon  Percy Deift
Courant Institute
251 Mercer St.
New York, NY 10012, USA<EMAIL_ADDRESS>Luen-Chau Li
Department of Mathematics
Pennsylvania State University
University Park, PA 16802, USA<EMAIL_ADDRESS>Herbert Spohn
Zentrum Mathematik
Technische Universität München
Boltzmannstrasse 3
D-85748, Garching, Germany<EMAIL_ADDRESS>Carlos Tomei
Depto. de Matemática
PUC-Rio
Rio de Janeiro, Brazil<EMAIL_ADDRESS>Thomas Trogdon
Department of Applied Mathematics
University of Washington
Seattle, WA 98195, USA<EMAIL_ADDRESS>In honor of Shmuel Agmon and his many
contributions to mathematics

###### Abstract.

We consider the open Toda chain with external forcing, and in the case when
the forcing stretches the system, we derive the longtime behavior of solutions
of the chain. Using an observation of Jürgen Moser, we then show that the
system is completely integrable, in the sense that the $\displaystyle
2N$-dimensional system has $\displaystyle N$ functionally independent Poisson
commuting integrals, and also has a Lax-Pair formulation. In addition, we
construct action-angle variables for the flow. In the case when the forcing
compresses the system, the analysis of the flow remains open.

###### 2010 Mathematics Subject Classification:

37J35

The Duck Test: ‘If it looks like a duck, walks like a duck and

quacks like a duck… it’s a duck!’

†† Keywords: Toda lattice, Liouville integrability, wave operator, action-
angle variables

In 1967, Morikazu Toda introduced [27] the eponymous Toda system with
Hamiltonian

(1) $H(q,p){\ =\ }\frac{1}{2m}\sum_{n\in{\mathbb{Z}}}\ p_{n}^{2}\
+\frac{a}{b}\sum_{n\in{\mathbb{Z}}}\ e^{-b(q_{n+1}-q_{n})}\ +\
a\sum_{n\in{\mathbb{Z}}}\ (q_{n+1}-q_{n}),\,\hbox{for \ }a,b>0$

for particles of equal mass $\displaystyle m>0$ with positions $\displaystyle
q=\\{q_{n}\\}$ on the line, and momenta $\displaystyle p=\\{p_{n}\\}$. The
Hamiltonian equations generated by $\displaystyle H$ have the form

(2) $\begin{array}[]{ll}\dot{q}_{n}&=\ \frac{\partial H}{\partial p_{n}}{\ =\
}\frac{1}{m}p_{n},\\\ \dot{p}_{n}&=\ -\ \frac{\partial H}{\partial q_{n}}{\ =\
}-a\ \big{(}e^{-b(q_{n+1}-q_{n})}-e^{-b(q_{n}-q_{n-1})}\big{)},\end{array}$

for $\displaystyle-\infty<n<\infty$, and so

(3) $\ddot{q}_{n}{\ =\ }-\frac{a}{m}\
\big{(}e^{-b(q_{n+1}-q_{n})}-e^{-b(q_{n}-q_{n-1})}\big{)}\ .$

Toda’s goal in considering $\displaystyle H$ was to investigate further the
observation of J. Ford and J. Waters [12], found by numerical computation,
that nonlinear systems have ‘normal modes where a normal mode is defined as a
motion in which each oscillator moves at essentially constant amplitude
(energy) and at a given frequency’. As Toda notes, the existence of such
normal modes implies, in particular, the non-ergodic character of the system.
Toda’s specific goal in [27] was to discover an exact, explicit form for such
normal modes for $\displaystyle H$, and he found (in the case $\displaystyle
m=1$) such modes in the form of a travelling wave solution for (3),

$q_{n+1}-q_{n}{\ =\
}-\frac{1}{b}\log\left(1+\frac{4(K\nu)^{2}}{ab}\left[\mathrm{dn}^{2}(2K(\nu
t\pm\frac{n}{\lambda}))-\frac{E}{K}\right]\right)\,,-\infty<n<\infty\ ,$

where for given any wave length $\displaystyle\lambda$ and modulus
$\displaystyle 0<k<1$, $\displaystyle K=K(k)$ and $\displaystyle E=E(k)$ are
complete elliptic integrals

$K{\ =\ }\int_{0}^{\pi/2}\ \frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}\
,\quad E{\ =\ }\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\theta}\ d\theta,$

and the frequency $\displaystyle\nu$ is given by

$\nu{\ =\ }\frac{1}{2K}\
\sqrt{\frac{ab}{\big{(}\frac{1}{\operatorname{sn}^{2}(2K/\lambda)}-1+\frac{E}{K}\big{)}}}\
.$

Here $\displaystyle\operatorname{sn}$ and $\displaystyle\operatorname{dn}$ are
the standard Jacobi elliptic functions.

In the case that $\displaystyle n$ runs over $\displaystyle{\mathbb{Z}}$ in
(1), the linear term $\displaystyle a\sum_{n\in{\mathbb{Z}}}\ (q_{n+1}-q_{n})$
plays no role. Scaling

$\displaystyle q_{n}\to bq_{n}\ ,\quad t\to t\sqrt{\frac{m}{ab}}$

in (3), we see that we can restrict our attention to the case

(4) $H{\ =\ }\frac{1}{2}\sum_{n\in{\mathbb{Z}}}\ p_{n}^{2}\ +\
\sum_{n\in{\mathbb{Z}}}e^{q_{n}-q_{n+1}}\ +\ c\sum_{n\in{\mathbb{Z}}}\
(q_{n}-q_{n+1})\ ,$

where $\displaystyle c$ is any constant.

In the periodic case,

$\displaystyle q_{n+N}{\ =\ }q_{n}+s\ ,\quad p_{n+N}{\ =\ }p_{n}$

for some $\displaystyle N\in{\mathbb{N}}$ and $\displaystyle
s\in{\mathbb{R}}$, equations (2) scaled as above take the form

(5) $\begin{array}[]{ll}\dot{q}_{n}&=\ p_{n}\\\ \dot{p}_{n}&=\
\big{(}e^{(q_{n+1}-q_{n})}-e^{(q_{n}-q_{n-1})}\big{)}\end{array}$

for $\displaystyle 1\leq n\leq N$, where $\displaystyle q_{N+1}{\ =\ }q_{1}+s$
as above and $\displaystyle-s=\sum_{n=1}^{N}(q_{n}-q_{n+1})$. In an extensive
numerical investigation in the cases $\displaystyle N=3$ and $\displaystyle
N=6$, J. Ford, S. Stoddard and J. Turner [11] found strong evidence that the
lattice was integrable. And indeed, inspired by [11], M. Hénon [14] and,
independently, H. Flaschka [9] showed that there are $\displaystyle N$
independent integrals for (5). Also independently, S. Manakov [20]111For the
record: the articles by Hénon and Flaschka were submitted on August 13, 1973
and August 22, 1973, respectively, and the article of Manakov was submitted on
February 8, 1974. Also, Henon, Flaschka and Manakov only considered the case
with $\displaystyle s=0$, but their methods go through for general
$\displaystyle s$.  proved the same result, viz., there are $\displaystyle N$
independent integrals for (5). The starting point of Hénon’s analysis was the
integrability of the hard-sphere gas, which is a limiting form of the Toda
lattice. On the other hand, Flaschka and Manakov based their analysis on the
observation that (5) can be written in Lax-pair form as follows: set

$a_{i}{\ =\ }-p_{i}/2\ ,\quad b_{i}{\ =\ }\frac{1}{2}e^{(q_{i}-q_{i+1})/2}\
,1\leq i\leq N\ ,$

where $\displaystyle q_{N+1}{\ =\ }q_{1}+s,\ p_{N+1}{\ =\ }p_{1}$, and in the
variables $\displaystyle a_{i},b_{i}$, equations (5) take the form

(6) $\dot{a}_{i}{\ =\ }2(b_{i}^{2}-b_{i-1}^{2})\ ,\quad\dot{b}_{i}{\ =\
}b_{i}(a_{i+1}-a_{i})\ ,\quad 1\leq i\leq N\ ,$

where $\displaystyle b_{0}=b_{N}$ and $\displaystyle a_{N+1}=a_{1}$. Note that
the stretch parameter $\displaystyle s$ is now part of the initial conditions.

Define the symmetric matrix

$L{\ =\ }\begin{pmatrix}a_{1}&b_{1}&0&b_{N}\\\ b_{1}&a_{2}&\ddots&0\\\
0&\ddots&\ddots&b_{N-1}\\\ b_{N}&0&b_{N-1}&a_{N}\end{pmatrix}\ {\ =\ }\ L^{T}$

and the skew-symmetric matrix

$B{\ =\ }\begin{pmatrix}0&-b_{1}&0&b_{N}\\\ b_{1}&0&\ddots&0\\\
0&\ddots&\ddots&-b_{N-1}\\\ -b_{N}&0&b_{N-1}&0\end{pmatrix}\ {\ =\ }-B^{T}\ .$

Then, if $\displaystyle q_{i}(t),p_{i}(t)$ solve (5), so that $\displaystyle
a_{i}(t),b_{i}(t)$ solve (6), then $\displaystyle L=L(t)$ solves

$\dot{L}{\ =\ }[L,B]{\ =\ }LB-BL$

with $\displaystyle L_{0}{\ =\ }L(t=0)$ given by $\displaystyle
q_{i}(0),p_{i}(0)$. By the general theorem of Lax [18], it follows immediately
that $\displaystyle t\mapsto L(t)$ is an isospectral deformation, i.e.,

$\displaystyle\operatorname{spec}L(t){\ =\
}\operatorname{spec}L_{0},\,\,\,t>0.$

In particular, the eigenvalues $\displaystyle\lambda_{1},\ldots,\lambda_{N}$
of $\displaystyle L_{0}$ are $\displaystyle N$ constants of the motion for
(5), and in [9], Flaschka relates the $\displaystyle\lambda_{i}$’s to the
integrals of the motion obtained by Hénon. Furthermore, in [20], Manakov
showed that the $\displaystyle\lambda_{i}$’s Poisson commute, so that the
Hamiltonian system (5) is completely integrable in the sense of Liouville. In
principle, this meant that the periodic Toda system could be solved up to
quadrature, and indeed in [16] M. Kac and P. van Moerbeke showed how to use
the Lax-pair formalism to partially solve (5) in terms of hyperelliptic
function theory: A full solution was given shortly thereafter by E.Date and
S.Tanaka [3].

The methods of Hénon, Flaschka and Manakov can also be used to prove the
integrability of the Toda lattice with other boundary conditions, particularly
the so-called open Toda lattice (sometimes referred to as Toda with “fixed-
ends”—see below), and also scattering systems with infinitely many particles
(see [14], [9], [20] and also [8]).

In this paper we are particularly interested in the open Toda lattice where
the Hamiltonian has the form

$H_{F}(q,p){\ =\ }\frac{1}{2}\sum_{n=1}^{N}\ p_{n}^{2}\ +\sum_{n=1}^{N-1}\
e^{(q_{n}-q_{n+1})}$

giving rise to the equations

(7) $\begin{array}[]{llll}\dot{q}_{n}&=\ p_{n},\quad 1\leq n\leq N,\\\
\dot{p}_{1}&=\ -e^{q_{1}-q_{2}}\\\ \dot{p}_{n}&=\
e^{q_{n-1}-q_{n}}-e^{q_{n}-q_{n+1}},2\leq n\leq N-1,\\\ \dot{p}_{N}&=\
e^{q_{N-1}-q_{N}}.\end{array}$

One can view (7) as arising from (5) with $\displaystyle N+2$ particles
$\displaystyle q_{0},q_{1},\ldots,q_{N},q_{N+1}$ by setting

$\displaystyle q_{N+1}{\ =\ }\infty\ ,\quad q_{0}=-\infty$

so that the ends are “fixed” at $\displaystyle\pm\infty$: For this reason the
open Toda lattice is sometimes referred to as Toda with “fixed-ends”.
Equations (7) can be written in Lax-pair form by setting

(8) $\begin{array}[]{ll}a_{i}{\ =\ }-p_{n}/2,\,1\leq n\leq N,\\\ b_{i}{\ =\
}\frac{1}{2}e^{(q_{n}-q_{n+1})/2},\,1\leq n\leq N-1,\end{array}$

and defining the symmetric matrix

$L_{F}{\ =\ }\begin{pmatrix}a_{1}&b_{1}&\ldots&0\\\ b_{1}&a_{2}&\ddots&0\\\
\vdots&\ddots&\ddots&b_{N-1}\\\ 0&0&b_{N-1}&a_{N}\end{pmatrix}\ {\ =\ }\
L_{F}^{T}$

and the skew-symmetric matrix

$B_{F}{\ =\ }\begin{pmatrix}0&-b_{1}&\ldots&0\\\ b_{1}&0&\ddots&0\\\
\vdots&\ddots&\ddots&-b_{N-1}\\\ 0&0&b_{N-1}&0\end{pmatrix}{\ =\ }-B_{F}^{T}\
.$

Then if $\displaystyle q_{n}(t),p_{n}(t)$ solve (7), $\displaystyle L_{F}(t)$
solves the Lax-pair equation

(9) $\dot{L}_{F}{\ =\ }[L_{F},B_{F}]$

with $\displaystyle L_{F,0}{\ =\ }L_{F}(t=0)$ given by $\displaystyle
q_{n}(0),p_{n}(0)$. Again, the eigenvalues $\displaystyle\lambda_{i}(t)$ of
$\displaystyle L_{F}(t)$ are constant and hence give $\displaystyle N$
integrals for the open Toda lattice. In what follows, we will often simply
refer to (7) as the Toda lattice, or the Toda system, or the Toda chain, and
in cases where the periodic problem is under discussion, we will specifically
refer to (5) as the periodic Toda system.

Just as in the periodic case, the Toda system can be solved explicitly, now in
terms of rational functions of exponentials, as shown by J. Moser in [22].
Furthermore, Moser showed that the system has the following remarkable long-
term scattering behavior:

(10) $q_{n}(t){\ =\ }\alpha_{n}^{\pm}t+\beta_{n}^{\pm}+O(e^{-\delta|t|}),\
t\to\pm\infty,\,\,\,\delta>0,1\leq n\leq N$

with

(11) $\alpha_{n}^{+}{\ =\ }-2\lambda_{n},\quad\alpha_{n}^{-}{\ =\
}-2\lambda_{N-n+1},\quad 1\leq n\leq N$

and scattering shift as $\displaystyle t$ goes from $\displaystyle-\infty$ to
$\displaystyle\infty,$ given by

(12) $\beta_{N-n+1}^{+}-\beta_{n}^{-}{\ =\ }\sum_{j\neq k}\
\ln(\alpha_{j}^{-}-\alpha_{k}^{+})^{2}\ .$

Explicit expressions for the $\displaystyle\beta_{n}^{\pm}$’s themselves were
derived later in the early 2000’s (see [4]), and subsequently in [19] ; see
(57) below. Here $\displaystyle\lambda_{1}>\lambda_{2}>\ldots>\lambda_{N}$ are
the eigenvalues of $\displaystyle L_{F,0}$.

When one of the authors (PD) came across formula (12), he was astounded: he
had just completed a PhD in abstract scattering theory in Hilbert space, and
the idea that one could compute the scattering shifts (equivalently, the
scattering matrix) for an $\displaystyle N$ particle system explicitly, was
beyond anything he had ever encountered. When he asked Moser how this was
possible, Moser replied, somewhat mysteriously, that ‘Every scattering system
is integrable!’

What Moser meant was the following (see [23], Integrals via Asymptotics: the
Störmer Problem): suppose one has the solution of a Hamiltonian system

$\displaystyle(q(t),p(t)){\ =\
}(q_{1}(t),\ldots,q_{N}(t),p_{1}(t),\ldots,p_{N}(t))\in{\mathbb{R}}^{2N}$

with Hamiltonian $\displaystyle H$ and with the property that, as
$\displaystyle t\to\infty$,

$\begin{array}[]{ll}p(t){\ =\ }p_{\infty}+o(1/t),\\\ q(t){\ =\
}q_{\infty}+tp_{\infty}+o(1),\end{array}$

for some constants $\displaystyle(q_{\infty},p_{\infty})$. Let $\displaystyle
U_{t}(q(0),p(0)){\ =\ }(q(t),p(t))$ be the solution of the system with initial
data $\displaystyle(q(0),p(0))$ and let $\displaystyle
U_{t}^{0}(q^{0}(0),p^{0}(0)){\ =\ }(q^{0}(t),p^{0}(t))$, where
$\displaystyle(q^{0}(t),p^{0}(t))$ solves the free particle motion with
Hamiltonian $\displaystyle H^{0}(q,p){\ =\ }p^{2}/2$, so

$\begin{array}[]{ll}p^{0}(t){\ =\ }p^{0}(0)\\\ q^{0}(t){\ =\
}q^{0}(0)+p^{0}(0)t\ .\end{array}$

Then

$\begin{array}[]{lll}U_{-t}^{0}\circ U_{t}(q_{0},p_{0})&{\ =\
}U_{-t}^{0}(q_{\infty}+p_{\infty}t+o(1),p_{\infty}+o(1/t))\\\ &{\ =\
}(q_{\infty}+p_{\infty}t+o(1)-(p_{\infty}+o(1/t))t,p_{\infty}+o(1/t))\\\ &{\
=\ }(q_{\infty}+o(1),p_{\infty}+o(1/t))\to(q_{\infty},p_{\infty})\
\quad\hbox{as}\ \ t\to\infty\ .\end{array}$

Thus the wave operator

$\displaystyle W(q_{0},p_{0})\equiv\lim_{t\to\infty}\ U_{-t}^{0}\circ
U_{t}(q_{0},p_{0}){\ =\ }(q_{\infty},p_{\infty})$

exists. But then

$\displaystyle U_{-t}^{0}\circ U_{t}\circ U_{s}{\ =\ }U^{0}_{s}\circ
U_{-(t+s)}^{0}\circ U_{t+s}$

implies

$\displaystyle W\circ U_{s}{\ =\ }U_{s}^{0}\circ W$

or, if $\displaystyle W^{-1}$ exists,

(13) $U_{s}{\ =\ }W^{-1}\circ U_{s}^{0}\circ W\ .$

Now $\displaystyle U_{-t}^{0}\circ U_{t}$ is symplectic for all $\displaystyle
t$ and so $\displaystyle W$, and hence $\displaystyle W^{-1}$, are symplectic.
Thus (13) shows us that $\displaystyle U_{s}$ is symplectically equivalent to
$\displaystyle U_{s}^{0}$, and hence is completely integrable. Indeed, if
$\displaystyle\alpha_{1},\ldots,\alpha_{N},$ are commuting integrals for
$\displaystyle H^{0}$, then $\displaystyle\beta_{i}=\alpha_{i}\circ
W,i=1,\ldots,N$ are commuting integrals for $\displaystyle H$:

$\displaystyle\beta_{i}\circ U_{t}(q(0),p(0)){\ =\ }\alpha_{i}\circ W\circ
U_{t}(q(0),p(0))=\alpha_{i}\circ U_{t}^{0}(W(q(0),p(0)){\ =\ }\hbox{constant}$

and as $\displaystyle W$ is symplectic,

(14) $\\{\beta_{i},\beta_{j}\\}{\ =\ }\\{\alpha_{i}\circ W,\alpha_{j}\circ
W\\}{\ =\ }\\{\alpha_{i},\alpha_{j}\\}\circ W{\ =\ }0\ .$

Said differently, the above calculation shows more generally that ‘if a system
behaves like an integrable system, then it is an integrable system!’ or, as in
the famous ‘Duck Test’, ‘if it looks like a duck, walks like a duck and quacks
like a duck… it’s a duck!’

In Moser’s argument in [22] one finds that in addition to (10) one also has

(15) $p_{n}{\ =\ }\alpha_{n}^{\pm}+O(e^{-\delta|t|})\ \quad\hbox{as}\ \
t\to\pm\infty$

and so by Moser’s integrability argument, the Toda lattice is integrable. Note
further that

$\displaystyle\alpha_{i}(q,p){\ =\ }p_{i}\ ,1\leq i\leq N$

are commuting integrals of the motion for $\displaystyle H^{0}$ and so

$\displaystyle\beta_{i}(q_{0},p_{0})=\alpha_{i}(W(q_{0},p_{0})){\ =\
}\alpha_{i}(q_{\infty},p_{\infty}){\ =\ }p_{\infty,i},\,\,1\leq i\leq N$

are commuting integrals of the motion for $\displaystyle H_{F}$. But, from
(10) and (11), up to a factor of $\displaystyle-2$, the $\displaystyle
p_{\infty,i}$’s are just the eigenvalues of $\displaystyle L_{F,0}$, as they
should be!

From the “duck” we learn that there is an interesting Catch 22 in the problem:
We could not have derived, by any means, utilizing any and all dynamical
tools, the asymptotic behavior of the system, unless it was integrable in the
first place!

Moser’s argument can be used to prove the integrability of a variety of
dynamical systems. For example, in [6] the authors showed, contrary to
expectations, that the perturbed defocusing nonlinear Schrödinger equation on
the line,

$\begin{array}[]{ll}iq_{t}+q_{xx}-2|q|^{2}q-\epsilon K(|q|^{2})q=0\\\
q(x,t=0)=q_{0}(x)\to 0\ ,\quad\hbox{as}\ \ |x|\to\infty\end{array}$

is integrable for $\displaystyle 0<\epsilon<\epsilon_{0}$, for some
$\displaystyle\epsilon_{0}>0$. Here $\displaystyle K(|q|^{2})=O(|q|^{\ell})$
as $\displaystyle|q|\to 0$ for suitably large $\displaystyle\ell>2$.

Wave operators were first introduced in the context of quantum mechanical
scattering theory, by C.Møller [21] . In particular, for two self-adjoint
operators $\displaystyle A$ and $\displaystyle B$ in Hilbert space, with
propagators $\displaystyle e^{iAt}$ and $\displaystyle e^{iBt}$, Møller
introduced the quantum mechanical wave operator

$\displaystyle Wf{\ =\ }\lim_{t\to\infty}e^{-iAt}e^{iBt}f$

for vectors $\displaystyle f$ in the Hilbert space. Cook [1], and shortly
thereafter Hunziker [15], were the first to use wave operators in the context
of classical dynamics. Wave operators can also be used more generally to
address problems in analysis: see, in particular, Nelson’s proof in [24] of
Sternberg’s Linearization Theorem for non-resonant systems, and also the proof
of Darboux’s Theorem establishing the existence of local canonical coordinates
for symplectic forms in [23]. A detailed treatment of classical two-body
scattering theory in three dimensions using wave operators is given in [25]. A
detailed treatment of classical and quantum mechanical N-particle scattering
theory is given in [7].

In this paper we consider Toda’s original system (4) in the finite fixed-end
case, with $\displaystyle c\neq 0$. The Hamiltonian for the system has the
form

$H_{c}(q,p){\ =\ }\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}+\sum_{n=1}^{N-1}\
e^{q_{n}-q_{n+1}}+c\ \sum_{n=1}^{N-1}(q_{n}-q_{n+1}),$

giving rise to the associated Hamiltonian equations

(16) $\begin{array}[]{llll}\dot{q}_{n}&{\ =\ }p_{n}\ ,\quad 1\leq n\leq N,\\\
\dot{p}_{1}&{\ =\ }-e^{q_{1}-q_{2}}-c,\\\ \dot{p}_{n}&{\ =\
}e^{q_{n-1}-q_{n}}-e^{q_{n}-q_{n+1}}\ ,\quad 2\leq n\leq N-1,\\\
\dot{p}_{N}&{\ =\ }e^{q_{N-1}-q_{N}}+c.\end{array}$

As opposed to the whole line case and the periodic case, the constant
$\displaystyle c$, and in particular the sign of $\displaystyle c$, now plays
a determining role. Note that

$\displaystyle c\sum_{n=1}^{N-1}(q_{n}-q_{n+1}){\ =\ }c(q_{1}-q_{N}),$

and we think of $\displaystyle H_{c}$ as the Hamiltonian of a lattice of
particles $\displaystyle q_{1},\ldots,q_{N}$ with external forces acting on
the endpoints of the lattice via the potential $\displaystyle cq_{1}-cq_{N}$.
When $\displaystyle c>0$, the forces

$\displaystyle-\frac{\partial}{\partial q_{1}}c(q_{1}-q_{N}){\ =\ }-c\ \
\quad-\frac{\partial}{\partial q_{N}}c(q_{1}-q_{N})=c$

stretch the lattice , and when $\displaystyle c<0$, they compress the lattice.

The study of $\displaystyle H_{c}$ is motivated in part by the statistical
mechanics of the Toda lattice. Here the statistics of the statistical
mechanical ensemble is given by the canonical measure

$\displaystyle\prod_{j=1}^{N}\mathrm{d}p_{j}\prod_{j=1}^{N}\mathrm{d}q_{j}\exp(-\beta
H_{c}(q,p))$

suitably normalized: This measure is clearly invariant under the flow
generated by $\displaystyle H_{c}$. From [26] we learn that the case
$\displaystyle c<0$ arises most naturally, but the case $\displaystyle c>0$ is
also of interest. In general, the analysis of the thermodynamic limit,
$\displaystyle N\to\infty$, of a statistical mechanical system with
Hamiltonian $\displaystyle H$, is greatly simplified if $\displaystyle H$ is
known to be integrable. This motivates, in particular, the study of the
integrability of $\displaystyle H_{c}$.

The numerical calculations below suggest strongly that in the case
$\displaystyle c>0$, $\displaystyle H_{c}$ is integrable. And indeed, the main
result in this paper is to show, using Moser’s integrability argument, that
this is the case. In the case $\displaystyle c<0$, we will argue below that
the numerical calculations suggest that also in this case there is integrable
structure, or near integrable structure, associated with the system.

Figure 1. Numerical computations for the open Toda lattice. The system is
integrated using a second-order accurate Störmer-Verlet method [2] with a time
step of $\displaystyle\Delta t=0.0001$. The initial conditions are generated
by sampling $\displaystyle p_{j}(0),q_{j+1}(0)-q_{j}(0)$, $\displaystyle
j=1,\ldots,N$ as independent and normally distributed random variables.

As a benchmark, Figure 1 displays the solution of the Toda lattice with
Hamiltonian $\displaystyle H_{F}$, $\displaystyle N=20$ particles and randomly
chosen initial data. As $\displaystyle t\to\infty$, $\displaystyle
p(t)=p_{\infty}+o(1)$ and $\displaystyle q(t)=q_{\infty}+p_{\infty}t+o(1)$ for
suitable constants $\displaystyle q_{\infty}$ and $\displaystyle p_{\infty}$
as in (10) and (15). Figure 2 displays the solution of the perturbed Toda
lattice with Hamiltonian $\displaystyle H_{c}$, $\displaystyle c=1,N=20$
particles and randomly chosen initial data. As $\displaystyle t\to\infty$,

$p_{i}(t){\ =\ }p_{i,\infty}+o(1)\ ,\quad q_{i}(t){\ =\
}q_{i,\infty}+tp_{i,\infty}+o(1)\ ,\quad 2\leq i\leq N-1$

for suitable constants $\displaystyle p_{i,\infty},q_{i,\infty}$. But

$\begin{array}[]{ll}p_{1}(t){\ =\ }-ct+O(1)\ ,\quad q_{1}(t){\ =\
}-ct^{2}/2+O(t)\ ,\\\ p_{N}(t){\ =\ }ct+O(1)\ ,\quad q_{N}(t){\ =\
}ct^{2}/2+O(t)\ .\end{array}$

This suggests that the solutions of the $\displaystyle H_{c}$ equations behave
like solutions of a system of $\displaystyle N$ particles $\displaystyle
q_{1},\ldots,q_{N},p_{1},\ldots,p_{N}$ consisting of a Toda lattice
$\displaystyle q_{2},\cdots,q_{N-1}$, $\displaystyle p_{2},\cdots,p_{N-1}$
decoupled from a pair of (decoupled) particles $\displaystyle q_{1},p_{1},$
$\displaystyle q_{N},p_{N}$ solving

$\begin{array}[]{ll}\dot{p}_{1}{\ =\ }-c\ ,\quad\dot{q}_{1}{\ =\ }p_{1}\\\
\dot{p}_{N}{\ =\ }c\ ,\quad\dot{q}_{N}{\ =\ }p_{N}\ .\end{array}$

Such a system of $\displaystyle N$ particles is clearly completely integrable.
What we will show is that solutions of the perturbed Toda system with
Hamiltonian $\displaystyle H_{c},c>0$, indeed behave asymptotically like
solutions of the decoupled system, and hence in view of Moser’s observation,
the perturbed system is integrable.

Figure 2. Numerical computations for the perturbed Toda lattice with
$\displaystyle c=1$. The system is integrated using a second-order accurate
Störmer-Verlet method with a time step of $\displaystyle\Delta t=0.0001$. The
initial conditions are generated by sampling $\displaystyle
p_{j}(0),q_{j+1}(0)-q_{j}(0)$, $\displaystyle j=1,\ldots,N$ as independent and
normally distributed random variables.

Figure 3 displays the solution of the perturbed Toda lattice with Hamiltonian
$\displaystyle H_{c}$ and $\displaystyle c=-1$. Here there are $\displaystyle
N$ particles and the initial conditions are

$\displaystyle q_{i}(0)=i\ ,\quad 1\leq i\leq 20\ ,$

with {$\displaystyle p_{i}(0)$} random. In Figure 4, we again have
$\displaystyle c=-1$, but now

$\displaystyle q_{i}(0)=-i\ ,\quad 1\leq i\leq 20\ ,$

again with {$\displaystyle p_{i}(0)$} random. In Figure 5, we again have
$\displaystyle c=-1$, but now {$\displaystyle q_{i}(0)$} and {$\displaystyle
p_{i}(0)$} are chosen randomly. In all three cases, the solution
$\displaystyle q_{i}(t)$ appears to evolve almost periodically in time, modulo
a slight gradient. In the first two cases, this behavior persists at least up
to times $\displaystyle t\approx 300$, but in the third case the almost
periodicity begins to unravel after $\displaystyle t\approx 200$.

This brings to mind the celebrated computations of E. Fermi, J. Pasta, S. Ulam
and M. Tsingou [10], in which the authors, anticipating ergodicity, found,
unexpectedly, almost periodic behavior in the solutions of a particular
nonlinear lattice system. This meant that in some sense the system was
‘remembering’ its past, and the only way a mechanical system can ‘remember’
its past is if it has many integrals of the motion. In this way, the discovery
was viewed as strong evidence for integrability and led eventually, and
famously, to the discovery by Kruskal-Zabusky and Gardner-Greene-Kruskal-Miura
that the Korteweg de Vries equation is completely integrable.

Over the years, as the power of computers grew, it became clear that Fermi et
al. had just not run their equations long enough: With longer computations,
they would have found that the almost periodicity unravelled and ergodicity
emerged. A very interesting understanding of Fermi et al. is given in [13]:
The lattice equations for unidirectional lattice waves can be written
schematically in the form

$\displaystyle\dot{x}=V(x)+O(h^{2})$

where $\displaystyle h^{2}$ is a continuum limit parameter, $\displaystyle
h^{2}\to 0$, and

$\displaystyle\dot{y}=V(y)$

is KdV. It follows that the solution of the lattice equation $\displaystyle
x(t)$ behaves like the (integrable) KdV equation for times $\displaystyle T$
of order $\displaystyle h^{-2}$, i.e., $\displaystyle Th^{2}$ = O(1), when
$\displaystyle x(t)$ begins to diverge from $\displaystyle y(t)$. Thus, the
lattice has many $\displaystyle h^{2}$-accurate integrals up to times of order
$\displaystyle h^{-2}$. It turns out, however, that the near-integrability
persists for much longer times $\displaystyle T$ of order $\displaystyle
h^{-4}$, and this they are elegantly able to explain by showing that in fact
$\displaystyle x(t)$ solves a system of the form

$\displaystyle\dot{x}=W(x)+O(h^{4})$

and now

$\displaystyle\dot{y}=W(y)$

is a solution of the KdV hierarchy, and hence, also, integrable.

We are led to the following speculation: Is the Fermi et al. problem a guide
to what we see for $\displaystyle c<0$? In Figure 5, in particular, when the
almost periodicity unravels on a moderate time scale (the same is likely true
regarding Figure 3 and Figure 4, but on a longer time scale), Fermi et al.
raises the issue of whether there is some integrable system associated with
the lattice, which describes the solutions of the lattice equations to high
accuracy for large, but not infinite, times? In this way, for large times, the
system would have excellent, but not perfect, ‘memory’.

One final comment: The Fermi-Pasta-Ulam-Tsingou paradox, as it is called, is a
modern illustration of the interesting phenomenon that sometimes science makes
progress, not because of the accuracy of its instruments, but rather because
of their inaccuracy. If computers in the 1950’s could have made longer
calculations, would KdV have been discovered as an integrable system? If
Copernicus had more accurate instruments, sensitive to the fluctuations in the
planetary orbits, would Kepler have been able to come up with his perfect
laws?

Figure 3. Numerical computations for the perturbed Toda lattice with
$\displaystyle c=-1$. The system is integrated using a second-order accurate
Störmer-Verlet method with a time step of $\displaystyle\Delta t=0.0001$. The
initial conditions are generated, for $\displaystyle j=1,\ldots,N$, by
sampling $\displaystyle p_{j}(0)$ as independent and normally distributed and
setting $\displaystyle q_{j+1}(0)-q_{j}(0)=1$. Figure 4. Numerical
computations for the perturbed Toda lattice with $\displaystyle c=-1$. The
system is integrated using a second-order accurate Störmer-Verlet method with
a time step of $\displaystyle\Delta t=0.0001$. The initial conditions are
generated, for $\displaystyle j=1,\ldots,N$, by sampling $\displaystyle
p_{j}(0)$ as independent and normally distributed and setting $\displaystyle
q_{j+1}(0)-q_{j}(0)=-1$. Figure 5. Numerical computations for the perturbed
Toda lattice with $\displaystyle c=-1$. The system is integrated using a
second-order accurate Störmer-Verlet method with a time step of
$\displaystyle\Delta t=0.0001$. The initial conditions are generated by
sampling $\displaystyle p_{j}(0),q_{j+1}(0)-q_{j}(0)$, $\displaystyle
j=1,\ldots,N$ as independent and normally distributed random variables.

We will prove the integrability of $\displaystyle H_{c}$ with $\displaystyle
c>0$ in steps. In Step 1, we prove that solutions of (16) with initial data
$\displaystyle q_{i}(0),p_{i}(0),1\leq i\leq N$ are unique and exist globally,
both for $\displaystyle c\geq 0$ and $\displaystyle c<0$. In the remainder of
the paper, we only consider the case $\displaystyle c>0$. In Step 2, we will
show that, as $\displaystyle t\to\infty$, the particle system under
$\displaystyle H_{c}$ splits up into two parts: a core Toda lattice
$\displaystyle q_{2},\ldots,q_{N-1},p_{2},\ldots,p_{N-1}$ obeying (7) up to
super-exponentially small errors,

$\begin{array}[]{llll}\dot{q}_{n}=p_{n},\ \quad 2\leq n\leq N-1,\\\
\dot{p}_{2}{\ =\ }-e^{q_{2}-q_{3}}+O_{2}(t),\\\ \dot{p}_{n}{\ =\
}e^{q_{n-1}-q_{n}}-e^{q_{n}-q_{n+1}},\quad 3\leq n\leq N-2,\\\ \dot{p}_{N-1}{\
=\ }e^{q_{N-2}-q_{N-1}}-O_{N-1}(t),\end{array}$

where $\displaystyle O_{2}(t)=e^{q_{1}-q_{2}}{\ =\ }O(e^{-\gamma
t^{2}}),O_{N-1}(t)=e^{q_{N-1}-q_{N}}{\ =\ }O(e^{-\gamma t^{2}})$ for some
$\displaystyle\gamma>0$, and a pair of decoupled particles $\displaystyle
q_{1},q_{N},p_{1},p_{N}$ separating from the core lattice, $\displaystyle
q_{1}(t)\to-\infty$ and $\displaystyle q_{N}(t)\to\infty$, as $\displaystyle
t\to\infty$. In Step 3, for solutions $\displaystyle
q_{1}(t),q_{2}(t),\ldots,q_{N}(t),p_{1}(t),p_{2}(t),\ldots,p_{N}(t)$ of (16),
we obtain precise asymptotics for the inner core $\displaystyle
q_{2}(t),\ldots,q_{N-1}(t),p_{2}(t),\ldots,p_{N-1}(t)$. Let $\displaystyle
U_{t}(q_{0}.p_{0})=((q(t),p(t))$ denote the solution of the equations (16)
generated by $\displaystyle H_{c}$. Let
$\displaystyle\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0})=(\hat{q}(t),\hat{p}(t))$
denote the solution generated by

$H^{d}_{c}(q,p){\ =\ }\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}+\sum_{n=2}^{N-2}\
e^{q_{n}-q_{n+1}}+c\ \sum_{n=1}^{N-1}(q_{n}-q_{n+1}),$

in which the inner Toda core
$\displaystyle(q_{2},...,q_{N-1},p_{2},...,p_{N-1})$ is decoupled from
particles $\displaystyle q_{1}$ and $\displaystyle q_{N}$. Finally, let
$\displaystyle U^{\\#}_{t}(q^{\\#}_{0},p^{\\#}_{0})=(q^{\\#}(t),p^{\\#}(t))$
denote the solution of the equation generated by the “free" decoupled
Hamiltonian

$H^{\\#}_{c}(q,p){\ =\ }\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}+c(q_{1}-q_{N}).$

Then in Step 4 we use the asymptotics obtained in Step 3 to show that as
$\displaystyle t\to\infty$, solutions of (16) behave like “free” particles,
and the convergence is sufficiently rapid so that Moser’s argument applied and
the wave operator

$\displaystyle W^{\\#}(q_{0}.p_{0}){\ =\ }\lim_{t\to\infty}U^{\\#}_{-t}\circ
U_{t}(q_{0},p_{0})$

exists. On the other hand, standard Toda asymptotics as in (10) and (15), also
show that as $\displaystyle t\to\infty$, the solution
$\displaystyle(\hat{q}(t),\hat{p}(t))$ of the equations generated by
$\displaystyle H^{d}_{c}$, also behave like “free” particles, and the
convergence is sufficiently rapid so that Moser’s argument again applied and
the wave operator

$\displaystyle\hat{W}^{\\#}(\hat{q}_{0},\hat{p}_{0}){\ =\
}\lim_{t\to\infty}U^{\\#}_{-t}\circ\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0})$

exists. A separate argument then shows that
$\displaystyle(\hat{W}^{\\#})^{-1}$ exists and a short calculation then shows
that

$\displaystyle W=(\hat{W}^{\\#})^{-1}W^{\\#}$

is an intertwining operator for $\displaystyle\hat{U}_{t}$ and $\displaystyle
U_{t}$,

$\displaystyle\hat{U}_{t}\circ W{\ =\ }W\circ U_{t}$

and as $\displaystyle H^{d}_{c}$ is integrable, the integrability of
$\displaystyle H_{c}$ follows. The intertwining relation is not enough,
however, to show that as $\displaystyle t\to\infty$, the solutions
$\displaystyle U_{t}$ behave, as advertised above, like solutions
$\displaystyle\hat{U}_{t}$ of the decoupled system: This is proved using a
separate argument.

Note that we do not construct $\displaystyle W$ directly as a wave operator

$\displaystyle\lim_{t\to\infty}\hat{U}_{-t}\circ U_{t}(q_{0},p_{0}).$

The technical reason for this is discussed at the end of the section, together
with a sketch of the argument that is needed to prove the existence of the
limit. We leave the details to the interested reader.

Finally in Step 5 we display $\displaystyle N$ independent, commuting
integrals for the $\displaystyle H_{c}$ flow and show how the flow can be
written in Lax-pair form.

Step 1.

Standard ODE methods show that (16) has unique local solutions
$\displaystyle(q(t),p(t))$ with $\displaystyle(q(0),p(0)){\ =\ }(q_{0},p_{0})$
for which $\displaystyle H_{c}(q(t),p(t)){\ =\ }h_{0}\equiv
H_{c}(q_{0},p_{0})$. Thus

(17)
$\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}(t)+\sum_{n=1}^{N-1}e^{q_{n}(t)-q_{n+1}(t)}=h_{0}-c(q_{1}(t)-q_{N}(t))$

and so in order to prove global existence it is enough to show, in particular,
that

$|q_{1}(t)-q_{N}(t)|\leq c_{1}t^{2}+c_{2}$

for some constants, $\displaystyle c_{1},c_{2}$. Indeed, by (17), we would
then have, for $\displaystyle 1\leq n\leq N$, $\displaystyle|p_{n}(t)|\leq
c^{\prime}_{1}t+c^{\prime}_{2}$ for some constants $\displaystyle
c^{\prime}_{1},c^{\prime}_{2}$ and so $\displaystyle|q_{n}(t)|\ \leq
c^{\prime\prime}_{1}t^{2}+c^{\prime\prime}_{2}$, again for some constants
$\displaystyle c^{\prime\prime}_{1},c^{\prime\prime}_{2}$. Global existence
for $\displaystyle(q(t),p(t))$ now follows, again by standard ODE methods. We
derive stronger bounds on $\displaystyle q_{n}(t),p_{n}(t)$ in Step 2 below.

The following elementary calculation plays a crucial role in our analysis.
From (17),

$\displaystyle(p_{1}-p_{N})^{2}\leq 2(p_{1}^{2}+p_{N}^{2})\leq
2\sum_{n=1}^{N}p_{n}^{2}\leq 4(h_{0}-c(q_{1}-q_{N}))\ .$

Setting $\displaystyle q_{1}-q_{N}=\Delta$, we have
$\displaystyle(\dot{\Delta})^{2}\leq 4(h_{0}-c\Delta)$ and so

(18) $-2\sqrt{h_{0}-c\Delta}\leq\dot{\Delta}\leq 2\sqrt{h_{0}-c\Delta}\ .$

Integrating we find

$\displaystyle-t+c_{3}\leq\frac{\sqrt{h_{0}-c\Delta}}{-c}\leq t+c_{4}$

for some constants $\displaystyle c_{3},c_{4}$. For $\displaystyle c>0$, this
implies

(19) $0\leq h_{0}-c\Delta\leq(ct+c^{\prime})^{2}$

where $\displaystyle c^{\prime}$ is some constant and so

$\frac{h_{0}-(ct+c^{\prime})^{2}}{c}\leq q_{1}-q_{N}\leq\frac{h_{0}}{c}\ .$

There are, of course, similar bounds for $\displaystyle c<0$.

Step 2.

From (16), we have

$\displaystyle\frac{d}{dt}(p_{N}-p_{1}){\ =\
}e^{q_{N-1}-q_{N}}+e^{q_{1}-q_{2}}+2c$

which implies

(20) $p_{N}(t)-p_{1}(t){\ =\
}p_{N}(0)-p_{1}(0)+\int_{0}^{t}(e^{q_{N-1}-q_{N}}+e^{q_{1}-q_{2}})\ ds+2ct.$

Now, from (18) and (19),

(21) $p_{N}-p_{1}{\ =\ }(\dot{q}_{N}-\dot{q}_{1})\leq
2\sqrt{h_{0}+c(q_{N}-q_{1})}\leq 2ct+2c^{\prime}\ .$

Inserting (20) into (21), we conclude that

(22) $\int_{0}^{\infty}(e^{q_{N-1}-q_{N}}+e^{q_{1}-q_{2}})\ ds<\infty\ .$

By (16),

$\displaystyle p_{1}(t){\ =\ }p_{1}(0)-\int_{0}^{t}e^{q_{1}-q_{2}}\ ds-ct$

and so, as $\displaystyle t\to\infty$,

(23) $p_{1}(t){\ =\ }p_{1,\infty}-ct+o(1)$

for some constant $\displaystyle p_{1,\infty}$. Similarly, as $\displaystyle
t\to\infty$,

(24) $p_{N}(t){\ =\ }p_{N,\infty}+ct+o(1)$

for some constant $\displaystyle p_{N,\infty}$ (note that $\displaystyle
p_{i,\infty}\neq\lim_{t\to\infty}p_{i}(t),$ $\displaystyle i=1,N$). We have

(25) $\frac{1}{2}p_{1}(t)^{2}{\ =\ }\frac{1}{2}c^{2}\ t^{2}-c\ t\
p_{1,\infty}+o(t)$

and

(26) $\frac{1}{2}p_{N}(t)^{2}{\ =\ }\frac{1}{2}c^{2}\ t^{2}+c\ t\
p_{N,\infty}+o(t)\ .$

Inserting (25) and (26) into

$\displaystyle\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}\leq h_{0}+c\ (q_{N}-q_{1}),$

and using (21), we find

$\displaystyle c^{2}\ t^{2}+c\ t\
(p_{N,\infty}-p_{1,\infty})+\frac{1}{2}\sum_{n=2}^{N-1}p_{n}^{2}+o(t)$
$\displaystyle\leq h_{0}+c(q_{N}-q_{1})\leq c^{2}\ t^{2}+2c\
c^{\prime}t+(c^{\prime})^{2}\ .$

We conclude that

(27) $|p_{n}|{\ =\ }O(t^{1/2})\ ,\quad 2\leq n\leq N-1$

and so

$|q_{n}|\leq O(t^{3/2})\ ,\quad 2\leq n\leq N-1\ .$

These bounds are sharper than those obtained in Step 1, and as

$q_{1}(t){\ =\ }-c\frac{t^{2}}{2}+O(t)\ ,\quad q_{N}(t){\ =\
}c\frac{t^{2}}{2}+O(t)$

by (23) and (24), we see that the particles $\displaystyle q_{1}$ and
$\displaystyle q_{N}$ separate from the core $\displaystyle
q_{2},\ldots,q_{N-1}$. Moreover,

(28) $e^{q_{1}-q_{2}}\ ,e^{q_{N-1}-q_{N}}{\ =\
}e^{-c\frac{t^{2}}{2}(1+O(t^{-1/2}))}{\ =\ }O(e^{-\gamma t^{2}})$

for some $\displaystyle\gamma>0$.

Step 3.

Consider $\displaystyle H_{F}(q_{2},\ldots,q_{N-1},p_{2},\ldots,p_{N-1}){\ =\
}\frac{1}{2}\sum_{n=2}^{N-1}p_{n}^{2}+\sum_{n=2}^{N-2}e^{q_{n}-q_{n+1}}$,
evaluated along the solutions $\displaystyle
q_{1}(t),q_{2}(t),\ldots,q_{N-1}(t),q_{N}(t),p_{1}(t),p_{2}(t),\ldots,p_{N-1}(t),p_{N}(t)$
of (16), the flow induced by $\displaystyle H_{c}$. Then

$\displaystyle\frac{d}{dt}H_{F}(q_{2},\ldots,q_{N-1},p_{2},\ldots,p_{N-1}){\
=\
}\sum_{n=2}^{N-1}p_{n}\dot{p}_{n}+\sum_{n=2}^{N-2}e^{q_{n}-q_{n+1}}(p_{n}-p_{n+1})$
$\displaystyle{\ =\ }\sum_{n=2}^{N-1}p_{n}\
e^{q_{n-1}-q_{n}}-\sum_{n=2}^{N-1}p_{n}\ e^{q_{n}-q_{n+1}}$
$\displaystyle+\sum_{n=2}^{N-2}p_{n}\
e^{q_{n}-q_{n+1}}-\sum_{n=2}^{N-2}p_{n+1}\ e^{q_{n}-q_{n+1}}$ $\displaystyle{\
=\ }-p_{N-1}\ e^{q_{N-1}-q_{N}}+p_{2}\ e^{q_{1}-q_{2}}\ .$

It follows from (27) and (28) that

(29) $H_{F}(q_{2},\ldots,q_{N-1},p_{2},\ldots,p_{N-1})\leq\hbox{const.}$

and hence

(30) $|p_{n}(t)|\leq\hbox{const.},\quad 2\leq n\leq N-1$

which is a further strengthening of (27).

The argument now follows in analogy with Moser’s convergence argument for the
Toda lattice in [22]. From (16),

$\displaystyle\frac{d}{dt}p_{2}{\ =\ }e^{q_{1}-q_{2}}-e^{q_{2}-q_{3}},$

and we obtain

$\displaystyle p_{2}(t){\ =\
}p_{2}(0)-\int_{0}^{t}e^{q_{2}-q_{3}}\,ds+\int_{0}^{t}e^{q_{1}-q_{2}}\,ds$
$\displaystyle{\ =\
}p_{2}(0)-\int_{0}^{t}e^{q_{2}-q_{3}}\,ds+\int_{0}^{\infty}e^{q_{1}-q_{2}}\,ds+O(e^{-\gamma
t^{2}})$

by (28). Hence, as $\displaystyle|p_{2}(t)|$ is bounded by (30), we conclude
that

$\int_{0}^{\infty}e^{q_{2}-q_{3}}\ ds<\infty\ .$

Thus

(31) $p_{2}(t){\ =\ }p_{2,\infty}+o(1)\ .$

Now

$\displaystyle\frac{d}{dt}(p_{2}+p_{3}){\ =\ }e^{q_{1}-q_{2}}-e^{q_{3}-q_{4}}$

and as $\displaystyle p_{2}(t)$ and $\displaystyle p_{3}(t)$ are bounded, we
conclude again as above that

$\int_{0}^{\infty}e^{q_{3}-q_{4}}\ ds<\infty$

and, using (31),

$p_{3}(t){\ =\ }p_{3,\infty}+o(1)\ .$

In particular, we conclude that

(32) $q_{2}(t)-q_{3}(t){\ =\ }t(p_{2,\infty}-p_{3,\infty})+o(t)\ .$

Using

$\displaystyle\frac{d}{dt}(p_{2}+\ldots+p_{n}){\ =\
}e^{q_{1}-q_{2}}-e^{q_{n}-q_{n+1}}\ ,\quad 2\leq n\leq N-1$

and proceeding by induction we find

(33) $\int_{0}^{\infty}e^{q_{n}-q_{n+1}}\ ds<\infty,\quad 2\leq n\leq N-2\ .$

(Of course the estimates

$\displaystyle\int_{0}^{\infty}e^{q_{N-1}-q_{N}}\
ds,\,\int_{0}^{\infty}e^{q_{1}-q_{2}}\ ds<\infty$

were obtained earlier.)

It follows that, as $\displaystyle t\to\infty$,

$p_{n}(t)=p_{n,\infty}+o(1),\quad 2\leq n\leq N-1$

and hence

$q_{n}(t){\ =\ }q_{n,\infty}+p_{n,\infty}t+o(t),\quad 2\leq n\leq N-1$

and so

(34)
$q_{n}(t)-q_{n+1}(t)=q_{n,\infty}-q_{n-1,\infty}+(p_{n,\infty}-p_{n+1,\infty})t+o(t),\quad
2\leq n\leq N-2\ .$

In particular, by (33), we must have

(35) $p_{n,\infty}-p_{n+1,\infty}\leq 0,\quad 2\leq n\leq N-2\ .$

We will show shortly that the inequality in (35) is strict. But note first
that by (29) and (30),

$\displaystyle\frac{d}{dt}e^{q_{n}-q_{n+1}}{\ =\
}e^{q_{n}-q_{n+1}}(p_{n}-p_{n+1}),\quad 2\leq n\leq N-2$

is bounded, and so $\displaystyle e^{q_{n}-q_{n+1}}$ is globally Lipschitz in
time, and in particular uniformly continuous in $\displaystyle t$, and it
follows from (33) that

(36) $e^{q_{n}(t)-q_{n+1}(t)}\to 0$

pointwise as $\displaystyle t\to\infty$.

We now prove that the inequality in (35) is strict.

In terms of the variables introduced in (8), $\displaystyle a_{n}{\ =\
}-p_{n}/2,1\leq n\leq N$, and $\displaystyle b_{n}{\ =\
}e^{(q_{n}-q_{n+1})/2}/2,\ 1\leq n\leq N-1$, the equations (16) take the form

(37) $\begin{array}[]{ll}\dot{a}_{n}{\ =\ }2(b_{n}^{2}-b_{n-1}^{2}),\quad
2\leq n\leq N-1,\\\ \dot{b}_{n}{\ =\ }b_{n}(a_{n+1}-a_{n}),\quad 2\leq n\leq
N-2\ .\end{array}$

Set $\displaystyle A_{n}=\lim_{t\to\infty}a_{n}{\ =\ }-p_{n,\infty}/2\ ,2\leq
n\leq N-1$, and consider

$S(t)\equiv\sum_{n=2}^{N-1}(a_{n}-A_{n})^{2}+2\sum_{n=2}^{N-2}b_{n}^{2}\ .$

From (37), we have

$\displaystyle b_{n}(t){\ =\ }b_{n}(0)\ e^{\int_{0}^{t}(a_{n+1}-a_{n})\ ds}\
,\quad 2\leq n\leq N-2$

and as $\displaystyle b_{n}(t)=\frac{1}{2}e^{(q_{n}(t)-q_{n+1}(t))/2}\to 0$ by
(36), we must have

(38) $\int_{0}^{\infty}(a_{n+1}-a_{n})\ ds{\ =\ }-\infty\ ,\quad 2\leq n\leq
N-2.$

Using (37), we find after summing by parts

$\displaystyle\frac{1}{4}\frac{d}{dt}S(t){\ =\
}\sum_{n=2}^{N-1}(a_{n}-A_{n})(b_{n}^{2}-b_{n-1}^{2})+\sum_{n=2}^{N-2}b_{n}^{2}(a_{n+1}-a_{n})$
$\displaystyle{\ =\ }(a_{N-1}-A_{N-1})\
\sum_{n=2}^{N-1}(b_{n}^{2}-b_{n-1}^{2})$
$\displaystyle+\sum_{n=2}^{N-2}(a_{n}-a_{n+1}-A_{n}+A_{n+1})\
\sum_{i=2}^{n}(b_{i}^{2}-b_{i-1}^{2})+\
\sum_{n=2}^{N-2}b_{n}^{2}(a_{n+1}-a_{n})$ $\displaystyle{\ =\
}(a_{N-1}-A_{N-1})(b_{N-1}^{2}-b_{1}^{2})$
$\displaystyle+\sum_{n=2}^{N-2}(a_{n}-a_{n+1}-A_{n}+A_{n+1})(b_{n}^{2}-b_{1}^{2})+\sum_{n=2}^{N-2}b_{n}^{2}(a_{n+1}-a_{n})$
$\displaystyle=\sum_{n=2}^{N-2}(A_{n+1}-A_{n})b_{n}^{2}+O(e^{-\gamma t^{2}}),$

where we have used (28),

$\displaystyle b_{1}^{2}{\ =\ }\frac{1}{4}\ e^{q_{1}-q_{2}}\ ,\quad
b_{N-1}^{2}{\ =\ }\frac{1}{4}\ e^{q_{N-1}-q_{N}}{\ =\ }O(e^{-\gamma t^{2}})\
.$

Now, if $\displaystyle A_{n+1}-A_{n}{\ =\
}-\frac{1}{2}(p_{n+1,\infty}-p_{n,\infty})>0$, then it follows from (34) that

$\displaystyle b_{n}(t){\ =\ }\frac{1}{2}e^{(q_{n}(t)-q_{n+1}(t))/2}{\ =\
}O(e^{-\mu t/2})$

for some $\displaystyle\mu>0$. As $\displaystyle A_{n+1}-A_{n}\geq 0$ by (35),
it follows that

$\displaystyle(A_{n+1}-A_{n})b_{n}^{2}(t){\ =\ }O(e^{-\mu t}),2\leq n\leq N-2\
.$

We conclude that, as $\displaystyle S(t)\to 0$ as $\displaystyle t\to\infty$,

$\displaystyle S(t){\ =\ }O(e^{-\mu t})$

and hence

$\displaystyle a_{n}{\ =\ }A_{n}+O(e^{-\mu t}),\quad 2\leq n\leq N-1$

as $\displaystyle t\to\infty$. Thus if $\displaystyle A_{n+1}-A_{n}=0$ for
some $\displaystyle 2\leq n\leq N-2$, we would have $\displaystyle
a_{n+1}(t)-a_{n}(t)=O(e^{-\mu t})$ for some $\displaystyle 2\leq n\leq N-2$,
which contradicts (38). Thus the inequality in (35) is strict,

(39) $p_{2,\infty}<p_{3,\infty}<\ldots<p_{N-1,\infty}\ .$

In particular, it follows (use (34), or more directly $\displaystyle S(t){\ =\
}O(e^{-\mu t})$) that, as $\displaystyle t\to\infty$,

(40) $e^{q_{n}(t)-q_{n+1}(t)}\equiv O(e^{-\mu t}),\quad 2\leq n\leq N-2$

and hence, by the above induction argument we also have

(41) $p_{n}(t){\ =\ }p_{n,\infty}+O(e^{-\mu t})\ ,\quad 2\leq n\leq N-1,$

and so

(42) $q_{n}{\ =\ }q_{n,\infty}+t\ p_{n,\infty}+O(e^{-\mu t})\ ,\quad 2\leq
n\leq N-1.$

Equations (31) and (36) already imply that the core Toda matrix

(43) $L_{F}{\ =\ }\begin{pmatrix}a_{2}&b_{2}&\ldots&0\\\
b_{2}&\ddots&\ddots&0\\\ \vdots&\ddots&\ddots&b_{N-2}\\\
0&0&b_{N-2}&a_{N-1}\end{pmatrix}$

converges to a diagonal matrix. By (40) and (41), the convergence is
exponential.

Remark: If $\displaystyle L_{F}$ evolved according to the (exact) Toda flow
(9), then, as $\displaystyle t\to\infty$, $\displaystyle p_{n}(t)\to
p_{n,\infty}=-2\lambda_{n}^{F},2\leq n\leq N-1$, where the
$\displaystyle\lambda_{n}^{F}$’s are the eigenvalues of $\displaystyle
L_{F}(t=0)$. As $\displaystyle L_{F}(t=0)$ is tridiagonal, the
$\displaystyle\lambda_{n}^{F}$’s are distinct and so the strict inequality in
(35) is immediate. As $\displaystyle L_{F}(t)$ solves only a perturbed Toda
flow, the strict inequality requires a more subtle analysis, as above.

Step 4.

From $\displaystyle p_{1}(t){\ =\ }p_{1}(0)-\int_{0}^{t}e^{q_{1}-q_{2}}\ ds-
ct$ and (28), we see that

(44) $p_{1}(t){\ =\ }p_{1,\infty}-ct+O(e^{-\gamma t^{2}})$

and similarly

$p_{N}(t){\ =\ }p_{N,\infty}+ct+O(e^{-\gamma t^{2}})$

from which it follows that

$q_{1}(t){\ =\ }q_{1,\infty}-\frac{ct^{2}}{2}+tp_{1,\infty}+O(e^{-\gamma
t^{2}})$

and

$q_{N}(t){\ =\ }q_{N,\infty}+\frac{ct^{2}}{2}+tp_{N,\infty}+O(e^{-\gamma
t^{2}})\ .$

From (41) and (42), we have for $\displaystyle 2\leq n\leq N-1$,

$p_{n}(t){\ =\ }p_{n,\infty}+O(e^{-\mu t})$

and

(45) $q_{n}(t){\ =\ }q_{n,\infty}+tp_{n,\infty}+O(e^{-\mu t})\ .$

Let $\displaystyle U_{t}(q_{0},p_{0}){\ =\ }(q(t),p(t))$ be the solution of
(16) with $\displaystyle(q(0),p(0){\ =\ }(q_{0},p_{0})$, and let
$\displaystyle\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0}){\ =\
}(\hat{q}(t),\hat{p}(t))$ be the solution of the Hamiltonian equations
generated by the decoupled Hamiltonian

(46) $H_{c}^{d}(q,p){\ =\
}\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}+\sum_{n=2}^{N-2}e^{q_{n}-q_{n+1}}+c(q_{1}-q_{N})\
.$

The Hamiltonian $\displaystyle H_{c}^{d}$ is clearly completely integrable. We
have

$\displaystyle\dot{\hat{q}}_{1}(t){\ =\ }\hat{p}_{1}(t)\
,\quad\dot{\hat{p}}_{1}{\ =\ }-c$

and so

$\hat{p}_{1}(t){\ =\ }\hat{p}_{1,0}-ct\ ,\quad\hat{q}_{1}(t){\ =\
}\hat{q}_{1,0}+\hat{p}_{1,0}t-\frac{ct^{2}}{2}$

and similarly

$\hat{p}_{N}(t){\ =\ }\hat{p}_{N,0}+ct\ ,\quad\hat{q}_{N}(t){\ =\
}\hat{q}_{N,0}+\hat{p}_{N,0}t+\frac{ct^{2}}{2}\ .$

By standard Toda asymptotics, (10) and (15), for $\displaystyle 2\leq n\leq
N-1$, as $\displaystyle t\to\pm\infty$,

$\hat{p}_{n}(t){\ =\ }\alpha_{n}^{\pm}+O(e^{-\gamma|t|})$

and

$\hat{q}_{n}(t){\ =\ }\beta_{n}^{\pm}+\alpha_{n}^{\pm}t+O(e^{-\gamma|t|})\ .$

Also let $\displaystyle
U_{t}^{\\#}(q_{0}^{\\#},p_{0}^{\\#})=(q^{\\#}(t),p^{\\#}(t))$ be the solution
of the equations generated by the “free" decoupled Hamiltonian

$H_{c}^{\\#}(q,p){\ =\ }\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}+c(q_{1}-q_{n})\ .$

Clearly

$p_{1}^{\\#}(t){\ =\ }p_{1,0}^{\\#}-ct\ ,\quad q_{1}^{\\#}(t){\ =\
}q_{1,0}^{\\#}+p_{1,0}^{\\#}t-\frac{ct^{2}}{2}$

and

$p_{N}^{\\#}(t){\ =\ }p_{N,0}^{\\#}+ct\ ,\quad q_{N}^{\\#}(t){\ =\
}q_{N,0}^{\\#}+p_{N,0}^{\\#}t+\frac{ct^{2}}{2}$

and

(47) $p_{n}^{\\#}(t){\ =\ }p_{n,0}^{\\#}\ ,\quad q_{n}^{\\#}(t){\ =\
}q_{n,0}^{\\#}+p_{n,0}^{\\#}t\ ,\quad 2\leq n\leq N-1\ .$

Now

$\displaystyle\hat{W}_{t}^{\\#}(\hat{q}_{0},\hat{p}_{0})\equiv
U_{-t}^{\\#}\circ\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0}){\ =\
}U_{-t}^{\\#}\big{(}\hat{q}_{1,0}+\hat{p}_{1,0}t-\frac{ct^{2}}{2},\hat{p}_{1,0}-ct,$
$\displaystyle\big{(}\beta_{n}^{+}+\alpha_{n}^{+}t+O(e^{-\gamma
t}),\alpha_{n}^{+}+O(e^{-\gamma
t})\big{)}_{n=2}^{N-1},\hat{q}_{N,0}+\hat{p}_{N,0}t+\frac{ct^{2}}{2},\hat{p}_{N,0}+ct\big{)}$

$\displaystyle{\ =\
}\big{(}(\hat{q}_{1,0}+\hat{p}_{1,0}t-\frac{ct^{2}}{2})+(\hat{p}_{1,0}-ct)(-t)-\frac{c(-t)^{2}}{2},(\hat{p}_{1,0}-ct)-c(-t)$

$\displaystyle\big{(}\beta_{n}^{+}+\alpha_{n}^{+}t+O(e^{-\gamma
t})+(\alpha_{n}^{+}+O(e^{-\gamma t}))(-t),\alpha_{n}^{+}+O(e^{-\gamma
t})\big{)}_{n=2}^{N-1},$
$\displaystyle(\hat{q}_{N,0}+\hat{p}_{N,0}t+\frac{ct^{2}}{2})+(\hat{p}_{N,0}+ct)(-t)+\frac{c(-t)^{2}}{2},(\hat{p}_{N,0}+ct)+c(-t)\big{)}$
$\displaystyle{\ =\
}\big{(}\hat{q}_{1,0},\hat{p}_{1,0},\big{(}\beta_{N}^{+}+O(e^{-\gamma
t}),\alpha_{n}^{+}+O(e^{-\gamma
t})\big{)}_{n=2}^{N-1},\hat{q}_{N,0},\hat{p}_{N,0}\big{)}$
$\displaystyle\to\big{(}\hat{q}_{1,0},\hat{p}_{1,0},\big{(}\beta_{N}^{+},\alpha_{n}^{+}\big{)}_{n=2}^{N-1},\hat{q}_{N,0},\hat{p}_{N,0}\big{)}\equiv(\hat{q}^{\\#},\hat{p}^{\\#}),$

where

(48) $\hat{q}^{\\#}{\ =\
}(\hat{q}_{1,0},\beta_{2}^{+},\ldots,\beta_{N-1}^{+},\hat{q}_{N,0})\
,\quad\hat{p}^{\\#}{\ =\
}(\hat{p}_{1,0},\alpha_{2}^{+},\ldots,\alpha_{N-1}^{+},\hat{p}_{N,0})\ .$

Thus

(49) $\hat{W}^{\\#}(\hat{q}_{0},\hat{p}_{0}){\ =\
}\lim_{t\to\infty}U_{-t}^{\\#}\circ\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0}){\ =\
}(\hat{q}^{\\#},\hat{p}^{\\#})$

exists. A similar argument shows that

(50) $W^{\\#}(q_{0},p_{0}){\ =\ }\lim_{t\to\infty}U_{-t}^{\\#}\circ
U_{t}(q_{0},p_{0}){\ =\ }(q^{\\#},p^{\\#})$

exists, where we now use (44) and (45),

$q^{\\#}{\ =\ }(q_{1,\infty},\ldots,q_{N,\infty})\ ,\quad p^{\\#}{\ =\
}(p_{1,\infty},\ldots,p_{N,\infty})\ .$

As

$I_{1}(q,p){\ =\ }\frac{1}{2}p_{1}^{2}+cq_{1}\ ,\quad I_{n}(q,p){\ =\ }p_{n},\
2\leq n\leq N-1\ ,\quad I_{N}(q,p){\ =\ }\frac{1}{2}p_{N}^{2}-q_{N}$

are commuting integrals for $\displaystyle U_{t}^{\\#}$, it follows from (49),
as in (14), that

$J_{n}(q,p){\ =\ }I_{n}\circ W^{\\#}(q,p){\ =\ }I_{n}(q^{\\#},p^{\\#})\ ,\quad
2\leq n\leq N-1$

are commuting integrals for $\displaystyle U_{t}$, i.e., for (16). In
particular, this shows that $\displaystyle H_{c}$ is integrable. However, we
want to show more: we want to show that there is an intertwining operator
$\displaystyle W$ for $\displaystyle\hat{U}_{t}$ and $\displaystyle U_{t}$,

(51) $\hat{U}_{t}\circ W=W\circ U_{t}\ .$

This will then allow us to display (16) in a convenient Lax-pair form, and
hence as an isospectral deformation similar to the (unperturbed) Toda case.
From (49) and (50), we have

(52) $U_{t}^{\\#}\circ\hat{W}^{\\#}{\ =\ }\hat{W}^{\\#}\circ\hat{U}_{t}\ \
\hbox{and}\ \ U_{t}^{\\#}\circ W^{\\#}{\ =\ }W^{\\#}\circ U_{t}$

and so, if $\displaystyle(\hat{W}^{\\#})^{-1}$ exists, then (modulo domain
issues, explained at the end of Step 4)

$\displaystyle\hat{U}_{t}\circ(\hat{W}^{\\#})^{-1}\circ W^{\\#}{\ =\
}(\hat{W}^{\\#})^{-1}\circ U_{t}^{\\#}\circ W^{\\#}{\ =\
}(\hat{W}^{\\#})^{-1}\circ W^{\\#}\circ U_{t}$

and so (51) holds with

(53) $W{\ =\ }(\hat{W}^{\\#})^{-1}\circ W^{\\#}\ .$

The proof that $\displaystyle(\hat{W}^{\\#})^{-1}$ exists requires detailed
knowledge of $\displaystyle(\hat{q}^{\\#},\hat{p}^{\\#})$.

To prove that $\displaystyle(\hat{W}^{\\#})^{-1}$ exists, we use the following
result from [4] (for an alternative proof, see Props. 5.2 and 3.6 in [19]).
Let

(54) $(q(t),p(t)){\ =\
}(q_{2}(t),\ldots,q_{N-1}(t),p_{2}(t),\ldots,p_{N-1}(t))$

solve the Toda equation generated by $\displaystyle H_{F}(q,p){\ =\
}\frac{1}{2}\sum_{n=2}^{N-1}p_{n}^{2}+\sum_{n=2}^{N-2}e^{q_{n}-q_{n+1}}$ with
initial data $\displaystyle(q_{0},p_{0})$. Then, as $\displaystyle t\to\infty$
(see (10)),

(55) $q_{n}(t){\ =\ }\alpha_{n}^{+}t+\beta_{n}^{+}+O(e^{-\gamma t})\
,\quad\gamma>0,\,\,2\leq n\leq N-1,$

where

(56) $\alpha_{n}^{+}{\ =\ }-2\lambda_{n}\ ,\quad 2\leq n\leq N-1$

and

(57) $\beta_{n}^{+}{\ =\
}\frac{1}{N-2}\sum_{j=2}^{N-1}q_{j,0}-\frac{2}{N-2}\sum_{j=2}^{N-1}\ln\left(\frac{u_{n}(2)}{u_{j}(2)}\frac{\Pi_{\ell=2}^{n-1}2(\lambda_{\ell}-\lambda_{n})}{\Pi_{\ell=2}^{j-1}2(\lambda_{\ell}-\lambda_{j})}\right)\
,$

where $\displaystyle 2\leq n\leq N-1$, and
$\displaystyle\Pi_{\ell=2}^{1}2(\lambda_{\ell}-\lambda_{2})\equiv 1$. Here
$\displaystyle\lambda_{2}>\lambda_{3}>\ldots>\lambda_{N-1}$ are the
eigenvalues of the core Toda matrix $\displaystyle L_{F}(q_{0},p_{0})$ in
(43), where $\displaystyle a_{i}{\ =\ }-p_{i,0}/2$, $\displaystyle b_{j}{\ =\
}\frac{1}{2}e^{(q_{j,0}-q_{j+1,0})/2}$, $\displaystyle 2\leq i\leq N-1$,
$\displaystyle 2\leq j\leq N-2,$ and $\displaystyle u_{2}(2),\ldots
u_{N-1}(2)$ are the first components of the normalized eigenvectors
$\displaystyle u_{n}=(u_{n}(2),\ldots,u_{n}(N-1))^{T}$ of $\displaystyle
L_{F}(q_{0},p_{0})$ corresponding to $\displaystyle\lambda_{n},\ 2\leq n\leq
N-1$. We have $\displaystyle\sum_{j=2}^{N-1}u_{n}^{2}(j)=1$ and $\displaystyle
u_{n}(2)>0$ for all $\displaystyle 2\leq n\leq N-1$. It is well known (see
e.g. [4]) that the map $\displaystyle\Phi$ from Jacobi matrices $\displaystyle
L_{F}$ with $\displaystyle b_{i}>0$, $\displaystyle 2\leq i\leq N-1$, is a
bijection onto

$\displaystyle\\{(\gamma_{2},\ldots,\gamma_{N-1},\mu_{2},\ldots,\mu_{N-1})\ :\
\gamma_{2}>\gamma_{3}>\ldots>\gamma_{N-1},$
$\displaystyle\sum_{i=2}^{N-1}\mu_{i}^{2}=1,\quad\mu_{i}>0,\quad 2\leq i\leq
N-1\\}.$

Let $\displaystyle U_{t}^{F}{\ =\ }(q(t),p(t))$ denote the solution of the
Toda equations in (54) above and let $\displaystyle U_{t}^{0}$ denote the
solution of the equations generated by $\displaystyle H^{0}(q,p){\ =\
}\frac{1}{2}\sum_{n=2}^{N-1}p_{n}^{2}$. Then the argument leading to (49)
shows that

$\displaystyle W^{F}(q_{0},p_{0}){\ =\ }\lim_{t\to\infty}U_{-t}^{0}\circ
U_{t}^{F}(q_{0},p_{0})$

exists and

(58) $W^{F}(q_{0},p_{0}){\ =\
}(\beta_{2}^{+},\ldots,\beta_{N-1}^{+},\alpha_{2}^{+},\ldots,\alpha_{N-1}^{+})\
.$

We show first that $\displaystyle W^{F}$ is one-to-one. Assume (58). From
(56), the eigenvalues $\displaystyle\\{\lambda_{n}\\}$ of $\displaystyle
L_{F}(q_{0},p_{0})$ are determined,

$\lambda_{n}{\ =\ }-\alpha_{n}^{+}/2\ ,\quad 2\leq n\leq N-1\ .$

From (57),

(59) $\sum_{n=2}^{N-1}\beta_{n}^{+}{\ =\
}\sum_{j=2}^{N-1}q_{j,0}-\frac{2}{N-2}\sum_{n=2}^{N-1}\sum_{j=2}^{N-1}\ln\left(\frac{u_{n}(2)}{u_{j}(2)}\frac{\Pi_{\ell=2}^{n-1}2(\lambda_{\ell}-\lambda_{n})}{\Pi_{\ell=2}^{j-1}2(\lambda_{\ell}-\lambda_{j})}\right){\
=\ }\sum_{i=2}^{N-1}q_{j,0}\ ,$

as the double sum vanishes by oddness. It follows then from (57) that, for
$\displaystyle 2\leq n\leq N-1$,

(60) $\sum_{j=2}^{N-1}\ln\frac{u_{n}(2)}{u_{j}(2)}{\ =\ }r_{n}\ ,$

where $\displaystyle r_{n}$ is a function of
$\displaystyle\\{\beta_{j}^{+}\\}_{i=2}^{N-1}$ and
$\displaystyle\\{\alpha_{n}^{+}{\ =\ }-2\lambda_{n}\\}$,

(61) $r_{n}{\ =\
}\frac{N-2}{2}\left(\frac{1}{N-2}\sum_{j=2}^{N-1}\beta_{j}^{+}-\beta_{n}^{+}-\frac{2}{N-2}\sum_{j=2}^{N-1}\ln\left(\frac{\Pi_{\ell=2}^{n-1}(\alpha_{n}^{+}-\alpha_{\ell}^{+})}{\Pi_{\ell=2}^{n-1}(\alpha_{j}^{+}-\alpha_{\ell}^{+})}\right)\right).$

Note that

(62) $\sum_{n=2}^{N-1}r_{n}{\ =\ }0\ .$

From (60),

$\displaystyle\prod_{j=2}^{N-1}\left(\frac{u_{n}(2)}{u_{j}(2)}\right){\ =\
}e^{r_{n}}$

and so

$\displaystyle u_{n}(2){\ =\
}\left(\prod_{j=2}^{N-1}u_{j}(2)\right)^{1/{(N-2)}}\ e^{r_{n}/(N-2)}\ .$

Since $\displaystyle\sum_{n=2}^{N-1}u_{n}(2)^{2}{\ =\ }1$, this gives

$\displaystyle\prod_{j=2}^{N-1}u_{j}(2){\ =\ }\left(\sum_{n=2}^{N-1}\
e^{2r_{n}/(N-2)}\right)^{(2-N)/2}\ .$

Thus

$u_{n}(2){\ =\
}\frac{e^{\frac{r_{n}}{N-2}}}{\big{(}\sum_{n=2}^{N-1}e^{\frac{2r_{n}}{N-2}}\big{)}^{1/2}}\
,\quad 2\leq n\leq N-1\ .$

As $\displaystyle L_{F}(q_{0},p_{0})$ is determined by its eigenvalues and the
first components of its normalized eigenvectors, it follows that
$\displaystyle L_{F}(q_{0},p_{0})$ is determined by
$\displaystyle\\{\beta_{i}^{+},\alpha_{j}^{+}\\},2\leq i,j,N-1$. But

$\displaystyle p_{n,0}{\ =\ }-2a_{n},\ \quad 2\leq n\leq N-1$

and

$\displaystyle q_{n,0}-q_{n+1,0}{\ =\ }2\ln 2b_{n}\ ,\quad 2\leq n\leq N-2\ .$

As $\displaystyle\sum_{n=2}^{N-1}q_{n,0}{\ =\ }\sum_{n=2}^{N-1}\beta_{n}^{+}$,
by (59), we thus see that $\displaystyle(q_{0},p_{0})$ is determined by

$\displaystyle(\beta_{2}^{+},\ldots,\beta_{N-1}^{+},\alpha_{2}^{+},\ldots\alpha_{N-1}^{+}){\
=\ }W^{F}(q_{0},p_{0})\ ,$

i.e., $\displaystyle W^{F}$ is one-to-one. We now show that $\displaystyle
W^{F}$ is onto

$X{\ =\ }\\{(x_{2},\ldots,x_{N-1},y_{2},\ldots,y_{N-1})\ :\
y_{2}<\ldots<y_{N-1}\\}\subset{\mathbb{R}}^{2(N-2)}\ .$

Let $\displaystyle r_{n}=r_{n}(x,y)$ in (61) with $\displaystyle\beta_{n}^{+}$
replaced by $\displaystyle x_{n}$ and $\displaystyle\alpha_{n}^{+}$ by
$\displaystyle y_{n},$ $\displaystyle 2\leq n\leq N-1.$ Set

(63) $u_{n}(2){\ =\ }\frac{e^{r_{n}/(N-2)}}{\big{(}\sum_{j=2}^{N-1}\
e^{2r_{j}/(N-2)}\big{)}^{1/2}}>0\ ,\quad 2\leq n\leq N-1\ .$

Then by (62),

$\prod_{n=2}^{N-1}u_{n}(2){\ =\
}\frac{e^{\frac{1}{N-2}\sum_{n=2}^{N-1}r_{n}}}{\big{(}\sum_{j=2}^{N-1}\
e^{2r_{j}/(N-2)}\big{)}^{(N-2)/2}}{\ =\ }\frac{1}{\big{(}\sum_{j=2}^{N-1}\
e^{2r_{j}/(N-2)}\big{)}^{(N-2)/2}}\ .$

From (63),

(64) $\sum_{n=2}^{N-1}u_{n}(2)^{2}{\ =\ }1\quad\hbox{and}\quad u_{n}(2)\geq
0,\,\,2\leq n\leq N-1,$

and using (63), we see that

$\displaystyle u_{n}(2){\ =\
}\big{(}\prod_{j=2}^{N-1}u_{j}(2)\big{)}^{1/(N-2)}\ e^{r_{n}/(N-2)}$

which implies

$\prod_{i=2}^{N-1}\ \left(\frac{u_{n}(2)}{u_{j}(2)}\right){\ =\
}e^{r_{n}(x,y)}$

from which we conclude that

$x_{n}{\ =\
}\frac{1}{N-2}\sum_{j=2}^{N-1}x_{j}-\frac{2}{N-2}\sum_{j=2}^{N-1}\ln\left(\frac{u_{n}(2)}{u_{j}(2)}\frac{\Pi_{\ell=2}^{n-1}(y_{n}-y_{\ell})}{\Pi_{\ell=2}^{n-1}(y_{j}-y_{\ell})}\right)\
.$

Now, as $\displaystyle\Phi$ is a bijection, there exists a unique Toda matrix
$\displaystyle L_{F}$ (see (43)) with spectrum
$\displaystyle\lambda_{2}=-y_{2}/2>\lambda_{3}=-y_{3}/2>\ldots>\lambda_{N-1}=-y_{N-1}/2$
and first components of the eigenvectors $\displaystyle
u_{2}(2),\ldots,u_{N-1}(2)$.

Set

$\begin{array}[]{ll}p_{n,0}{\ =\ }-2a_{n}\ ,\quad 2\leq n\leq N-1,\\\
q_{n,0}-q_{n+1,0}{\ =\ }2\ln(2b_{n})\ ,\quad 2\leq n\leq N-1\ .\end{array}$

Then determine the $\displaystyle q_{n,0}$’s uniquely by requiring

$\sum_{n=2}^{N-1}q_{n,0}{\ =\ }\sum_{n=2}^{N-1}x_{n}\ .$

It then follows from the above calculations that

$\displaystyle W^{F}(q_{0},p_{0}){\ =\
}(x_{2},\ldots,x_{N-1},y_{2},\ldots,y_{N-1})$

which completes the proof that $\displaystyle W^{F}$ is a bijection from
$\displaystyle{\mathbb{R}}^{2(N-2)}$ to $\displaystyle X$. Finally, we
conclude from (48) that $\displaystyle\hat{W}^{\\#}$ is a bijection from
$\displaystyle{\mathbb{R}}^{2N}$ to

$\displaystyle\hat{X}^{\\#}{\ =\
}\\{(x_{1},x_{2},\ldots,x_{N-1},x_{N},y_{1},y_{2},\ldots,y_{N-1},y_{N})\ :\
y_{2}<y_{3}<\ldots<y_{N-1}\\}\ .$

In order to derive (51) with $\displaystyle W$ as in (53), we need to verify
certain domain issues. For $\displaystyle(x,y)\in\hat{X}^{\\#}$, we have from
(52)

$\displaystyle\hat{W}^{\\#}\circ\hat{U}_{t}\circ(\hat{W}^{\\#})^{-1}(x,y){\ =\
}U_{t}^{\\#}\circ\hat{W}^{\\#}\circ(\hat{W}^{\\#})^{-1}(x,y){\ =\
}U_{t}^{\\#}(x,y)$

from which we see necessarily that $\displaystyle
U_{t}^{\\#}(x,y)\in\hat{X}^{\\#}$, a fact that can be seen directly from (47).
Hence

$\displaystyle\hat{U}_{t}\circ(\hat{W}^{\\#})^{-1}{\ =\
}(\hat{W}^{\\#})^{-1}\circ U_{t}^{\\#}\ \ \hbox{on}\ \ \hat{X}^{\\#}\ .$

But it follows from (39) that for any
$\displaystyle(x,y)\in{\mathbb{R}}^{2N}$, $\displaystyle
W^{\\#}(x,y)\in\hat{X}^{\\#}$, and so

$\displaystyle\hat{U}_{t}\circ(\hat{W}^{\\#})^{-1}\circ W^{\\#}(x,y){\ =\
}(\hat{W}^{\\#})^{-1}\circ U_{t}^{\\#}\circ W^{\\#}{\ =\
}(\hat{W}^{\\#})^{-1}\circ W^{\\#}\circ U_{t}(x,y)$

which verifies, indeed, that $\displaystyle W{\ =\ }(\hat{W}^{\\#})^{-1}\circ
W^{\\#}$ mapping $\displaystyle{\mathbb{R}}^{2N}$ to itself intertwines
$\displaystyle U_{t}$, the propagator for the equations generated by
$\displaystyle H_{c}$, and $\displaystyle\hat{U}_{t}$, the propagator for the
equations generated by the completely integrable Toda-core Hamiltonian
$\displaystyle H_{c}^{d}$.

As noted earlier, although the intertwining relation
$\displaystyle\hat{U}_{t}\circ W=W\circ U_{t}$ is enough to prove
integrability, it is not sufficient to prove that solutions generated by
$\displaystyle H_{c}$ behave asymptotically like solutions generated by the
decoupled Toda core Hamiltonian $\displaystyle H^{d}_{c}$. In the quantum
mechanical case , the fact that $\displaystyle e^{iAt}$ is linear and unitary
implies that $\displaystyle||e^{-iAt}e^{iBt}f-Wf||=||e^{iBt}f-e^{iAt}Wf|$, and
so the convergence of the wave operator $\displaystyle W$ is equivalent to
showing that a solution $\displaystyle e^{iBt}f$ generated by the operator B,
behaves like a solution $\displaystyle e^{iAt}g$ generated by the operator A,
where $\displaystyle g:=Wf$. In the case at hand, as
$\displaystyle\hat{U}^{\\#}_{t}$, is neither linear nor bounded, we cannot, in
particular, immediately infer from the convergence $\displaystyle
U^{\\#}_{-t}\circ\hat{U}_{t}(q_{0},p_{0})-\hat{W}^{\\#}(q_{0},p_{0})\rightarrow
0$, that
$\displaystyle\hat{U}_{t}(q_{0},p_{0})-U^{\\#}_{t}\hat{W}^{\\#}(q_{0},p_{0})\rightarrow
0$, as desired. However, as we see from the calculations following (47), as
$\displaystyle t\to\infty$,

$\displaystyle
U^{\\#}_{-t}\circ\hat{U}_{t}(q_{0},p_{0})=\hat{W}^{\\#}(q_{0},p_{0})+O(e^{-\gamma
t})$

it then follows from the explicit form, and polynomial growth, of
$\displaystyle U^{\\#}_{t}$ that

$\displaystyle\hat{U}_{t}(q_{0},p_{0})=U^{\\#}_{t}(\hat{W}^{\\#}(q_{0},p_{0})+O(e^{-\gamma
t}))=(U^{\\#}_{t}\hat{W}^{\\#}(q_{0},p_{0}))+O(e^{-\gamma t/2}).$

A similar argument shows that

$\displaystyle
U_{t}(q_{0},p_{0})=U^{\\#}_{t}(W^{\\#}(q_{0},p_{0})+O(e^{-\gamma
t}))=(U^{\\#}_{t}W^{\\#}(q_{0},p_{0}))+O(e^{-\gamma t/2}).$

Now as $\displaystyle W^{\\#}(q_{0},p_{0})\in\hat{X}^{\\#}$, and as
$\displaystyle\hat{W}^{\\#}$ is a bijection onto $\displaystyle\hat{X}^{\\#}$,
it follows that there exist $\displaystyle(\hat{q}_{0},\hat{p}_{0})$ such that
$\displaystyle\hat{W}^{\\#}(\hat{q}_{0},\hat{p}_{0})=W^{\\#}(q_{0},p_{0}).$
Substitution into the above two relations shows that

$\displaystyle
U_{t}(q_{0},p_{0})=\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0})+O(e^{-\gamma t/2})$

where $\displaystyle(\hat{q}_{0},\hat{p}_{0})=W(q_{0},p_{0})$ as desired.

Finally, as noted before, we do not construct $\displaystyle W$ directly as a
wave operator $\displaystyle\lim_{t\to\infty}\hat{U}_{-t}\circ
U_{t}(q_{0},p_{0}).$ The reason for this is the following. In evaluating

$\displaystyle\hat{U}_{-t}\circ U_{t}(q_{0},p_{0})=\hat{U}_{-t}(q(t),p(t))$

we are facing a double scaling limit. The asymptotics of
$\displaystyle\hat{U}_{-t}(\hat{q}_{0},\hat{p}_{0})$ as $\displaystyle
t\to\infty$ is known for $\displaystyle(\hat{q}_{0},\hat{p}_{0})$ fixed, or in
a compact set, but $\displaystyle q(t)$, in particular, grows linearly. This
considerably complicates the analysis. The difficulty is avoided when we
evaluate

$\displaystyle\hat{U}^{\\#}_{-t}\circ
U_{t}(q_{0},p_{0})=\hat{U}^{\\#}_{-t}(q(t),p(t))$

as we have an explicit, and simple, formula for
$\displaystyle\hat{U}^{\\#}_{-t}(\hat{q}^{\\#},\hat{p}^{\\#})$ for all
$\displaystyle(\hat{q}^{\\#},\hat{p}^{\\#})$, and so the double scaling limit
is avoided. To avoid the problem of the double scaling limit in evaluating
$\displaystyle\hat{U}_{-t}(q(t),p(t))$, we need to use an explicit formula for
$\displaystyle\hat{U}_{-t}(\hat{q}_{0},\hat{p}_{0})$ for all
$\displaystyle(\hat{q}_{0},\hat{p}_{0})$. This is most conveniently done by
mapping the solution
$\displaystyle(q_{1}(t),...,q_{N}(t),p_{1}(t),...,p_{N}(t))$ onto the
eigenvalues $\displaystyle(\lambda_{2}(t),...,\lambda_{N-1}(t))$ and first
components of the associated normalized eigenvectors
$\displaystyle(u_{2}(2)(t),...,u_{N-1}(2)(t))$ of the core Toda matrix
$\displaystyle L_{F}=L_{F}(t)$ in (43) . In [22], Moser used the Lax-Pair form
(9) to show that under the Toda flow

$\displaystyle\hat{\lambda}_{i}(t)=\hat{\lambda}_{i,0},\ i=2,...,N-1$

and

$\displaystyle\hat{u}_{i}(2)(t)=\frac{\hat{u}_{i,0}(2)e^{\hat{\lambda}_{i,0}t}}{(\sum_{j=2}^{N-1}(\hat{u}_{j,0}(2))^{2}e^{2\hat{\lambda}_{j,0}t)})^{1/2}},\
i=2,...,N-1$

for any initial conditions
$\displaystyle\hat{\lambda}_{i,0}$,$\displaystyle\hat{u}_{j,0}(2)$. The
evolution of $\displaystyle L_{F}(t)$ is almost the same as in (9), except
there is an additional diagonal driving term

$\displaystyle\dot{L}_{F}{\ =\
}[L_{F},B_{F}]+\operatorname{diag}(-2b_{1}^{2},0,0,...,0,2b_{N-1}^{2})$

where $\displaystyle b_{1}=\frac{1}{2}e^{(q_{1}(t)-q_{2}(t))/2}$ and
$\displaystyle b_{N-1}=\frac{1}{2}e^{(q_{N-1}(t)-q_{N}(t))/2}.$ Now Moser’s
method to obtain the above formulae for
($\displaystyle\hat{\lambda}_{i}(t)$,$\displaystyle\hat{u}_{j}(2)(t)$), can be
extended, using the driven Lax-Pair for $\displaystyle L_{F}(t)$, to obtain
the asymptotics of $\displaystyle\lambda_{i}(t)$ and $\displaystyle
u_{j}(2)(t)$. Then when we evaluate

$\displaystyle\hat{U}_{-t}\circ U_{t}(q_{0},p_{0})=\hat{U}_{-t}(q(t),p(t))$

now in the ($\displaystyle\lambda,u(2)$) variables, rather than the original
$\displaystyle(q,p)$ variables, we can use Moser’s explicit formulae for
($\displaystyle\hat{\lambda}_{i}(t)$,$\displaystyle\hat{u}_{j}(2)(t)$), and
the double scaling limit is avoided. Using the bijection between
$\displaystyle L_{F}$ and the ($\displaystyle\lambda,u(2)$) variables, we can
then assert, after some algebra, the existence of

$\displaystyle\lim_{t\to\infty}\hat{U}_{-t}\circ U_{t}(q_{0},p_{0})$

directly in the original $\displaystyle(q,p)$ variables. We leave the details
to the interested reader.

Finally we note that the core Toda flow can also be solved explicitly for any
time $\displaystyle t$ in the bidiagonal formalism of [19], so that in
evaluating

$\displaystyle\lim_{t\to\infty}\hat{U}_{-t}\circ U_{t}(q_{0},p_{0})$

we could just as easily have worked in bidiagonal coordinates.

Step 5.

We assert that $\displaystyle W=(\hat{W}^{\\#})^{-1}\circ W^{\\#}$ is
$\displaystyle C^{1}({\mathbb{R}}^{2N})$. Indeed, $\displaystyle\hat{W}^{\\#}$
is a diffeomorphism from $\displaystyle{\mathbb{R}}^{2N}$ onto
$\displaystyle\hat{X}^{\\#}$, as can be seen directly from (56) and (57) and
the fact that $\displaystyle\Phi$ is a diffeomorphism, coupled with the fact
that

$\displaystyle(q_{0},p_{0})\mapsto\left(a_{j}{\ =\ }-p_{j,0}/2,b_{j}{\ =\
}\frac{1}{2}e^{(q_{j,0}-q_{j+1,0})/2},2\leq j\leq
N-1,\sum_{i=2}^{N-1}q_{i,0}\right),$

is a diffeomorphism. At the analytical level, the fact that
$\displaystyle\hat{W}^{\\#}{\ =\
}\lim_{t\to\infty}U_{-t}^{\\#}\circ\hat{U}_{t}$ is $\displaystyle C^{1}$
follows, alternatively, from the fact that $\displaystyle(q_{0}(t),p_{0}(t)){\
=\ }U_{-t}^{\\#}\circ\hat{U}_{t}(q_{0},p_{0})$ is a $\displaystyle C^{1}$
function of $\displaystyle(q_{0},p_{0})$ for any finite $\displaystyle t$ by
standard ODE methods, and then noting from (7) that the derivative of
$\displaystyle(q_{0}(t),p_{0}(t))$ with respect to
$\displaystyle(q_{0},p_{0})$ is a linear system with exponentially decaying
coefficients. But the same is true for (16), and so $\displaystyle W^{\\#}{\
=\ }\lim_{t\to\infty}U_{-t}^{\\#}\circ U_{t}$ is also $\displaystyle C^{1}$,
and hence $\displaystyle W=(\hat{W}^{\\#})^{-1}\circ W^{\\#}$ is
$\displaystyle C^{1}$. We leave the details to the interested reader.

For the core Toda system
$\displaystyle(q,p)=(q_{2},\ldots,q_{N-1},p_{2},\ldots,p_{N-2})$, let
$\displaystyle\lambda_{2}>\ldots>\lambda_{N-1}$ be the eigenvalues and
$\displaystyle u_{2}(2),\ldots,u_{N-1}(2)$ be the first components of the
normalized eigenvectors of the associated Toda matrix $\displaystyle
L_{F}(q,p)$. Let $\displaystyle Z(\lambda){\ =\ }\det(L_{F}(q,p)-\lambda)$.
Then, as shown in [5], and also, more directly in [4] (action-angle variables
for the Toda flow are also given in [17]),

$\begin{array}[]{llll}\theta_{k}{\ =\ }\ln\left(\frac{u_{k}(2)}{u_{N-1}(2)}\
\Big{|}\frac{Z^{\prime}(\lambda_{k})}{Z^{\prime}(\lambda_{N-1})}\Big{|}^{1/2}\right)\
,\quad 2\leq k\leq N-2\\\ \theta_{N-1}{\ =\
}\frac{1}{N-2}(q_{2}+\ldots+q_{N-1}),\\\ \tilde{\lambda}_{k}{\ =\
}\lambda_{k}-\frac{1}{N-2}\sum_{j=2}^{N-1}\lambda_{j}\ ,\quad 2\leq k\leq
N-2,\\\ \tilde{\lambda}_{N-1}{\ =\ }p_{2}+\ldots+p_{N-1}\end{array}$

are action-angle variables for the core Toda flow,

(65) $\\{\theta_{i},\theta_{j}\\}{\ =\ }0\
,\\{\tilde{\lambda}_{i},\tilde{\lambda}_{j}\\}{\ =\
}0,\\{\theta_{i},\tilde{\lambda}_{j}\\}=\delta_{ij}\ ,2\leq i,j\leq N-1\ .$

On the other hand, for additional variables $\displaystyle
q_{1},p_{1},q_{N},p_{N}$, if

$\begin{array}[]{ll}\theta_{1}{\ =\ }-\frac{1}{2}p_{1}\
,\quad\theta_{N}=\frac{1}{c}p_{N},\\\ \tilde{\lambda}_{1}{\ =\
}\frac{1}{2}p^{2}+cq_{1}\ ,\quad\tilde{\lambda}_{N}{\ =\
}\frac{1}{2}p_{N}^{2}-cq_{N},\end{array}$

it follows that (65) holds for $\displaystyle 1\leq i,j\leq N$. Also from
(46),

(66) $H_{c}^{d}{\ =\
}\tilde{\lambda}_{1}+\tilde{\lambda}_{n}+2\sum_{i=2}^{N-1}\lambda_{k}^{2}\ .$

Now, as
$\displaystyle\tilde{\lambda}_{N-1}=-2(a_{2}+\ldots+a_{N-1})=-2\operatorname{tr}L_{F}{\
=\ }-2(\lambda_{2}+\ldots+\lambda_{N-1})$, we see that
$\displaystyle\lambda_{k}{\ =\
}\tilde{\lambda}_{k}-\frac{1}{2(N-2)}\tilde{\lambda}_{N-1},2\leq k\leq N-2$,
and then solving for $\displaystyle\lambda_{N-1}$, we find

$\displaystyle\lambda_{N-1}{\ =\
}-\frac{1}{2(N-2)}\tilde{\lambda}_{N-1}-\sum_{j=2}^{N-2}\tilde{\lambda}_{i}\
.$

Substitution into (66) gives

$\displaystyle H_{c}^{d}{\ =\
}\tilde{\lambda}_{1}+\tilde{\lambda}_{N}+2\sum_{k=2}^{N-2}(\tilde{\lambda}_{k})^{2}+2\left(\sum_{k=2}^{N-2}\tilde{\lambda}_{k}\right)^{2}+\frac{1}{2(N-2)^{2}}\tilde{\lambda}_{N-1}^{2}$

for which

(67) $\begin{array}[]{lll}\\{\theta_{i},H_{c}^{d}\\}{\ =\ }1\ ,\quad i=1\
\hbox{or}\ N\\\ \\{\theta_{i},H_{c}^{d}\\}{\ =\
}4\left(\tilde{\lambda}_{i}+\sum_{j=2}^{N-2}\tilde{\lambda}_{j}\right)\ ,\quad
2\leq i\leq N-2,\\\ \\{\theta_{N-1},H_{c}^{d}\\}{\ =\
}\frac{1}{(N-2)^{2}}\tilde{\lambda}_{N-1}\end{array}$

so that the $\displaystyle\theta_{i}$’s move linearly under $\displaystyle
H_{c}^{d}$. Also clearly,

(68) $\\{\tilde{\lambda}_{i},H_{c}^{d}\\}{\ =\ }0\ ,\quad 1\leq i\leq N\ .$

Now for the flow $\displaystyle\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0}){\ =\
}(\hat{q}(t),\hat{p}(t))$ ,

$\displaystyle\frac{d}{dt}f(\hat{q}(t),\hat{p}(t)){\ =\
}\\{f,H_{c}^{q}\\}(\hat{q}(t),\hat{p}(t))$

for any $\displaystyle f:{\mathbb{R}}^{2N}\to{\mathbb{R}}$ and it follows from
(67) and (68) that
$\displaystyle\\{\theta_{i}\\}_{i=1}^{N},\\{\tilde{\lambda}_{i}\\}_{i=1}^{N}$
are action-angle variables for $\displaystyle H_{c}^{d}$.

Set

$\Theta_{i}{\ =\ }\theta_{i}\circ W\ ,\quad\Lambda_{i}{\ =\
}\tilde{\lambda}_{i}\circ W\ ,\quad 1\leq i\leq N.\\\ $

Then as in (14), $\displaystyle\Theta_{i}$ and $\displaystyle\Lambda_{i}$ are
canonically conjugate. Furthermore, under the flow $\displaystyle U_{t}$
generated by $\displaystyle H_{c}$, for any $\displaystyle
f:{\mathbb{R}}^{2N}\to{\mathbb{R}}$ and $\displaystyle F{\ =\ }f\circ W$,

(69) $\displaystyle\frac{d}{dt}F(U_{t}(q_{0},p_{0})){\ =}$
$\displaystyle\frac{d}{dt}(f\circ W\circ U_{t}(q_{0},p_{0}))$ $\displaystyle=$
$\displaystyle\frac{d}{dt}(f\circ\hat{U}_{t}(\hat{q}_{0},\hat{p}_{0}))\ ,\
(\hat{q}_{0},\hat{p}_{0}){\ =\ }W(q_{0},p_{0})$ $\displaystyle=$
$\displaystyle\frac{d}{dt}f(\hat{q}(t),\hat{p}(t))\ .$

Hence under $\displaystyle U_{t}$,

(70) $\displaystyle\frac{d}{dt}\Theta_{i}(q_{0},p_{0}){\ =\
}\frac{d}{dt}\theta_{i}(W(q_{0},p_{0})){\ =\ }1\ ,\quad i=1,N,$
$\displaystyle\frac{d}{dt}\Theta_{i}(q_{0},p_{0}){\ =\
}\frac{d}{dt}\theta_{i}(W(q_{0},p_{0})){\ =\
}4\left(\Lambda_{i}(q_{0},p_{0})+\sum_{j=2}^{N-2}\Lambda_{j}(q_{0},p_{0})\right),\,\,\quad
2\leq i\leq N-2,$ $\displaystyle\frac{d}{dt}\Theta_{N-1}(q_{0},p_{0}){\ =\
}\frac{1}{(N-2)^{2}}\Lambda_{N-1}(q_{0},p_{0})\ .$

Also,

$\frac{d}{dt}\Lambda_{i}(q_{0},p_{0}){\ =\ }0\ ,\quad 1\leq i\leq N\ .$

Thus $\displaystyle\\{\Theta_{i}\\}_{i=1}^{N}$,
$\displaystyle\\{\Lambda_{i}\\}_{i=1}^{N}$ are action-angle variables for
$\displaystyle H_{c}$.

Note that by a general and simple argument, the relations
$\displaystyle\\{\Theta_{i},\Lambda_{j}\\}{\ =\ }\delta_{ij}$ imply that
$\displaystyle\Lambda_{1},\ldots,\Lambda_{n}$ are functionally independent,
which is equivalent to the statement that their gradients are linearly
independent on an open, dense set. Thus $\displaystyle H_{c}$ is integrable in
the sense of Liouville.

Finally note that every Hamiltonian of the form

$\displaystyle H{\ =\ }\frac{1}{2}p^{2}+V(q)\ \ (q,p)\in{\mathbb{R}}^{2}$

generates a flow that can be expressed in Lax-pair form. Indeed, for

(71) $\hat{L}{\ =\ }\begin{pmatrix}p&2V(q)\\\ 1&-p\end{pmatrix}\
,\quad\hat{B}{\ =\ }\begin{pmatrix}0&V^{\prime}(q)\\\ 0&0\end{pmatrix},$

a simple computation shows that

(72) $\dot{q}{\ =\ }p\ ,\quad\dot{p}{\ =\
}-V^{\prime}(q)\quad\Leftrightarrow\quad\frac{d}{dt}\hat{L}{\ =\
}[\hat{L},\hat{B}]\ .$

Let $\displaystyle(\hat{q}_{k}(t),\hat{p}_{k}(t),1\leq k\leq N$, solve the
flow generated by $\displaystyle H_{c}^{d}$ as above. Set

$Q_{k}{\ =\ }\hat{q}_{k}\circ W\ ,\quad P_{k}{\ =\ }\hat{p}_{k}\circ W\ ,\quad
1\leq k\leq N\ .$

Then, for $\displaystyle 2\leq k\leq N-1$,

$\displaystyle\frac{d}{dt}Q_{k}{\ =\ }\hat{p}_{k}\circ W{\ =\ }P_{k},$
$\displaystyle\frac{d}{dt}P_{k}{\ =\
}\big{(}e^{\hat{q}_{k-1}-\hat{q}_{k}}-e^{\hat{q}_{k}-\hat{q}_{k+1}}\big{)}\circ
W$ $\displaystyle{\ =\ }e^{Q_{k-1}-Q_{k}}-e^{Q_{k}-Q_{k+1}},\,$

where $\displaystyle e^{Q_{1}-Q_{2}}=e^{Q_{N-1}-Q_{n}}=0$. Also

$\displaystyle\frac{d}{dt}Q_{1}{\ =\ }P_{1}\ ,\quad\frac{d}{dt}P_{1}{\ =\ }-c$
$\displaystyle\frac{d}{dt}Q_{N}{\ =\ }P_{N}\ ,\quad\frac{d}{dt}P_{N}{\ =\ }c\
.$

Finally, set

${\mathcal{L}}{\ =\ }\begin{pmatrix}A_{1}&B_{2}&&&&&&\\\
B_{2}&\ddots&\ddots&&&0&&\\\ &\ddots&&B_{N-2}&&&&\\\ &&B_{N-2}&A_{N-1}&0&&&\\\
&&&0&P_{1}&2cQ_{1}&&\\\ &&&&1&-P_{1}&0&0\\\ &&0&&&&P_{N}&-2cQ_{N}\\\
&&&&&&1&-P_{N}\\\ \end{pmatrix},$

and

${\mathcal{B}}{\ =\ }\begin{pmatrix}0&-B_{2}&&&&&&\\\
B_{2}&\ddots&\ddots&&&0&&\\\ &\ddots&&-B_{N-2}&&&&\\\ &&B_{N-2}&0&0&&&\\\
&&&0&0&c&&\\\ &&&&0&0&0&\\\ &0&&&&&0&-c\\\ &&&&&&0&0\\\ \end{pmatrix},$

where

$\begin{array}[]{ll}A_{k}{\ =\ }-\frac{1}{2}P_{k}\ ,\quad 2\leq k\leq N-1,\\\
B_{k}{\ =\ }\frac{1}{2}e^{(Q_{k}-Q_{k+1})/2}\ ,\quad 2\leq k\leq N-2\
.\end{array}$

Then

$\displaystyle(q_{k}(t),p_{k}(t))_{k=1}^{N}$ solve (16)

$\displaystyle\Leftrightarrow$ $\displaystyle\frac{d}{dt}{\mathcal{L}}{\ =\
}[{\mathcal{L}},{\mathcal{B}}]\ .$

Thus the Hamilton equations for $\displaystyle H_{c}$ have a Lax-pair form.

Remark: Instead of using (51), we could use (52), $\displaystyle
U_{t}^{\\#}\circ W^{\\#}{\ =\ }W^{\\#}\circ U_{t}$ to display (16) as a Lax-
pair in another form. But now the analog $\displaystyle{\mathcal{L}}^{\\#}$
and $\displaystyle{\mathcal{B}}^{\\#}$ of $\displaystyle{\mathcal{L}}$ and
$\displaystyle{\mathcal{B}}$ convey little information,

$\displaystyle{\mathcal{L}}^{\\#}{\ =\ }\begin{pmatrix}P_{1}&0&&&&&&\\\
0&P_{2}&\ddots&&&0&&\\\ &\ddots&\ddots&0&&&&\\\ &&0&P_{N-1}&0&&&\\\
&&&0&P_{1}&2cQ_{1}&&\\\ &0&&&1&-P_{1}&0&\\\ &&&&&&P_{N}&-2cQ_{n}\\\
&&&&&&1&-P_{N}\\\ \end{pmatrix}$

and

$\displaystyle{\mathcal{B}}^{\\#}{\ =\ }\begin{pmatrix}0&0&&&&&&\\\
0&\ddots&\ddots&&&0&&\\\ &\ddots&&0&&&&\\\ &&0&0&0&&&\\\ &&&0&0&c&&\\\
&&&&0&0&&\\\ &0&&&&&0&-c\\\ &&&&&&0&0\\\ \end{pmatrix}$

where

(73) $Q_{k}{\ =\ }q_{k}^{\\#}\circ W^{\\#}\ ,\quad P_{k}{\ =\
}p_{k}^{\\#}\circ W^{\\#}\ ,\quad 1\leq k\leq N\ .$

Here
$\displaystyle(q^{\\#}(t),p^{\\#}(t))=U^{\\#}_{t}(q^{\\#}_{0},p^{\\#}_{0})$.

Acknowledgments. Luen-Chau Li acknowledges the support from the Simons
Foundation through grant #585813. Tom Trogdon acknowledges the support of NSF
grant DMS-1945652 and Carlos Tomei from CNPq and FAPERJ, Brazil. The authors
would also like to thank Barry Simon for some helpful information.

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