Lecture 0 showed three Toda particles leaving a collision with the momenta they came in with, rearranged, and promised the reason. So the question is

The answer: the set of momenta, which are the eigenvalues of the Lax matrix (Theorem 12.2), and the scattering map is a product of two-body maps, one per pair; for three particles the two ways of composing them agree (Figure 12.6). The worked example is two particles, in closed form (Proposition 12.1).

Lecture 0 told the story of the Fermi–Pasta–Ulam–Tsingou chain, whose spring energy (0.2) is weakly nonlinear and whose energy came back to the lowest mode instead of spreading (a reading at the end recalls it).

In 1967 Toda looked for a spring law for which the chain has exact periodic nonlinear waves, trying elliptic functions as the nonlinear analogue of sines, and found the exponential (Toda, §2.1). With the constants set to one and $r_n=q_{n+1}-q_n$, $$V_{\mathrm{Toda}}(r)=e^{-r}+r-1 =\frac{r^{2}}2-\frac{r^{3}}6+\frac{r^{4}}{24}-\cdots ,$$(12.1) which to cubic order is the FPUT spring (0.2), with $\alpha=-\tfrac12$ (Figure 12.1). Compressed ($r<0$) it is a steep exponential wall; stretched, it softens to the constant force of the linear term.

For a ring of springs the linear terms add up to $\sum_nr_n=$ the circumference, a constant, and can be dropped; the result is the Toda ring of Lecture 9, whose Poincaré section is Figure 7.5. For an open chain nothing holds the ends, and we drop the linear terms by fiat.¹ What is left is the open chain of Lecture 8, $$H=\frac12\sum_{n=1}^{N}p_n^{2}+\sum_{n=1}^{N-1}e^{q_n-q_{n+1}},$$(12.2) $N$ unit masses on a line, each repelled by its neighbours (Figure 12.2). The repulsion is felt only when particle $n$ comes close to, or overtakes, particle $n+1$. So the open chain has no FPUT oscillations: its particles fly apart. It is a nonlinear many-body system simple enough to take apart completely.

It is Lecture 9’s Toda ring with its closing spring removed, and its Lax matrix is the tridiagonal one of Proposition 8.3.

Take $N=2$: $H=\tfrac12(p_1^{2}+p_2^{2})+e^{q_1-q_2}$. The mean momentum $\bar p=\tfrac12(p_1+p_2)$ is conserved, and the centre of mass $\tfrac12(q_1+q_2)$ moves at the constant speed $\bar p$. The relative coordinate $x=q_1-q_2$ obeys $$\dot x=p_1-p_2,\qquad \ddot x=\dot p_1-\dot p_2=-2e^{x},$$ and since $\tfrac12(p_1^{2}+p_2^{2})=\bar p^{2}+\tfrac14(p_1-p_2)^{2}$, the energy of the relative motion is $$H-\bar p^{2}=\frac{\dot x^{2}}4+e^{x}=:\frac{v^{2}}4,\qquad v>0 .$$(12.3) It is positive because $e^{x}$ is, and it fixes $v$. Particle $1$ starts behind particle $2$ ($x\to-\infty$, where the potential is negligible) and approaches at relative speed $v$; the wall $e^{x}$ stops it at $e^{x}=v^{2}/4$ and sends it back.

Read Figure 12.3 as a scattering experiment. In the far past there are two free particles with momenta $\lambda_2>\lambda_1$, the faster one behind; in the far future, two free particles with the same two momenta, the faster one in front. Following the momentum rather than the particle, it is as if they had passed through each other, except that each straight line is displaced: the fast one by $-2\ln v$, a time delay $2\ln v/\lambda_2$ when $\lambda_2>0$. This displacement is the classical counterpart of the two-body phase shift of the delta gas in Lecture 11. For $v<1$ its sign reverses: slow particles turn round while still far apart, and the momentum is handed forward across the gap.

For $N$ particles the general solution has determinant formulas (Moser; Babelon–Bernard–Talon, §4.8), which we omit: the asymptotics are all we need.

Given the separation, (12.5) is one line. If $q_{n+1}-q_n\to\infty$ then every $a_n\to0$ and $L(t)\to\operatorname{diag}(p^{\pm})$. Eigenvalues depend continuously on the matrix and do not change in time, so $\{p_n^{\pm}\}$ is the spectrum $\{\lambda_k\}$. The separation fixes the order: $q_{n+1}-q_n\to+\infty$ as $t\to+\infty$ needs $p_{n+1}^{+}\ge p_n^{+}$, so the outgoing momenta increase along the chain, and as $t\to-\infty$ they decrease. This is what the eigenvalues of $L$ mean, which is what Figure 8.3 was hinting at: they are the momenta the particles have when they are free. The separation and (12.6) are Moser’s (1975, pp. 480–481, in his normalisation; Toda, §3.10, gives (12.5) with $L$ halved); scripts/check-l9-algebra.py integrates $N=3,4$ in ours and finds both to within $10^{-8}$.

Two words describe this (Figure 12.4). The scattering is elastic: the set of outgoing momenta equals the set of incoming ones, so each particle’s kinetic energy is handed on intact rather than shared. And it is non-diffractive: the outgoing momenta are not merely constrained by energy and momentum, which for $N\ge3$ would allow a continuum of outcomes; they are exactly the incoming ones, permuted. $N$ conserved eigenvalues leave room for nothing else.

Look at (12.6) term by term. The line carrying $\lambda_k$ picks up $+2\ln(\lambda_j-\lambda_k)$ from every faster momentum and $-2\ln(\lambda_k-\lambda_j)$ from every slower one: exactly the two-body displacements of Proposition 12.1, one per pair, added. The total depends on the eigenvalues only. In particular it is the same whichever pair the incoming data bring together first.

Figure 12.5 makes it concrete. With incoming momenta $3,0,-3$ on particles $1,2,3$, started well apart, the run looks like three pairwise exchanges of momentum, which reverse the order, in the order $(12),(23),(12)$ or $(23),(12),(23)$. The forces never switch off, so the exchanges are only nearly separate; the net result, (12.6), is exact. The two runs start from different incoming lines and look different throughout, but each outgoing line is displaced from its own incoming line by the same amount in both. Written as permutations of the three momentum lines, $s_1s_2s_1=s_2s_1s_2$ (Figure 12.6), and the displacements respect it.

This is the concrete content of the phrase factorised scattering: the $N$-body scattering map is a product of two-body maps, one per pair, taken along a route of crossings of adjacent lines from the incoming order of the lines to the outgoing one, and every such route gives the same map. The general theorem, that the $N$-body scattering map of the open chain factorises this way, is Moser’s cited. It is a statement about the asymptotic motion: the particles scatter as if they met in pairs, although for $N\ge3$ more than two of them can interact at once. What we have seen is the two-body map in closed form and the three-body composition checked. Its quantum form is Lecture 11’s and Lecture 13’s: the two-body S-matrix has a phase whose derivative gives the packet displacement, and the braid relation becomes the Yang–Baxter equation.

• The meaning of the Lax eigenvalues: the momenta the particles carry when they are free, which is what Lecture 0’s collision figure showed.

• Scattering without diffraction, the second face of integrability: $N$-body scattering only permutes the momenta and shifts each line by a sum of two-body shifts.

• The classical shadow of Part II’s key structures: the two-body shift against Lecture 11’s phase, and the braid relation, which with matrices at the crossings is Lecture 13’s Yang–Baxter equation.

• Status. Proposition 12.1 is proved. Moser’s theorem, the separation of the particles and the $N$-body displacements (12.6), is cited; given the separation, the identification of the outgoing momenta with the eigenvalues of $L$ is one line. That the $N$-body scattering map is a product of two-body maps is Moser’s too, a statement about the asymptotic motion, not a claim that the particles meet only in pairs; here the three-body composition is checked numerically.

• A classical counterpart, not a limit, for the delta gas: classical point particles with a contact interaction show no displacement at all (Sutherland §4.2).

• The linear terms of the spring are dropped by fiat on the open chain; the end particles go to infinity, where they would be an infinite constant.

• The open chain’s own integrability is not used here, only its Lax matrix (Lecture 8) and Moser’s theorem. Its proof, and the reason its fibres are not compact, are in Lecture 9’s readings.

In 1955 Fermi, Pasta, Ulam and Tsingou put a chain of masses joined by weakly nonlinear springs on a computer, with spring energy $$V_{\mathrm{FPUT}}(r)=\frac{r^{2}}2+\alpha\frac{r^{3}}3 ,$$(12.7) $r$ the extension of the spring. They started it with all the energy in the lowest normal mode and expected the nonlinearity to share it out among all the modes. It did not: the energy wandered into a few neighbouring modes and came back, almost completely (Figure 1). A system that behaves this regularly is behaving as if it had more integrals than energy.

Lecture 9’s readings prove the open chain Liouville integrable and each connected component of its level sets a cylinder $T^{k}\times\R^{N-k}$ (Corollary 9.24), with the number $k$ of periodic directions left open. The scattering picture settles it.

On a torus the commuting flows have a full lattice of periods, even though a single irrational flow need never return (Figure 5.2). On the open chain’s fibre no combination of the flows ever returns; the proof used the physical flow of $H$, whose late-time configurations are free particles moving apart. In effect $\Sigma\cong\R^{N}$ is coordinatised by the incoming offsets $\beta_n^{-}$ of Theorem 12.2, while the asymptotic momenta are fixed on it (Figure 12.7). They are the integrals, and they are also the obstruction: as long as they are distinct, the particles separate.

Close the chain into a ring and this changes. The whole ring can still drift (the flow of $F_1$ is still a translation), but the particles can no longer separate. Once the total momentum is fixed and the common translation split off, the relative level sets are compact, and each regular connected component is a torus $T^{N-1}$, whose actions are read off a curve. That is Lecture 9.