"Goldfish'' equations for infinitely many particles

Technical Summary: "Goldfish" Equations for Infinitely Many Particles

Problem Statement
The "goldfish" equations describe a system of $N$ coupled non-linear ordinary differential equations (ODEs) for complex variables $z_j(t)$, originally formulated as:
$$\ddot{z}_j = 2 \dot{z}_j \sum_{k=1, k \neq j}^N \frac{\dot{z}_k}{z_j - z_k}$$
These equations are exactly solvable for finite $N$ by mapping the particle positions to the coefficients of a monic polynomial $p(z)$ and an auxiliary polynomial $q(z)$. The solution is given by the zeroes of $\Phi(z, t) = p(z) + tq(z)$.

This paper addresses the extension of this solvability to the case where $N \to \infty$. The primary challenge lies in the transition from the algebraic theory of polynomials to the analytic theory of entire functions. Specifically, the concepts of "degree" and the unique solvability of interpolation problems (determining $p$ and $q$ from initial conditions) do not have direct analogues for entire functions, creating significant obstacles in defining the dynamics and ensuring the existence of solutions.

Methodology
The author generalizes the finite-$N$ solution method by redefining the normalization of the functions involved. Instead of monic polynomials, the study utilizes entire functions of order $\rho < 1$ normalized such that $p(0)=1$ and $q(0)=0$. The infinite system is defined via the zeroes $z_k(t)$ of the function:
$$\Phi(z, t) = p(z) + tq(z) = \prod_{k=1}^\infty \left(1 - \frac{z}{z_k(t)} \right)$$
The methodology proceeds through the following steps:

1. Regularization: To avoid singular behaviors where zeroes diverge or collide (analogous to polynomials of different degrees), the author imposes a condition $q(z) \ll p(z)$. This relation requires that there exists a sequence of radii $R_j \to \infty$ such that $\lim_{j \to \infty} \max_{|z|=R_j} |q(z)/p(z)| = 0$. This ensures that $p(z)$ dominates $q(z)$ at infinity in a specific sense, preserving the count of zeroes via Rouché's theorem.
2. Approximation by Polynomials: The infinite system is approached as the limit of finite systems. Let $p_N(z)$ and $q_N(z)$ be polynomials constructed from the first $N$ zeroes of $p(z)$ and $q(z)$. The zeroes $z_k^{(N)}(t)$ of the corresponding finite $\Phi_N$ are known to satisfy the finite goldfish equations.
3. Convergence Analysis: Using the uniform convergence of analytic functions on compact sets (Cauchy's theorem), the author demonstrates that $z_k^{(N)}(t)$ converges to $z_k(t)$ and their derivatives converge similarly. This allows the transfer of the differential equation satisfaction from the finite approximations to the infinite limit.
4. Regularity Conditions: The existence and analyticity of $z_k(t)$ with respect to time $t$ are established using the implicit function theorem. This requires that $\Phi(z, t)$ does not possess double zeroes along the path of time evolution. The author argues that double zeroes occur only at isolated points in the complex $t$-plane, allowing for the definition of a contour $C$ where the solution is analytic.

Key Contributions and Results

• Derivation of Infinite Equations: The paper derives the infinite version of the goldfish equations:
  $$-z_j \frac{d^2}{dt^2} [z_j]^{-1} = \frac{\ddot{z}_j}{z_j} - \frac{2\dot{z}_j^2}{z_j^2} = 2 \dot{z}_j \sum_{k=1, k \neq j}^\infty \frac{\dot{z}_k}{z_k(z_j - z_k)}$$
  It is proven that the zeroes of $\Phi(z, t) = p(z) + tq(z)$ satisfy this system, provided $p$ and $q$ are entire functions of order $\rho < 1$ satisfying $q(z) \ll p(z)$.
• Singular Behavior Analysis: The paper highlights that the "degree" of an entire function is not a sufficient substitute for polynomial degree. A counter-example is provided where $p(z) = \cos(\sqrt{z})$ and $q(z) = 1 - \cos(2\sqrt{z})$ (both order $1/2$). Despite having the same order, the zeroes of the sum exhibit singular behavior (divergence of some zeroes as $t \to 0$) because the "type" of $q(z)$ is higher than that of $p(z)$. This motivates the strict $q(z) \ll p(z)$ condition.
• Initial Value Problem Limitations: The paper establishes a mapping from specific initial conditions (derived from $p$ and $q$) to solutions. However, it explicitly states that the converse is not guaranteed. It is not proven that all possible initial conditions for the infinite system can be generated by such entire functions. The interpolation problem of determining $p$ and $q$ from arbitrary infinite sets of $z_k(0)$ and $\dot{z}_k(0)$ remains unsolved and is likely ill-posed for general data.

Significance and Scope
The paper claims to establish a formal connection between the dynamics of an infinite system of interacting particles and the zero sets of specific entire functions. The author notes that while connecting polynomial zeroes to ODE solutions represents genuine progress, extending this to entire functions is "distinctly less impressive" because the null sets of entire functions are less well-understood than those of polynomials.

The work is presented as a partial solution. It successfully defines a class of solutions for the infinite goldfish equations under strict regularity conditions ($q \ll p$) but admits that the full dynamics, including the characterization of the set of all possible initial conditions and the behavior of the system under periodic forcing (analogous to the finite periodic case), remain open problems. The author speculates that the periodic infinite system might exhibit complex or chaotic behavior, a topic left for future investigation. The paper concludes by dedicating these speculations to the memory of Francesco Calogero, who sought systems combining chaos and integrability.