Integrable nonlinear chain — exact soliton solutions via Lax pairs
ṁ_n = e^(q_{n-1}-q_n) - e^(q_n-q_{n+1})
Toda lattice is exactly integrable — infinite conserved quantities via Lax pair L,M. Solitons pass through each other with only a phase shift. Continuum limit → KdV equation.
Toda Lattice (Morikazu Toda, 1967): The exponential nearest-neighbor potential V(r) = e^{-r} + r − 1 makes the lattice exactly solvable. The Lax representation dL/dt = [M,L] ensures that eigenvalues of L are constants of motion — providing as many conserved quantities as degrees of freedom. N-soliton solutions are known exactly. The Toda lattice interpolates between hard-sphere (α→∞) and harmonic (small amplitude) limits and is a foundational example in the theory of integrable systems. Fermi-Pasta-Ulam recurrences are explained by near-integrability.