Lax-pair spectral curve: L*psi = lambda*psi defines Riemann surface C_g with dim Omega^1(C_g) = g

Lax-pair spectral-curve framework (Lax 1968 Commun Pure Appl Math 21, 467; Novikov-Manakov-Pitaevskii-Zakharov Theory of Solitons 1984). Setup: an integrable PDE (KdV, sine-Gordon, Toda lattice, NLS) is expressed as d/dt L = [P, L] for a…