The inverse scattering of the Zakharov–Shabat system solves the weak noise theory of the Kardar–Parisi–Zhang equation

The paper rigorously formulates the AKNS/ZS inverse-scattering solution of the {P,Q} system, gives the GLM/Fredholm determinant reconstruction, proves the rank-one derivative identity, derives the conserved-charge machinery, and obtains Ψ(z) = −∫(dq/2π) Li2(−z b(q) e−q2) (and its droplet specialization), as well as Φ(Hz) = Ψ(z) − zΨ′(z) = g2∫P2Q2. The candidate solution follows the same backbone (Lax pair, GLM operators, derivative identity, and the Legendre structure) and arrives at the same formulas, but uses an additional spectral-diagonalization claim for the Hankel product to jump directly to zΨ′(z) = ∫ log(1 + z b e−q2). Aside from a minor sign/g-factor slip in the written Wronskian relation, the candidate’s argument matches the paper’s results and logic. Hence both are correct, with slightly different proof routes.