The authors consider the \(N\)th order spectral problem of Gel’fand, Dikij, Zakharov and Shabat \[ (\partial^ N+V_{N-1}(x,t)d^{N-1}+...+V_ 1\partial +V_ 0)\psi =\Lambda^ N\psi. \tag{1.1} \] The spectral problem is invariant under the transformation \(\psi (x,t;\lambda)\to \psi'(x,t;\lambda)=g(x,t)\psi (x,t;\lambda),\) where \(g(x,t)\) is any smooth function such that \(\lim_{| x| \to \infty}g(x,t)=1,\) \[ V_ k(x,t)\to V'\!_ k(x,t)=g(x,t)\sum^{N-k}_{n=0}C^ k_{k+n}\cdot V_{k+n}(x,t)\partial^ n(1/g(x,t)). \] These transformations form an infinite dimensional abelian group. There exist \(N-1\) independent invariants \(W_ 0(V_ 0...V_{N-1}),...,W_{N-2}(V_ 0,...,V_{N-1})\) under the action of this group. For example, if \(N=2\), let \(V_{0\infty}=V_{1\infty}=0\). Then \(W_ 0=V_ 0-()\partial V_ 1- (1/4)V^ 2_ 1\) is an invariant and \(V'\!_ 0=(1/2)\partial V_ 1- (1/4)V^ 2_ 1,\) which is the Miura transformation.
The authors rewrite the equation (1.1) in the matrix Frobenius form \(\partial F/\partial x=(A+P(x,t))F,\) where \(F=(\psi,\partial \psi,...,\partial^{N-1}\psi)\). A standard scattering matrix is introduced \(S(\Lambda,t): F^+(x,t,\lambda)=F^-(x,t,\lambda)\cdot S(\Lambda,t).\) It is shown that the scattering matrix is gauge invariant. Using the AKNS technique the authors establish the relations between the evolution equations for the potentials and the evolution equation for the scattering matrix.
A general form of integrable equations is derived and several gauge invariance properties are established. The role played by the so called recursion operator turns out to be central in this development. The authors prove that the general form of the evolution equation derived from (1.1) may be represented as an \(N-1\)-dimensional Hamiltonian system, after deleting the pure gauge degrees of freedom. Some interesting examples conclude this paper.