Abstract
In this paper, taking the Lax pairs of KdV equation as an illustrative example, with the help of a transformation of the spectral problem, another form of spectral function is used to give finite dimensional integrable systems. These integrable systems are generated by two different kinds of polynomial expansion of spectral function on spectral parameter. Further, the root variables of the spectral function are introduced to study the canonical equations of Hamilton. Finally, based on the Hamilton-Jacobi theory, the action-angle variables are built and the soliton solutions of KdV equation are obtained by inversion.
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