KdV Flow I: Reflectionless Case

In this chapter we rewrite Sato’s theory for \(\nu =2\) in terms of the spectral quantities of the underlying Schrödinger operator L( \(=L_{2}\) ). One of the features in this case is the self-adjointness of L for which the spectral property is well investigated. Moreover, Lax observation teaches us the unitary equivalence between two Schrödinger operators with potentials of initial data and the solution. Therefore it is natural to ask if it is possible to state \( Gr ^{\left( 2\right) }\) completely by spectral terminologies of Schrödinger operators.