On the structure and properties of the singularity manifold equations of the KP hierarchy

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Abstract
This paper is concerned with the Painlevé expansion and the singularity manifold (SM) equation of the Kadomtsev–Petviashvili (KP) equation. Several aspects of the interrelation between the SM equation and the KP auxiliary linear system are studied. It is shown that the simultaneous Painlevé expansion for the KP potential u and the KP eigenfunction ψ can be treated as a Bäcklund-gauge transformation. Two methods for the derivation of the SM equation based on this treatment are proposed and their equivalence is proved. The interrelation between the SM equation and the vertical hierarchy of the KP eigenfunction equations is discussed. The explanation of the coincidence of the KP eigenfunction equation of the second level and the KP SM equation is given. Compact forms of the hierarchy of SM equations of the KP hierarchy are presented. The connection between the KP singularity manifold function φ and the KP eigenfunctions ψ and the adjoint KP eigenfunctions ψ* is derived. The bilinear-bilocal description of the hierarchy the KP SM equations is given within the framework of Sato’s τ-function theory.