THE SPECTRAL PROBLEM ASSOCIATED WITH THE KADOMTSEV-PETVIASHVILI EQUATION-II

To the Kadomtsev-Petviashvili equation (KPII): q(t) + 6qq(x) + q(xxx) = -3D(-1)q(yy), where D-1 = integral(o)(x) . ds. is associated the liner operator L(q) = D-x(2) - D-y + q. We prove the existence of a discrete complex-valued spectrum of L(q), in a Sobolev space of L(2) (T), where T is the torus [0, 2] x [0, 2] and q is a smooth period one function in x and y. We show that the set of potentials for which simplicity of the spectrum occurs, is dense in the space L(2) (T). The asymptotic behavior of the eigensolutions of L(q) is then discussed. We also show that the eigenvalues of L(q) forms a complete set of functionals along the KPII flow. We then derive a relation between these functionals and the ones found by Fokas in XII. This leads us to prove, in parallel to the work done by Lax on KdV, the compactness of the isospectral set M(qo) in L(o)(2) (T), for a fixed potential q(o) with simple spectrum.