Periodic Soliton Solutions as Imbricate Series of Rational Solitons: Solutions to the Kadomtsev-Petviashvili Equation with Positive Dispersion

The recent development of the nonlinear wave theory clarifies the role of a soliton in various systems. The inverse scattering theory shows that the time-asymptotic state of any initial conditions consists of solitons and ripples under the boundary condition that amplitudes tend to zero as x→±∞. Solitons are stable and the interaction between them affects only phase shifts [1]. Therefore, solitons are regarded as fundamental structures in nonlinear integrable systems. Spatial structures of solitons are usually solitary waves whose amplitudes tend to zero as x→±∞. It is known that soliton equations often allow an exact nonlinear superposition principle [2]-[8]. Let U(x − cst) be a solitary wave solution of a nonlinear equation which is invariant under the group of translations, then the function