A Finite-Dimensional Integrable System Related to the Kadometsev–Petviashvili Equation

Abstract

In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem Lφ=(∂2−v∂−λu)φ=λφx. By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao’s method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived.

Keywords

• nonlinearization of Lax pairs
• Kadometsev–Petviashvili equation
• involutive solution