A nonconfocal involutive system and constrained flows associated with the MKdV − equationThe special quasiperiodic solution of the (2ϩ1)-dimensional Kadometsev-Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov-Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev-Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.Hence c 1 ϭ¯ϭc NϪ1 ϭ0 since the coefficient determinant is equal to 1 by Eq. ͑4.6͒. Thus c 0 dF 0 ϭ0. And c 0 ϭ0 since dF 0 ϭϪ ͚ ͑ q k dp k ϩ p k dq k ͒ 0. 3955

A generalization of integrable peakon equations with cubic nonlinearity and the Degasperis-Procesi equation with peakon solutions is proposed, which is associated with a 3×3 matrix spectral problem with two potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The generalization is exactly a negative flow in the hierarchy and admits exact solutions with N -peakons and an infinite sequence of conserved quantities. Moreover, a reduction of this hierarchy and its Hamiltonian structures are discussed.

No abstract

No abstract