The Two-order Eigenvalue Problem And Its Integerable System In Bargmann Constraint

This paper first introduces the related theoretical background and the research status of soliton and Integerable system. Then the two-order eigenvalue problem which the energy is dependent on speed and potential is mainly discussed.Lφ=(λ~2＋λv+u)+vφ_x And its Hamilton integrable system under the Bargmann constraint condition is also given.According to the compatible condition of the main spectral and auxiliary spectral problem, the bi-Hamilton operator K 、 J are obtained by the means of Lie algebra transposition operator. Then based on the Lenart recursive sequence{G_j-1,2,…} and functional gradient, the evolution equations of the eigenvalue problem has been found. By the constraint relation between the potentials(u,v) and the eigenvector Φ,Ψ, the associated Lax pairs are nonlineared, with the equivalent Bargmann system producing. Afterwards based on the Hamilton mechanics and Euler-Lagrange function, a reasonable Jacobi-Ostrongradsky coordinate system has been established. The Bargmann system can be equal to the Hamilton canonical system. Finally, the Liouville theorem are applied to prove the integrability of Hamilton canonical system, the involutive representations of the solutions for the nonlineared evolution equation are obtained.