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Applied Mathematics & Information Sciences

Author Country (or Countries)

China

Abstract

In this paper, the Bargmann system for the second-order eigenvalue problem with the energy is dependent on potential and speed Lj −(¶ 2+l u+v)j −ljx is discussed. First of all, some basic concepts are introduced. After that, by means of the auxiliary problem and the isospectral compatible condition, the reasonable bi-Hamilton operator K and J are defined, and the evolution equations related to the spectral problem are obtained. Using the functional gradient and Lenard recursive sequence, the Bargmann constraint is given. By the constraint relation between the potential function and the eigenvector, the associated Lax pairs are nonlineared, then the Bargmann system of the eigenvalue problem is found. According to the viewpoint of Hamilton mechanics, a reasonable coordinate system has been found. The Bargmann system is transformed into the Hamilton canonical equations in the coordinate system. Finally, based on the Liouville theorem and confocal involutive system, the integrability of the Hamilton system is proved. A new integrable system is found. Moreover, the involutive representations of the solutions for the evolution equations are generated.

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