Integrable couplings of a generalized AKNS hierarchy

Abstract

A generalized AKNS isospectral problem where the trace of corresponding spectral matrix is not zero, is transformed to a new isospectral problem where the trace of the resulting matrix is zero, by using transformation of Lax pairs, and these two spectral problems lead to the same hierarchy of equations. The authors started from the transformed spectral problem and constructed a new loop algebra which has not appeared before, and obtained the integrable coupling of the generalized AKNS hierarchy. Specially, the integrable couplings of the KdV equation and MKdV equation are obtained.

Keywords

integrable coupling / AKNS hierarchy / spectral problem

References

[1] FussteinerBClarksonP A. Coupling of completely integrable systems: the perturbation bundle[A]. Analytic and Geometric Methods to Nonlinear Differential Equations[C], 1993, Dordrecht, Kluwer: 125-138

[2] LakshmananM, TamizhaniK M. Lie-Backlund symmetries of certain nonlinear evolution equations under perturbation around solutions[J]. J Math Phys, 1985, 26: 1189-1200

[3] MaW X, FuchssteinerB. Integrable theory of the perturbation equation[J]. Chaos, Soliton & Fractals, 1996, 7: 1227-1250

[4] TuGuizhang, MengDazhi. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems (II) [J]. Acta Mathematicae Applicatae Sinica, 1989, 5(1): 89-95

[5] TuGuizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems(I) [J]. J of Math Phys, 1989, 30: 330-338