Based on the fundamental commutator representation proposed by Cao et al. we established two expl... more Based on the fundamental commutator representation proposed by Cao et al. we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel'fand-Dickey hierarchy.
Solutions of a classical nonlinear differential recurrence equation is investigated for explicit ... more Solutions of a classical nonlinear differential recurrence equation is investigated for explicit expressions. Normalization of infinitely many conserved densities of KdV equation is further discussed. Closed formulas of integrals involving $sech^2(x)$ are derived as an application.
We establish in this paper an infinitely dimensional Lie algebraic structure of the AKNS hierarch... more We establish in this paper an infinitely dimensional Lie algebraic structure of the AKNS hierarchy which is connected with anN × N matrix nonisospectral problem.
An isospectral problem with four potentials is discussed. The corresponding hierarchy of nonlinea... more An isospectral problem with four potentials is discussed. The corresponding hierarchy of nonlinear evolution equations is derived. It is shown that the AKNS, Levi,D-AKNS hierarchies and a new one are reductions of the above hierarchy. In each case the relevant Hamiltonian form is established by making use of the trace identity.
Based on the fundamental commutator representation proposed by Cao et al. we established two expl... more Based on the fundamental commutator representation proposed by Cao et al. we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel'fand-Dickey hierarchy.
Solutions of a classical nonlinear differential recurrence equation is investigated for explicit ... more Solutions of a classical nonlinear differential recurrence equation is investigated for explicit expressions. Normalization of infinitely many conserved densities of KdV equation is further discussed. Closed formulas of integrals involving $sech^2(x)$ are derived as an application.
We establish in this paper an infinitely dimensional Lie algebraic structure of the AKNS hierarch... more We establish in this paper an infinitely dimensional Lie algebraic structure of the AKNS hierarchy which is connected with anN × N matrix nonisospectral problem.
An isospectral problem with four potentials is discussed. The corresponding hierarchy of nonlinea... more An isospectral problem with four potentials is discussed. The corresponding hierarchy of nonlinear evolution equations is derived. It is shown that the AKNS, Levi,D-AKNS hierarchies and a new one are reductions of the above hierarchy. In each case the relevant Hamiltonian form is established by making use of the trace identity.
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