INTEGRABLE COUPLINGS AND BI-HAMILTONIAN STRUCTURES OF GENERALIZED ISOSPECTRAL-NONISOSPECTRAL C-KDV HIERARCHIES

In this paper, we derive a combined isospectral-nonisospectral C-KdV hierarchy, which generates some physically significant equations such as the generalized variable-coefficient C-KdV and Burgers equations. For the nonisospectral C-KdV equation, we establish infinite conservation laws and analyze invariant solutions and point symmetries for both the Burgers equation and the classical C-KdV equation. A key contribution is the construction of a new $ 4\times 4 $ Lie algebra $ g $ which enables the formulation of two linear problems. Their compatibility conditions yield isospectral and nonisospectral C-KdV integrable couplings. Using the quadratic-form identity, we derive the bi-Hamiltonian structures for the isospectral hierarchy, while the component-trace identity provides bi-Hamiltonian structures for the nonisospectral hierarchy.