Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties

In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product.Several classes of spectral problems are introduced by threedimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra.It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out.The derived integrable hierarchies are reduced to the modified Korteweg-de Vries (mKdV) equation, generalized coupled mKdV integrable system and nonisospectral mKdV equation under specific parameter selection.Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit N -fold Darboux transformation for integrable system (2.8) with the help of a gauge transformation of the corresponding spectral problem.At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper.It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.