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| Generalized Isospectral-NonisospectralModified Korteweg-de Vries IntegrableHierarchies and Related Properties |
| Huanhuan Lu,Xinan Ren,Yufeng Zhang,Hongyi Zhang |
| (School of Mathematics, China University of Mining and Technology, Xuzhou,Jiangsu 221116, China) |
| DOI： |
| Abstract: |
| 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 |
| Key words: Darboux transformation, nonlocal symmetry, covering, manifold, contact
structure. |