Abstract
Based on some known loop algebras with finite dimensions, two different negative-order integrable couplings of the negative-order Korteweg-de Vries (KdV) hierarchy of evolution equations are generated by making use of the Tu scheme, from which the corresponding negative-order integrable couplings of the negative-order KdV equations are followed to be obtained. The resulting Hamiltonian structure of one negative integrable coupling is derived from the variational identity.
Highlights
Since the concept on integrable couplings was proposed [1], some integrable couplings of the known integrable systems, such as the AKNS system and the KN system, were obtained
Reference [3] employed a simple finite-dimensional Lie algebra to present a method for generating integrable couplings of integrable hierarchies of evolution equations
Ma and Chen [5] further generalized the quadratic-form identity and completely improved it to obtain the variational identity for deducing the Hamiltonian structures of integrable couplings which is more convenient
Summary
Since the concept on integrable couplings was proposed [1], some integrable couplings of the known integrable systems, such as the AKNS system and the KN system, were obtained. Reference [3] employed a simple finite-dimensional Lie algebra to present a method for generating integrable couplings of integrable hierarchies of evolution equations. We know that some interesting related negativeorder integrable equations including the negative-order KdV equation and some associated properties were obtained, such as the results in [13,14,15,16,17,18] Their negative-order integrable couplings have not been discussed. Enlightened by this work, we will generate the negative-order KdV hierarchy and its integrable couplings by enlarged Lie algebras and the enlarged Lax pairs. The Hamiltonian structure of one negative-order integrable coupling in the negative-order KdV hierarchy is obtained by the variational identity
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