Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations. Integrable couplings [1,2] are associated with non-semisimple Lie algebras [3,4] and variational identities provide tools to generate Hamiltonian structures of integrable couplings, in both continuous and discrete cases [5,6]. Most of the presented integrable couplings are linear with respect to the supplementary variables (see, e.g., [1,7–13]). For example, the spectral matrices of the form U ¼ UðuÞ U 0 ½vŠ 0 UðuÞ ! ; where the sub-spectral matrix U is associated with a given integrable equation u t = K(u) and U 0 denotes its Gateaux derivative , lead to integrable couplings of the perturbation type. In such resulting integrable couplings, the equation for the supplementary variable v is linear with respect to v. If the second equation of an integrable coupling u t ¼ KðuÞ; v t ¼ Sðu; vÞ; & defines a nonlinear equation for v, then the whole system is called a nonlinear integrable coupling. The two variables u and v above can be either scalars or vectors. Linear integrable couplings contain extensions of symmetry equations [1,7] and are important in classifying integrable equations, but definitely, nonlinear ones have much richer structures. There are a few systematical ways to construct linear integrable couplings, starting from the perturbed spectral matrices [2,7], defined as before, and the amended spectral matrices [8,10]: U ¼ UðuÞ U a ðvÞ 0 0 ! ; where U a may not be a square matrix. However, there is no feasible way which allows us to construct nonlinear integrable couplings.