TAILIEUCHUNG - A novel kind of AKNS integrable couplings and their Hamiltonian structures

We present a novel hierarchy of AKNS integrable couplings based on a specific semidirect sum of Lie algebras associated with sl. By applying the variational identity, we derive a bi-Hamiltonian structure of the resulting coupling systems, thereby showing their Liouville integrability. | Turk J Math (2017) 41: 1467 – 1476 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article A novel kind of AKNS integrable couplings and their Hamiltonian structures 3 ¨ ¨ Yu-Juan ZHANG1∗, Wen-Xiu MA2 , Omer UNSAL School of Mathematics and Statistics, Xidian University, Xian, . China 2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA 3 Department of Mathematics and Computer Science, Eski¸sehir Osmangazi University, Eski¸sehir, Turkey 1 Received: • Accepted/Published Online: • Final Version: Abstract: We present a novel hierarchy of AKNS integrable couplings based on a specific semidirect sum of Lie algebras associated with sl (2) . By applying the variational identity, we derive a bi-Hamiltonian structure of the resulting coupling systems, thereby showing their Liouville integrability. Key words: Integrable couplings, semidirect sum, Hamiltonian structure 1. Introduction The concept of integrable couplings has been introduced and various integrable couplings have been studied systematically. This originated from an investigation on centerless Virasoro symmetry algebras of integrable systems [4, 10]. Generally, for a given integrable system ut = K(u) , an integrable coupling is a triangular integrable system { ut = K(u), vt = S(u, v), () where the potentials u and v can be either scalar functions or vector functions of the dependent variables x and t, and ∂S/∂[u] ̸= 0, such that the whole system is not separated. At the beginning, integrable couplings were constructed by using perturbations [4, 5, 10], or enlarging spectral problems and Lie algebras [6, 21]. Later, it was found that integrable couplings have a close connection with semidirect sums of Lie algebras, and a new method to construct integrable couplings was presented [13, 14]. The existing approaches are just specific examples of the semidirect sum

• Toán học
• Integrable couplings
• Semidirect sum
• Hamiltonian structure
• AKNS integrable couplings
• Lie algebras associate
• Liouville integrability