The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

Abstract

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ -symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.

Multi‐component integrable couplings for the Ablowitz–Kaup–Newell–Segur and Volterra hierarchies

“…Because of rich structures of non‐semisimple Lie algebras, various bi‐integrable and tri‐integrable couplings have been obtained , . Actually, once a generating scheme associated with a non‐semisimple Lie algebra is established, it can be used to construct integrable couplings for different soliton hierarchies.…”

“…Therefore, zero curvature equations over semi‐direct sums of Lie algebras, that is, non‐semisimple Lie algebras, lay the foundation for generating integrable couplings, which provide valuable new insights into the classification of multi‐component integrable systems. A general structure of integrable couplings connected with these kinds of algebras has been recognized recently, and some examples have been presented such as the Ablowitz–Kaup–Newell–Segur (AKNS), Wadati–Konono–Ichikawa, Kaup–Newell, Korteweg–de Vries, Boiti–Pempinelli–Tu and Volterra integrable couplings , . From the previous discussions, we can find that similar to integrable systems derived from semisimple Lie algebras s l (2, R ) and s o (3, R ), integrable couplings usually show various specific mathematical structures, such as block matrix type Lax representations, bi‐Hamiltonian structures, infinitely many symmetries and conservation laws of triangular form.…”

“…The standard Volterra lattice spectral problem reads where u = u ( n , t ) is a dependent variable. Suppose that the associated temporal spectral problem is as follows: The compatibility condition U t −( E V ) U + U V = 0 in this case gives equivalently The previously mentioned system can be reduced to …”

“…In this section, we shall illustrate our construction process for semi-discrete case by establishing the N-coupled integrable couplings of the Volterra lattice hierarchy. The standard Volterra lattice spectral problem reads [3,17]…”

The Bi-Integrable Couplings of Two-Component Casimir-Qiao-Liu Type Hierarchy and Their Hamiltonian Structures

“…In recent years, the construction of soliton hierarchies and integrable couplings have become important research fields in soliton theory [7][8][9][10]. The soliton hierarchies are generated from the zero curvature equations [11,12] which are based on semisimple Lie algebras, while the integrable couplings are generated from the zero curvature equations based on semidirect sums of Lie algebras [13][14][15].…”