A systematic approach to the soliton equations of a discrete eigenvalue problem

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Abstract
The Ablowitz–Ladik (AL) problem is a linear vector difference equation whose isospectral flow equations include several important soliton equations; e.g., the discrete nonlinear Schrödinger equation: iq̇n=qn−1−2qn+qn+1 +‖qn‖2(qn−1+qn+1). There is an established procedure for describing the soliton hierarchy of the more familiar AKNS (Ablowitz, Kaup, Newell and Segur)problem. It is based on the notion of a generator for the hierarchy. In this paper the soliton equations of the AL hierarchy are described and characterized by a generator pair. A new continuous spectral problem is introduced and the AKNS hierarchy is embedded in its hierarchy as a specialization.