A Chain of Type II and Its Exact Solutions

The exact solutions of a chain of type II are investigated. The chain of type II is first transformed to an integrable differential-difference equation, which has the Kaup–Newell spectral problem as its continuous spatial spectral problem and a Darboux transformation of the Kaup–Newell equation as its discrete temporal spectral problem. Then, with these spectral problems, a Darboux transformation of the transformed equation is constructed. Finally, as an application of the Darboux transformation, an exact solution of the transformed equation and thus the chain of type II are presented.