Higher-order Solitons Of A Discrete Ablowitz-Ladik Equation

The regularization of the Riemann-Hilbert problem is applied for studying a discrete Ablowitz-Ladik equation and the soliton and higher-order soliton solutions of the discrete Ablowitz-Ladik equation are obtained. By virtue of the inverse spectral transform method,the analytic properties of the spectral problem associated with the discrete Ablowitz-Ladik equation are discussed, and then a nonregular martrix Riemann-Hilbert problem with zero is constructed, and the relationship between the solution of a discrete Ablowitz-Ladik equation and that of the Riemann-Hilbert problem is established. The nonregular RiemannHilbert problem discussed in this article involves the simple zeros and higher-order zeros.It is noted that it is very di?cult to solve a nonregular martrix Riemann-Hilbert problem.The author constructs a representation of the solution of discrete Ablowitz-Ladik equation about the soliton matrix, and finally obatainds the soliton solutions, including the simple soliton and higher-order soliton solutions.