Abstract
In this paper, we seek connections between the Sylvester equation and the Ablowitz—Kaup—Newell—Segur (AKNS) system. By the Sylvester equation KM — MK = r sT, we introduce master function S(i, j) = sT Kj (I + M)−1Kir. This function satisfies some recurrence relations. By imposing dispersion relations on r and s, we study the constructions of the AKNS system, where some AKNS type equations are investigated emphatically, including second-AKNS equation, second-modified AKNS (mAKNS) equation, third-AKNS equation, third-mAKNS equation and (—1)st-AKNS equation. The reductions of these equations to complex Korteweg—de Vries (KdV) equation, real and complex modified Korteweg—de Vries (mKdV) type equations, nonlinear Schrodinger (NLS) type equations and sine-Gordon (sG) equation are discussed.
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