Kuznetsov–Ma rogue wave clusters of the nonlinear Schrödinger equation

Abstract In this work, we investigate rogue wave (RW) clusters of different shapes, composed Kuznetsov–Ma solitons (KMSs) from the nonlinear Schrödinger equation (NLSE) with Kerr nonlinearity. We present three classes exact higher-order solutions on uniform background that are calculated using Darboux transformation (DT) scheme precisely chosen parameters. The first solution class is characterized by strong intensity narrow peaks periodic along evolution x -axis, when eigenvalues in DT generate KMSs commensurate frequencies. second exhibits a form elliptical clusters; it derived m shifts n th-order nonzero and equal. show high-intensity built order $$n-2m$$ n - 2 m periodically appear -axis. This structure, considered as central wave, enclosed ellipses consisting certain number first-order determined ellipse index order. third KMS obtained purely imaginary tend to some preset offset value higher than one, while keeping -shifts unchanged. at (0, 0) always retains its tails formed above below maximum. When even, more complicated patterns generated, $$m-1$$ 1 loops RW, respectively. Finally, compute an additional wavy background, defined Jacobi elliptic dnoidal function, which displays specific consistent perturbation. work demonstrates incredible power creating new NLSE tremendous richness function those solutions.