Akhmediev and Kuznetsov–Ma rogue wave clusters of the higher-order nonlinear Schrödinger equation

In this work, we analyze three types of rogue wave (RW) clusters for the quintic nonlinear Schrödinger equation (QNLSE) on a flat background. These exact QNLSE solutions, composed of higher-order Akhmediev breathers (ABs) and Kuznetsov–Ma solitons (KMSs), are generated using the Darboux transformation (DT) scheme. We analyze the dependence of their shapes and intensity profiles on the three real QNLSE parameters, eigenvalues, and evolution shifts in the DT scheme. The first type of RW clusters, characterized by the periodic array of peaks along the transverse or evolution axis, is obtained when the condition of commensurate frequencies of DT components is applied. The elliptical RW clusters are computed from the previous solution class when the first m evolution shifts in the DT scheme of order n are equal and nonzero. For both AB and KMS solutions a periodic structure is obtained with the central RW and m ellipses, containing the first-order maxima that encircle the central peak. We show that RW clusters built on KMSs are significantly more vulnerable to the application of high values of QNLSE parameters, in contrast to the AB case. We next present non-periodic long-tail KMS clusters, characterized by the rogue wave at the origin and n tails above and below the central point containing first-order KMSs. We finally show that the breather-to-soliton conversion, enabled by the QNLSE system, can transform the shape of RW clusters, by setting the real parts of DT eigenvalues to particular values, while keeping all other DT parameters intact.