Akhmediev breathers and Kuznetsov-Ma rogue wave clusters of the higher-order nonlinear Schr\"odinger equation

Abstract

In this work, we analyze three types of the rogue wave (RW) clusters for the quintic nonlinear Schrödinger equation (QNLSE) on a flat background. These exact QNLSE solutions, composed of the higher-order Akhmedieav 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, the 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 on 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, the periodic structure is obtained with the central RW of order n-2m 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 sufficiently high values of QNLSE parameters, in contrast to the AB case. We next present the non-periodic long-tail KMS clusters characterized by the central rogue wave at the origin and $n$ tails above and below the central (0,0) point containing the first-order KMS. We finally show that the breather-to-soliton conversion, enabled by the QNLSE system, can completely transform the shape of RW clusters, by simply setting the real parts of DT eigenvalues to particular values, while keeping the three quintic parameters, imaginary parts and evolution shifts in the DT scheme intact.