Abstract
In this paper, we construct the discrete higher-order rogue wave (RW) solutions for a generalized integrable discrete nonlinear Schrödinger (NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation is constructed. Second, the dynamical behaviors of first-, second- and third-order RW solutions are investigated in corresponding to the unique spectral parameter, higher-order term coefficient, and free constants. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz–Ladik (AL) equation are illustrated in figures. Moreover, we explore the numerical experiments, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.
Highlights
Rogue wave was founded in many fields, such as nonlinear optics, fluid mechanics, and even finance[1,2,3]
We study the dynamical behaviors of the first-order RW solutions by numerical simulation with the initial conditions and perturbation for the Eq(2)
We have studied a higher-order integrable discrete NLS equation by the generalized discrete DT
Summary
Rogue wave was founded in many fields, such as nonlinear optics, fluid mechanics, and even finance[1,2,3]. Breather solutions, rogue wave and modulation instability of this integrable three-parameter fifth-order nonlinear Schrodinger equation are analytically studied based on DT and robust inverse scattering transform [21, 22]. As we know, there is little work on rogue wave solutions and breather solutions of this higher-order integrable discrete NLS equation (2). This is the main motivation for us to investigate the higher-order RWs of the discrete integrable NLS equation (2) with higher-order excitations in this paper. In Sect., by using the modified discrete Lax pairs, we apply the generalized (1,N-1)-fold Darboux transformation [4, 9] to construct higher-order discrete RW solutions of Eq(2).
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