Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability

Abstract

An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.

Higher-Order Rogue Wave and Rational Soliton Solutions of Discrete Complex mKdV Equations

“…(1.1) in Section 5, and a higher-order discrete rogue wave solution that differs from the rogue wave solutions of discrete coupled Ablowitz-Ladik equations in Ref. [43] is obtained. Moreover, we also derive higher-order discrete rational soliton solutions.…”

“…Section 4 is devoted to a discrete version of the generalised perturbation (n, N − n)-fold Darboux transformation of Eq. (1.1), used to study integrable continuous NPDEs [41,42,44,45] and the discrete coupled Ablowitz-Ladik equation [43]. These ideas are applied to Eq.…”

A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane

“…5Ð9 Extension to discrete systems and rogue wave pairs with elevations and depressions have been performed. 10,11 A benchmark entity in the theory of nonlinear waves is the soliton, which preserves its identity after collisions with other solitons. 4 An illuminating theoretical perspective is to describe the collisions of solitons in terms of interactions of poles by analytic continuation of the spatial or temporal coordinate to a complex variable.…”

Dynamics of bright and dark multi-soliton solutions for two higher-order Toda lattice equations for nonlinear waves

“…Exact solutions can make people understand deeply the physical mechanism of the natural phenomena described by NLEs. Some methods for constructing explicit exact solutions of NLEs, such as the inverse scattering method [1-4, 18, 19], Hirota method [13,20], Bäcklund transformation [21][22][23], and Darboux transformation (DT) [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], have been developed. Among them, the DT based on the Lax pair is an algebraic method which is an effective tool to solve the Lax integrable NLEs.…”

“…[31][32][33]), and the DT with the N -order Darboux matrix whose elements are the polynomials of the spectral parameters (see, e.g., Refs. [34][35][36][37][38][39][40][41]), etc. Theoretically, the solution obtained by a single DT (i.e., 1-fold DT which is the N -fold DT when N = 1) can be taken as the new starting point from which to derive another solution by making a DT again, hence a series of explicit solutions can be generated by iteration step by step [35][36][37].…”