General rogue wave solutions of the coupled Fokas–Lenells equations and non-recursive Darboux transformation

Abstract

We formulate a non-recursive Darboux transformation technique to obtain the general nth-order rational rogue wave solutions to the coupled Fokas-Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and secondorder rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.

Peregrine Solitons on a Periodic Background in the Vector Cubic-Quintic Nonlinear Schrödinger Equation

“…52-56 for more details. Intended for rogue wave states only, a generalized or nonrecursive Darboux transformation method can be developed, which can give the nth-order rogue wave solutions without any iteration operation [42,57,58,59].…”

“…Also, we will assume below the initial constant phases ϕ j to be zero, without loss of generality. Then, with the help of the Darboux transformation technique outlined above [42,58,59] followed by tedious algebraic manipulations, we obtain the exact fundamental rogue wave solutions on a periodic background, expressed by…”

Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces

“…The interest to the equations from the "negative" flows is to a big extent related to the variety and complexity of their solutions, [38,40,42,43,6,34,31]. Multi-component generalizations of the FL equation have appeared recently in numerous studies like [14,15,29,31,41,50,51,52,53,44] and this naturally leads to the need of their classification from the viewpoint of the simple Lie algebras, the associated symmetric spaces and their reductions. The other multi-component integrable equations in a non-evolutionary form include for example the massive Thirring-like model, whose integrability was shown by Kuznetsov and Mikhailov [37]; its multicomponent extensions were proposed in [48].…”

Modulation instability, conservation laws and localized waves for the generalized coupled Fokas-Lenells equation

“…The coupled FL equation shares the same spatial part of the spectral problem with the coupled derivative NLS equation [31], which is the first nontrivial negative flow of the vector Kaup-Newell hierarchy and relevants in the theory of polarized Alfvén waves and the propagation of the ultra-short pulse. Many effective methods, such as Riemann-Hilbert method [32], DT [33], non-recursive DT [34], generalized DT [35], etc., have been developed to study on the coupled FL equation. The coupled FL equation is one of the integrable systems as shown in [36] and of course admit other integrable properties including multi-Hamiltonian structure, infinitely many conservation laws, the general soliton solutions [37] and optical soliton [38].…”