[Paper Review] The Darboux transformation and higher-order rogue wave modes for a derivative nonlinear Schr\"odinger equation

Yongshuai Zhang, Lijuan Guo+2 more|arXiv (Cornell University)|Sep 28, 2014

Nonlinear Waves and Solitons75 references3 citations

TL;DR

This paper derives n-th order solutions of the derivative nonlinear Schrödinger equation (CLL-NLS) using a novel Darboux transformation (DT) technique that eliminates iterative integrals in the transformation matrix, enabling explicit determinant-form expressions. The method yields higher-order rogue wave solutions via Taylor expansion at a specific eigenvalue, with analytical and graphical analysis showing that the self-steepening effect alters the localization length and width of first-order rogue waves.

ABSTRACT

Abstract. We derive the n-th order solution of the mixed Chen-Lee-Liu derivative nonlin-ear Schrödinger equation (CLL-NLS) by applying the Darboux transformation (DT). Such solutions together with the n-fold DT, represented by Tn, are given in terms of determinant representation, whose entries are expressed by eigenfunctions associated with the initial “seed” solutions. This kind of DT technique is not common, since Tn is related to an overall factor expressed by integrals of previous potentials in the procedure of iteration. As next step, we annihilate these integrals in the overall factor of Tn, except the only one depending on the initial “seed ” solution, which can be easily calculated under the reduction condition. Furthermore, the formulae for higher-order rogue wave solutions of the CLL-NLS are obtained according to the Taylor expansion, evaluated at a specific eigenvalue. As possible applications, the expres-sions and figures of non-vanishing boundary solitons, breathers and a hierarchy of rogue wave solutions are presented. In addition, we discuss the localization characters of rogue wave by defining their length and width. In particular, we show that these localization characters of the first-order rogue wave can be changed by the self-steepening effect in the CLL-NLS by use of an analytical and a graphical method.

Motivation & Objective

• To develop a practical n-fold Darboux transformation for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation (CLL-NLS) with reduced iterative complexity.
• To eliminate non-local integral dependencies in the DT iteration process, retaining only the initial seed solution's contribution in the overall factor.
• To construct explicit higher-order rogue wave solutions using Taylor expansion around a specific eigenvalue.
• To analyze the localization characteristics (length and width) of rogue waves and investigate the influence of the self-steepening effect.
• To present visual and analytical evidence of how the self-steepening parameter modifies rogue wave structure.

Proposed method

• The n-fold Darboux transformation is formulated using determinant representations, where entries are eigenfunctions of the initial seed solution.
• An iterative procedure is modified to remove integrals of previous potentials, retaining only the initial seed solution in the overall transformation factor.
• The transformation matrix Tn is derived in closed-form determinant expression, enabling systematic construction of higher-order solutions.
• Higher-order rogue wave solutions are generated by Taylor expansion of the Darboux transformation around a specific eigenvalue.
• The self-steepening effect is incorporated into the CLL-NLS model to study its impact on rogue wave localization.
• Localization parameters—length and width—are analytically defined and numerically evaluated for first-order rogue waves.

Experimental results

Research questions

• RQ1 How can the n-fold Darboux transformation be simplified to eliminate non-local integral dependencies in the iteration process?
• RQ2 What is the explicit form of higher-order rogue wave solutions for the CLL-NLS equation?
• RQ3 How does the self-steepening effect influence the spatial localization (length and width) of first-order rogue waves?
• RQ4 Can the determinant-based Darboux transformation be applied to generate non-vanishing boundary solitons and breathers?
• RQ5 What analytical and graphical methods can be used to characterize rogue wave localization under varying self-steepening parameters?

Key findings

• The n-fold Darboux transformation for the CLL-NLS equation is successfully derived in determinant form, with all non-local integrals removed except the initial seed solution contribution.
• Higher-order rogue wave solutions are explicitly constructed using Taylor expansion at a specific eigenvalue, enabling systematic analysis of multi-peak structures.
• The self-steepening effect is shown to significantly alter the localization length and width of first-order rogue waves, as confirmed by both analytical derivation and numerical visualization.
• Non-vanishing boundary solitons and breathers are presented as part of the solution hierarchy, demonstrating the method's broad applicability.
• The localization characteristics of rogue waves are quantitatively defined through analytically derived length and width parameters, which vary with the self-steepening coefficient.
• The proposed DT framework enables the generation of complex nonlinear wave patterns, including higher-order rogue waves, with full control over initial conditions and nonlinear parameters.