Miura transformation in bidifferential calculus and a vectorial Darboux transformation for the Fokas-Lenells equation

TL;DR

The paper develops a vectorial binary Darboux transformation for the first member of the negative Kaup-Newell (KN) hierarchy, cast in the bidifferential calculus framework, and applies it to the coupled Fokas-Lenells equations. It derives Miura transformations linking the FL system to the first negative AKNS member and its pseudodual, and then implements the reduction v = u^* to obtain a vectorial Darboux transformation for the Fokas-Lenells equation, including both vanishing and plane-wave seeds. Using a plane-wave seed, it yields a rich zoo of exact solutions—breathers, single and multi-solitons, dark/bright solitons, and rogue waves—via Lyapunov/Sylvester equations that determine the necessary transformation data. The approach unifies DT methods with bidifferential calculus to produce nontrivial multi-component FL solutions in a single framework, and it clarifies the compatibility of reductions and spectral conditions needed to preserve the physical constraint v = u^*. This provides a powerful, algebraic route to generating and analyzing large families of soliton solutions for the FL system and related hierarchies.

Abstract

Using a general result of bidifferential calculus and recent results of other authors, a vectorial binary Darboux transformation is derived for the first member of the "negative" part of the potential Kaup-Newell hierarchy, which is a system of two coupled Fokas-Lenells equations. Miura transformations are found from the latter to the first member of the negative part of the AKNS hierarchy and also to its "pseudodual". The reduction to the Fokas-Lenells equation is implemented and exact solutions with a plane wave seed generated.

Figures (5)

• Figure 1: Plot of the absolute value of a "positon" solution of the Fokas-Lenells equation from the class in Example \ref{['ex:2x2Jordan']}. Here we chose $\gamma=a_{11}=a_{12}=a_{21}=a_{22}=1$.
• Figure 2: Plot of the absolute value of a dark soliton solution of the Fokas-Lenells equation from the class in Section \ref{['subsec:FLeq_1sol']}(2). Here we chose $\alpha=a_1=c=1,b_1=0,A=2,k=1,$ so that $r=\sqrt{3}\,\mathrm{i} \,$.
• Figure 3: Plot of the absolute value of a dark-dark (left plot) and a dark-bright (right plot) soliton pair solution of the Fokas-Lenells equation from the class in Section \ref{['subsec:dark-bright_solitons']}. Here we chose $\alpha=a_1=a_2=c_1 = k_1 =1$, $k_2=10$, $A=2$ and $c_2 =1$, respectively $c_2=-1$.
• Figure 4: Plot of the absolute value of a solution of the Fokas-Lenells equation from the class in Example \ref{['ex:rw']}. Here we chose $\alpha=a=b=1,A=\frac{1}{2}$, so that $\gamma=\frac{\sqrt{3}}{2}+\frac{1}{2}\,\mathrm{i} \,$. The deformation of the constant density background is localized in both dimensions.
• Figure 5: Plot of the absolute value of a solution of the Fokas-Lenells equation from the class in Example \ref{['ex:rw2']}. Here we chose $\alpha=1/4$ and $A=a_0=a_1=b_0=b_1=1$.

Theorems & Definitions (30)

• Theorem 2.1
• Corollary 2.2
• Corollary 2.3
• Theorem 2.4
• proof
• Proposition 3.1: LLZ24
• proof
• Remark 3.2
• Lemma 3.3
• proof