Matrix Fokas–Lenells Equation

• The paper presents a novel integrable matrix extension of the Fokas–Lenells equation, uniting discrete lattice and continuous models through Bäcklund transformations.
• It employs tau-function determinants to construct explicit N-dark soliton solutions, generalizing classical hierarchies such as AL, NLS, and DNLS.
• This framework supports multicomponent applications in fiber optics and Bose–Einstein condensates, maintaining integrability across complex systems.

The integrable matrix version of the Fokas–Lenells equation (FLE) is a nonlinear evolution equation forming the first negative flow in the derivative nonlinear Schrödinger (DNLS) hierarchy. It provides a unifying integrable framework bridging scalar and matrix systems, discrete (lattice) and continuous representations, and enables explicit construction of N-dark soliton solutions in determinant form. This matrix FLE encapsulates the essential features of the Ablowitz–Ladik (AL), NLS, DNLS, and Merola–Ragnisco–Tu (MRT) hierarchies and generalizes the integrability and solitonic structure to physical contexts requiring multicomponent modeling (e.g., nonlinear fiber optics, Bose–Einstein condensates).

1. Definition and Lattice Construction

The Fokas–Lenells equation can be written in a coupled form for functions $u(x, y)$ and $v(x, y)$: $$\begin{cases} u_{xy} + u - 2i\, u v u_x = 0 \ v_{xy} + v + 2i\, u v v_x = 0 \end{cases}$$ The key innovation is the introduction of a pair of commuting Bäcklund transformations $T$ and $T^{-1}$. Explicitly,

$$T: (u, v) \mapsto (\hat{u}, \hat{v}),\quad \hat{u} = -i u_y + u^2 v - v^{-1}$$

$$T^{-1}: (u, v) \mapsto (\check{u}, \check{v}),\quad \check{u} = +i u_x + u^2 v - v^{-1}$$

These transformations map solutions to new solutions and are mutual inverses on-shell. Iterating $T$ defines a discrete lattice of solutions $(u_n, v_n) = T^n(u, v)$, $n\in\mathbb{Z}$.

This “Bäcklund lattice” admits difference-differential (lattice) equations, such as \begin{align*} -i \partial_x v_n & = v_{n+1} - v_n\, v^2 \ -i \partial_y v_n & = v_{n-1} - v_n\, u z \end{align*} These relations, upon grouping and bilinearization, are identified as equations belonging to the Merola–Ragnisco–Tu (MRT) lattice hierarchy, and, via tau-function formalism, to the discrete Ablowitz–Ladik (AL) hierarchy.

2. Connections to DNLS, NLS, MRT, and AL Hierarchies

The matrix FLE is deeply intertwined with other well-studied integrable models:

• As the first negative flow of the DNLS hierarchy, it connects directly to the broad AKNS landscape.
• The lattice representation generated by Bäcklund transformations reduces to the MRT equations, which are discrete AKNS systems.
• Bilinearization via tau functions transforms the MRT/AL representation into a form directly permitting application of AL results. For bilinear tau functions $T_n, P_n, N_n$, the AL-like relations include

$T_n^2 - T_{n-1} T_{n+1} = P_n N_n$

and Hirota bilinear forms,

$i D_x O_n \cdot T_n - T_{n-1} O_{n+1} = 0$

This web of links demonstrates that the matrix FLE and its soliton solutions are embedded in a universal integrable hierarchy, with reduction and mapping procedures to NLS, DNLS, MRT, and AL models.

3. N–Dark Soliton Solutions and Determinant Formulas

The AL hierarchy connection furnishes explicit N-dark soliton solutions for the FLE. These solutions are constructed using tau-functions in determinant form, with parameters encoding background amplitude and individual soliton properties. For $n=0$ (physical field),

$$u(x, y) = is\, e^{-ip(x, y)}\, \frac{\det\left|d_{jk} + C_{jk}(x, y) e^{i\alpha_k}\right|}{\det\left|8_{jk} + C_{jk}(x, y) e^{i\beta_k}\right|}$$

$v(x, y) = -u(x, y)$

where $p(x, y)$ is a linear function of $(x, y)$ encoding boundary conditions, $C_{jk}$, $\alpha_k$, and $\beta_k$ are solution parameters, and the determinants are over $j, k = 1, ..., N$. Each soliton represents an intensity dip (dark soliton) on a non-vanishing background—a structure only feasible within this integrable framework.

The determinant structure, already prominent in the AL system, generalizes naturally to the matrix case, with matrix-valued tau functions and determinants, allowing for dark multi-soliton solutions in the multicomponent FLE.

4. Integrability and Matrix Generalization

All essential integrability features—Lax pairs, Bäcklund transformations, tau-function representations, and determinant formulae—lift to the matrix context. The methods described yield

• Matrix-valued Lax pairs, whose compatibility condition yields the matrix FLE.
• Matrix-valued Bäcklund transformations and mutual invertibility.
• Matrix tau-functions, enabling the bilinearization and determinant solution construction.
• Direct applicability to physically motivated systems, such as birefringent fiber models or spinor condensates, where multiple interacting field components are indispensable.

Integrability is preserved through the matrix extension, as the underlying algebraic (hierarchical) structure remains intact due to its formulation via universal objects (e.g., Lax pairs, tau-functions, bilinear operators).

5. Physical and Mathematical Significance

The matrix FLE and the rich solution space constructed through this framework address multiple needs:

• Enabling the paper of multicomponent soliton interactions (e.g., for higher-order polarization effects in optics).
• Exploring the universality and translation of soliton dynamics across hierarchically-related integrable systems.
• Providing explicit analytic formulas suitable for investigating stability, interaction, and modulation phenomena in matrix (vectorial) nonlinear wave equations.
• Facilitating the extension to generalized AL/MRT types, offering a consistent structure for discrete to continuous, scalar to matrix, and negative to positive flow interpolations.

6. Broader Hierarchical Context

The integrable matrix FLE, as constructed, represents a central node in the integrable systems landscape. Its ability to realize, unify, and “lift” solutions across DNLS, NLS, MRT, and AL hierarchies, while maintaining determinant-based explicitness at the matrix level, makes it a powerful and universal tool for both mathematical analysis and applications in nonlinear wave science.

The explicit use of N-dark soliton determinant formulas and the Bäcklund lattice approach enable further semi-discrete and nonlocal generalizations as well as the paper of higher-dimensional or coupled models. The tau-function structure ensures that both analytical and algebraic methods from the AL and AKNS frameworks remain operative in the multicomponent Fokas–Lenells equation setting (Vekslerchik, 2011).

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In summary, the integrable matrix version of the Fokas–Lenells equation is established through its lattice (Bäcklund–tau–function) representation, connection to a universal set of integrable hierarchies, and explicit soliton solutions. The methodology provides a pathway for the systematic paper and construction of matrix soliton solutions, and demonstrates the AL framework as the encompassing environment for understanding and expanding upon the FLE’s integrable structure.