Two-Component Vector Fokas-Lenells Equation

• The paper introduces an integrable system that generalizes the scalar Fokas–Lenells equation by coupling two complex fields with derivative nonlinear interactions.
• It utilizes a Lax pair representation, bi-Hamiltonian structure, and Bäcklund transformations to link the model with DNLS, AL, and other integrable hierarchies.
• Exact solutions, including dark solitons, rogue waves, and matrix soliton configurations, demonstrate its applicability to femtosecond optics, spinor Bose–Einstein condensates, and coupled wave dynamics.

The Two-Component Vector Fokas–Lenells Equation (VCFLE) is an integrable system generalizing the scalar Fokas–Lenells equation by coupling two complex fields through derivative nonlinear interactions. The VCFLE arises in the negative flow of the derivative nonlinear Schrödinger (DNLS) hierarchy and inherits exact solvability, rich solution structure, and connections to several prominent integrable models, including the Ablowitz–Ladik (AL), nonlinear Schrödinger (NLS), and Merola–Ragnisco–Tu (MRT) hierarchies. It is physically motivated by phenomena in femtosecond optics, spinor Bose–Einstein condensates, and coupled wave dynamics.

1. Mathematical Formulation and Integrable Structure

The canonical VCFLE is presented as a system of two coupled nonlinear PDEs: $$u_{xy} + u - 2i\,u\,v\,u_x = 0,\qquad v_{xy} + v + 2i\,u\,v\,v_x = 0,$$ where $u(x,y)$ and $v(x,y)$ are complex-valued fields. The system is integrable and admits a Lax pair representation with suitable spectral parameter dependence, often as the first negative flow of the Kaup–Newell hierarchy (Vekslerchik, 2011). The vector character arises naturally from reductions imposed on larger matrix systems or as a special case of FL hierarchies on Hermitian symmetric spaces (Gerdjikov et al., 2021).

Derivations in Lie algebra settings define the VCFLE via potentials $Q(x,t)$ in graded Lie algebras, typically with $Q(x,t)$ lying in the “off-diagonal” (odd) part selected by the Cartan involution. For two components, the principal Hermitian symmetric space is $SU(2)/S(U(1)\times U(1))$ with the two fields $u,v$ associated to basis elements of $g(1)$ (Gerdjikov et al., 2021).

The bi-Hamiltonian structure of the VCFLE follows from compatibility of two Hamiltonian operators $\mathcal{D}^{(0)}, \mathcal{D}^{(1)}$ and distinct Hamiltonian functionals, e.g.,

$$m_t = \mathcal{D}^{(0)} \frac{\delta H_1}{\delta n},\qquad n_t = \mathcal{D}^{(0)} \frac{\delta H_1}{\delta m},$$

where $m = u + i u_x$, $n = v - i v_x$, ensuring recursion and integrability (Gerdjikov et al., 2021).

2. Lattice and Bilinear Representations: Connections to Ablowitz–Ladik Hierarchy

The VCFLE inherits discrete (lattice) dynamics through mutually inverse Bäcklund transformations: $$T:(u, v) \mapsto (\hat{u}, \hat{v}), \quad \hat{u} = -i v_y + u v^2 - \frac{1}{v} u_x,$$

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

where iterating $T$ generates a lattice-indexed sequence of solutions $(u_n, v_n)$ (Vekslerchik, 2011). These variables satisfy discrete relations (MRT equations), which become bilinear upon expressing $u_n$ and $v_n$ via tau-functions, e.g.,

$$i D_x O_n \cdot T_{n-1} - T_n T_{n+1} = 0, \quad i D_x T_n \cdot P_n - P_{n-1} T_{n+1} = 0,$$

where $D_x$ is the Hirota bilinear operator (Vekslerchik, 2011).

The VCFLE is reducible to the AL hierarchy through appropriate identification of tau-functions and parameter reductions. This connection is pivotal; since AL equations possess determinant-form N-soliton solutions, it permits direct construction of VCFLE soliton and multi-soliton solutions using well-understood algebraic techniques.

3. Exact Solutions: Dark Solitons, Rogue Waves, and Multi-soliton Excitations

(A) Dark Soliton Configurations

The explicit form of N-dark soliton solutions for VCFLE is realized through reduction $v = -u$ and determinant expressions (after certain involutions on spectral parameters): $$u(x, y) = i s e^{-ip(x, y)} \frac{\det[\delta_{jk} + C_{jk}(x, y) e^{i\alpha_k}]}{\det[\delta_{jk} + C_{jk}(x, y) e^{i\beta_k}]}, \quad v(x, y) = -u(x, y),$$ where $C_{jk}(x, y)$, $\alpha_k$, $\beta_k$ encode soliton parameters arising from reductions in the AL tau-functions (Vekslerchik, 2011).

(B) Rogue Waves and Breathers

DT and Taylor expansion methods yield explicit, rational rogue wave solutions as higher-order limits of breather profiles. For the two-component system, the construction employs vector-valued Darboux data enabling simultaneous generation of coupled rogue waves: $q^{[1]}_{rw} = \alpha_1 \frac{s_2}{s_3} e^{s_1},$ with background normalization yielding $$|q_{rw}^{[1]}|^2 \to \alpha_1^2 / (\alpha_1^2 + \beta_1^2)^2$$, and peak amplitude nine times background (He et al., 2012). The determinant representation of multi-rogue wave solutions extends via n-fold Darboux transformations and degenerate eigenvalue limits, granting control over location, profile, and interaction parameters (Xu et al., 2012).

(C) Binary Darboux Transformation and Matrix Soliton Solutions

Recent developments establish a unified vectorial binary Darboux transformation (bDT) for VCFLE: $u' = u - \theta_2 \Omega^{-1} \eta_1,$ where $\theta_{1,2}$, $\eta_{1,2}$ solve the Lax pair for the seed solution (e.g., a plane wave), and $\Omega$ is determined by a Sylvester or Lyapunov equation. By suitable reduction (e.g., $v = u^*$), the transformation produces scalar, vector, and matrix soliton configurations: breathers, rogue waves, and "beating solitons," the latter typified by energy exchange between components (Müller-Hoissen et al., 17 Apr 2025, Müller-Hoissen et al., 24 Aug 2025).

4. Spectral Theory, Riemann–Hilbert Problem, and Soliton Interactions

The VCFLE admits a matrix-valued Lax pair (typically $3\times3$ for $N=2$), underlying a robust spectral theory. The associated inverse scattering transformation (IST) method, developed for single-component FL with NZBC (Zhao et al., 2019) and extended to vector systems (Xun et al., 2020), leads to Riemann–Hilbert formulations for soliton reconstruction:

• The jump relations and analyticity domains for the vector Jost solutions yield determinant formulas for N-soliton solutions.
• Soliton interactions are characterized by phase shifts, amplitude redistribution, and polarization dynamics absent in scalar cases.
• Reflectionless solutions correspond to multi-soliton states, while the inclusion of continuous spectrum yields radiation effects.

In higher-component cases ($N>2$), the complexity increases (Lax operator size $N+1$), with richer possibilities for breather–like structures and collisional dynamics (Xun et al., 2020).

5. Reductions, Generalizations, and Interconnections

(A) Algebraic Reductions

The VCFLE is obtained from larger multicomponent FL equations via algebraic reductions (e.g., imposition of Hermitian conjugation or complex involution $p = q^*$ (Gerdjikov et al., 2021)). Nonlocal reductions further generalize the system, admitting solutions that depend on spatial or temporal reflection.

(B) Relation to Other Integrable Models

The integrable nature of VCFLE is underscored by its intersection with:

• DNLS hierarchy (VCFLE corresponds to first negative flow).
• Standard NLS equation.
• AL and MRT hierarchies (through bilinearization and tau-function correspondence).
• AKNS hierarchy via Miura transformation, elucidating further integrable reductions (Müller-Hoissen et al., 17 Apr 2025).

Importantly, reductions from the self-dual Yang–Mills equation via Cauchy matrix schemes produce the pKN(–1) system, whose solutions furnish those of VCFLE (Li et al., 16 Nov 2024). The two solution schemes (KP-type, AKNS-type) are reflection-equivalent.

6. Physical Context and Applications

The VCFLE is motivated by femtosecond optical pulse propagation, where higher-order nonlinear effects and vector interactions (e.g., polarization mixing) are relevant (Vekslerchik, 2011, He et al., 2012). Other contexts include:

• Ion-cyclotron plasma waves (DLFL equation).
• Ultrafast optics and nonlinear fiber dynamics.
• Spinor Bose–Einstein condensates, described via extended FL equations.
• Fluid dynamics (interfacial waves), DNA double-strand dynamics, and analog black hole surface geometries (Gerdjikov et al., 2021, Talukdar et al., 5 Jun 2024).

The availability of bright, dark, rogue, beating, and composite vector soliton solutions enhances modeling fidelity for coupled wave phenomena.

7. Open Problems and Further Directions

Several promising research directions remain:

• Extension of IST, spectral analysis, and Riemann–Hilbert techniques to nontrivial boundary and nonlocal reductions in VCFLE.
• Systematic construction and classification of multi-component rogue waves via matrix Darboux transformations.
• Application of geometric methods (e.g., Sym–Tafel formula) to multi-component soliton surfaces, including investigation of differential geometric invariants such as Gaussian curvature (Talukdar et al., 5 Jun 2024).
• Numerical studies and data-driven approaches (modified PINN) for soliton parameter estimation, stability analysis, and solution discovery (Saharia et al., 2023).

Recent advances in matrix Darboux transformations and bidifferential calculus provide unifying frameworks for exact solution generation, algebraic linkages among hierarchies, and applications extending across nonlinear physics domains.

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Summary Table: Core Mathematical Objects in VCFLE Theory

Object	Definition / Role	Hierarchy/Model
$(u, v)$	Coupled fields, vector components	VCFLE
$\text{Bäcklund transformations}$	Generate lattice representation, recursive solutions	MRT hierarchy, VCFLE
$\text{Tau-functions}$	Determinant-based solution representation	AL hierarchy, VCFLE
$\text{Lax pair}$	Encodes integrability via spectral parameter	DNLS/AKNS/AL hierarchy
$\text{Miura transformation}$	Maps VCFLE to negative-flow AKNS hierarchy	AKNS, VCFLE
$\text{Darboux transformation (DT)}$	Solution-generating via auxiliary linear system	Generates VCFLE, multicomponent solitons
$\text{Riemann–Hilbert problem}$	IST, soliton reconstruction	Single-/multi-component FL, coupled systems

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The Two-Component Vector Fokas–Lenells Equation thus serves as a cornerstone example of modern integrable systems theory, integrating discrete, algebraic, spectral, and geometric techniques with broad relevance for coupled nonlinear wave phenomena.