Nonlocal Fokas–Lenells Equation

• Nonlocal Fokas–Lenells equation is an integrable nonlinear evolution equation characterized by a space-time reflection reduction that couples u(x,t) with u(-x,-t).
• The bilinear formulation using Hirota’s method and double Wronskian determinants yields exact solutions, demonstrating soliton, breather, and algebraic decay behaviors.
• Extensions to multicomponent and symmetric-space frameworks confirm the equation’s integrability through Lax pairs and bi-Hamiltonian structures, broadening its applications in integrable systems.

Searching arXiv for recent and foundational papers on the nonlocal Fokas–Lenells equation to ground the article in published work. The nonlocal Fokas–Lenells equation is an integrable nonlinear evolution equation in which the field at $(x,t)$ is coupled to its reflected value at $(-x,-t)$. In the scalar form emphasized in the bilinear treatment, it is

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$

with nonlocality carried by the term $u(-x,-t)$ (Liu et al., 2021). Within the broader integrable-systems literature, this equation arises as a nonlocal reduction of a coupled system and admits exact construction by Hirota bilinearization and double Wronskian techniques; in parallel, multicomponent analogues have been formulated on Hermitian symmetric spaces, where the same space-time reflection mechanism yields vector and matrix nonlocal Fokas–Lenells models compatible with Lax and bi-Hamiltonian structures (Liu et al., 2021, Gerdjikov et al., 2021).

1. Definition and reduction structure

In the scalar setting, the nonlocal Fokas–Lenells equation is obtained by imposing the reduction

$v(x,t)=\delta\,u(-x,-t)$

on the coupled pKN$(-1)$ system, so the reflected field enters directly into the nonlinear term (Liu et al., 2021). This reduction is a space-time reflection rather than a local conjugation constraint, and it distinguishes the nonlocal model from the classical Fokas–Lenells equation.

The multicomponent formulation generalizes the same principle. There, the generic fields are $u^\alpha(x,t)$ and $v^\alpha(x,t)$, and one of the two principal reductions is the nonlocal constraint

$v^\alpha(x,t)=\eta\,u^\alpha(-x,-t)^*,$

with $\eta=\pm 1$ (Gerdjikov et al., 2021). The paper on Hermitian symmetric spaces describes this as the reduction denoted R2 and states that it produces nonlocal integrable models akin to PT- and CPT-symmetric integrable systems.

For the A.III vector case, the nonlocal equation is written as

$(-x,-t)$0

where $(-x,-t)$1 (Gerdjikov et al., 2021). In the one-component reduction this becomes a scalar nonlocal Fokas–Lenells equation with reflected complex-conjugate dependence. A common misconception is that “nonlocal” here means merely an integro-differential modification; in this literature it specifically denotes the reflected dependence on $(-x,-t)$2 imposed by a reduction symmetry.

2. Bilinear formulation from the unreduced system

The bilinear approach begins from the unreduced pKN$(-x,-t)$3 system and introduces the dependent-variable transformation

$(-x,-t)$4

Under this substitution, the system is rewritten in Hirota bilinear form as

$(-x,-t)$5

$(-x,-t)$6

$(-x,-t)$7

$(-x,-t)$8

(Liu et al., 2021). These are the unreduced bilinear equations from which both classical and nonlocal reductions are derived.

The corresponding exact solutions are expressed by double Wronskian determinants. In the notation of that construction,

$(-x,-t)$9

The Wronskian entries are generated from basic column vectors

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$0

where $$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$1 is a constant invertible matrix and $$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$2 are constant vectors (Liu et al., 2021). This determinant framework is the constructive core of the scalar exact-solution theory.

3. Nonlocal reduction in the bilinear framework

To pass from the unreduced double Wronskian solution to the nonlocal Fokas–Lenells equation, the bilinear construction imposes

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$3

with invertible $$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$4 satisfying

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$5

(Liu et al., 2021). This is the reduction step that inserts the reflected arguments directly into the determinant data.

For the case $$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$6, the solution is written as

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$7

where

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$8

$$u_{xt} + u - 2i\delta\, u\,u(-x,-t)u_x = 0, \qquad \delta=\pm 1,$$9

The reduced bilinear system then becomes

$u(-x,-t)$0

$u(-x,-t)$1

$u(-x,-t)$2

(Liu et al., 2021). The appearance of both $u(-x,-t)$3 and $u(-x,-t)$4 is the bilinear signature of nonlocality in this formulation.

For explicit realization, the parameter matrix $u(-x,-t)$5 may be chosen diagonal or as a Jordan block. In the diagonal case, the elementary vector components can be written

$u(-x,-t)$6

with arbitrary $u(-x,-t)$7 (Liu et al., 2021). Distinct spectral choices then generate different classes of exact solutions.

4. Solution families and asymptotic behavior

The bilinear paper states that the reduction yields a full profile of solutions of the classical and nonlocal Fokas–Lenells equations, and it emphasizes as a notable new result the construction of solutions related to real discrete eigenvalues that had not been reported before in analytic approaches (Liu et al., 2021). The solution taxonomy depends on the spectral structure encoded in $u(-x,-t)$8.

Choice of $u(-x,-t)$9	Eigenvalue type	Solution nature
Diagonal	Complex	Multi-soliton, breathers
Diagonal	Real	Multi-periodic or double periodic
Jordan block	Real	Algebraically decaying solitary waves

When the eigenvalues are complex and nonreal, the resulting solutions include localized solitary pulses, breathers, and their interactions. When the eigenvalues are real and distinct, the solutions are described as multi-periodic or “double periodic” in nature. For repeated real eigenvalues represented by Jordan blocks, the resulting waves have algebraic decay rather than exponential localization.

The algebraically decaying solutions are singled out in the paper as a new feature. Their asymptotics are described by a decay law of the type

$v(x,t)=\delta\,u(-x,-t)$0

and the discussion states that such waves decay like $v(x,t)=\delta\,u(-x,-t)$1 and exhibit no phase shift after interaction (Liu et al., 2021). By contrast, the summary table in that work associates exponential decay and phase shift with diagonal complex-eigenvalue solutions.

The same source also remarks that, in the nonlocal case, the decoupled eigenvalue sets between $v(x,t)=\delta\,u(-x,-t)$2 and $v(x,t)=\delta\,u(-x,-t)$3 make two-soliton and higher interactions significantly more diverse and complex than in local FL/NLS equations (Liu et al., 2021). This suggests that the nonlocal reduction does not merely constrain the local solution space; it also changes the interaction geometry available to exact solutions.

5. Multicomponent and symmetric-space generalizations

A separate development formulates multicomponent Fokas–Lenells equations associated with irreducible Hermitian symmetric spaces of types A.III, BD.I, C.I, and D.III (Gerdjikov et al., 2021). In that framework the equations are constructed through Lax pairs valued in the corresponding symmetric spaces, and two reductions are considered, one of which yields a nonlocal integrable model.

For the A.III matrix case with $v(x,t)=\delta\,u(-x,-t)$4, the nonlocal equation is

$v(x,t)=\delta\,u(-x,-t)$5

For the BD.I case, the reduced equation contains additional terms involving a signature matrix $v(x,t)=\delta\,u(-x,-t)$6 dictated by the Lie-algebraic structure. These examples show that the scalar reflected coupling extends naturally to vector and matrix nonlinearities built from $v(x,t)=\delta\,u(-x,-t)$7 and $v(x,t)=\delta\,u(-x,-t)$8 or $v(x,t)=\delta\,u(-x,-t)$9.

The same paper explicitly states that it does not present explicit soliton or general solutions for the nonlocal multicomponent equations, but instead establishes their integrability through the underlying formalism (Gerdjikov et al., 2021). This corrects a possible misunderstanding: the scalar bilinear paper develops concrete exact solutions, whereas the symmetric-space paper primarily develops the structural framework in which nonlocal multicomponent models are defined.

6. Integrability, related models, and scope

The symmetric-space treatment provides the general Lax representation

$(-1)$0

and states that under the nonlocal reduction the Lax operators remain the same while the potential satisfies

$(-1)$1

with $(-1)$2 an involutive automorphism depending on the symmetric space (Gerdjikov et al., 2021). The result is that the Lax pair is compatible with the nonlocal reduction, ensuring integrability of the reduced equation.

The same work further states that the Hamiltonian functionals and operators survive the nonlocal reduction and that the nonlocal Fokas–Lenells equations inherit the bi-Hamiltonian nature of the unreduced models (Gerdjikov et al., 2021). In this sense, nonlocality is implemented as a symmetry reduction of an existing integrable hierarchy rather than as an ad hoc modification that destroys its algebraic structure.

The bilinear paper adds a further connection by obtaining solutions to the two-dimensional massive Thirring model from those of the Fokas–Lenells equation (Liu et al., 2021). Within the data presented, this places the nonlocal Fokas–Lenells equation at an intersection of several standard integrable-systems themes: bilinearization, determinant solutions, reduction groups, symmetric-space Lax structures, and model-to-model correspondence. A plausible implication is that the nonlocal Fokas–Lenells equation functions less as an isolated PDE than as a reduction node within a broader web of integrable negative-flow and symmetric-space constructions.