In this article, we focus on the inverse scattering transformation for the Fokas–Lenells (FL) equation with nonzero boundary conditions via the Riemann–Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity, symmetry and asymptotic behavior of the Jost solutions and scattering matrix are discussed in detail. With these results, we further present a generalized RH problem, from which a reconstruction formula between the solution of the FL equation and the Riemann–Hilbert problem is obtained. The N-soliton solutions of the FL equation is obtained by solving the RH problem.
with ν = ±1 is a significant mathematical and physical model for the optical fibers, deep water waves and plasma physics . The NLS equation is a completely integrable system which admits Lax pair and bi-Hamiltonian formulation [21,22].
In the late 70s, standard bi-Hamiltonian formulation was used to obtain an integrable generalization of a given equation . For instance, the Camassa–Holm equation is derived from the KdV equation via the bi-Hamiltonian structure , and the same mathematical trick can be applied to the NLS equation yields Fokas–Lenells (FL) equation 
where α, β > 0. In this paper, we consider the focusing case with σ = −1.
In recent years, much work has been done on the FL equation (1.3). For example, Lax pair for FL equation was obtained via the bi-Hamiltonian structure by Fokas and Lenells . They further considered the initial-boundary problem for the FL equation (1.3) on the half-line by using the Fokas unified method . The dressing method is applied to obtain an explicit formula for bright-soliton and dark soliton solutions for Eq. (1.3) [11,15]. The bilinear method was used to obtain bright and dark soliton solution are obtained [12,13]. The Darboux transformation is used to obtain rogue waves of the FL equation [10,19]. The Deift–Zhou nonlinear steepest decedent method was used to analyze long-time asymptotic behavior for FL equation with decaying initial value . Riemann–Hilbert (RH) approach was adopted to construct explicit soliton solutions under zero boundary conditions . As far as we know, the soliton solutions for the FL equation (1.3) with Nonzero Boundary Conditions (NZBCs) have not been reported. In this paper, we apply RH approach to study the inverse transformation of the FL equation (1.3) with the following NZBCs
and assume that u(x, t) − q± ∈ L1(R±); with respect to x for all t ≥ 0. In next section, we will see that this kind of NZBCs (1.4) avoids the discussion on a multi-valued function as the case of the NLS equation .
The inverse scattering transform is an important method to study important nonlinear wave equations with Lax pair such as the NLS equation, the modified KdV equation, Sine-Gordan equation [5,14]. As an improved version of inverse scattering transform, the RH method has been widely adopted to solve nonlinear integrable systems [2,4,16,17,20,23].
The paper is organized as follows. In Section 2, by introducing appropriate transformations, we change the asymptotic boundary conditions (1.4) into constant boundary conditions. Furthermore, we analyze the analyticity, symmetry and asymptotic behavior of eigenfunctions and scattering matrix associated with the Lax pair. In Section 3, a generalized RH problem for the FL equation is constructed, and the distribution of discrete spectrum and residue conditions associated with RH problem are discussed. Based on these results, we reconstruct the potential function from the solution of the RH problem. In Section 4, we give the N-soliton solutions via solving RH problem under reflectionless case.
Take A = 2C and B = 2D in the above equation and consider the limit as x → ±∞, one gets
These two matrices are proportional if
Moreover, T± and X± are proportional
which implies that their eigenvalues are proportional and they share the same eigenvector matrices.
To diagonalize the matrices X± and T± and further obtain the Jost solutions, we need to get the eigenvalues and eigenvector matrices of them. Direct calculation shows that the eigenvalues of the matrices X± are
; and the eigenvalues of the matrices T± are ±iηλ, where
and the corresponding eigenvector matrices are
With the above results, the matrices can be diagonalized as:
Thus, the asymptotic Lax pair
can be written as
which has a solution
Therefore, the asymptotic of the eigenfunctions φ± are
Therefore, we obtain the limit of the Jost solutions as k → ∞:
then we have
In addition, the asymptotic property of φ± follows:
Consider the wronskians expression of the scattering coefficients (2.17), we find that
2.3.2. Asymptotic as k → 0
We first assume that J± admit a Lorrent expansion
Substituting into (2.13) and comparing the coefficients of the same order of k gives that
. Thus we expend solution of (2.13) of the form
then we obtain the comparison results:
It is easy to check that
is a constant independent on x. This implies that the following limit
is uniformly convergent for x ∈R. While the following limit
exists for every fixed k ∈C. Thus the following limit is commutative, and using (2.32) and (2.33) yields
which leads to
The asymptotics of functions φ± and μ± can be obtained by transformations (2.12) and (2.26). Consider the wronskian expressions of scattering coefficients and note the boundary conditions, we find that
Under the transformation (2.26), we write the Lax pair as
We rewrite the Eq. (2.35) into a full differential form
which implies μ± can be formally integrated to obtain the integral equations for the eigenfunctions:
By using a similar way to Appendix A in Biondini and Kovačič , under mild integrability conditions on the potential, the eigenfunctions (2.37) and (2.38) can be analytically extended in the complex k-plane into the following regions:
where μ± = (μ±,1,μ±,2), the subscript 1, 2 denote the first and second column of μ±.
Apparently, the functions φ± and μ± share the same analyticity, hence from the wronskians expression of the scattering coefficients, we know that s11 is analytic in D−, and s22 is analytic in D+.
To investigate the discrete spectrum and residue conditions in the Riemann–Hilbert problem, one needs to analyze the symmetric property for the solutions φ± and the scattering matrix S(k).
The Jost eigenfunctions satisfy the following symmetric relations
and the scattering matrix satisfies
Proof. We only prove (2.40) and (2.42), (2.41) and (2.43) can be shown in a similar way. The functions φ± are the solutions of the spectral problem
Conjugating and multiplying left by σ on both sides of the equation gives
Note that σX*(k*) = X(k)σ, hence
is also a solution of (2.44). By using relations
, we obtain
TY - JOUR
AU - Yi Zhao
AU - Engui Fan
PY - 2020
DA - 2020/12/10
TI - Inverse Scattering Transformation for the Fokas–Lenells Equation with Nonzero Boundary Conditions
JO - Journal of Nonlinear Mathematical Physics
SP - 38
EP - 52
VL - 28
IS - 1
SN - 1776-0852
UR - https://doi.org/10.2991/jnmp.k.200922.003
DO - 10.2991/jnmp.k.200922.003
ID - Zhao2020