In this article we derive the triangular representations of the fundamental eigensolutions of the focusing 1 + 1 AKNS system with symmetric nonvanishing boundary conditions. Its continuous spectrum equals
, where μ is the absolute value of the AKNS solution at spatial infinity. We also study the behavior of the scattering coefficients near the endpoints ±iμ of the branch cut, where we distinguish between the generic case and the exceptional case.
Nonlinear Schrödinger (NLS) equations have been fundamental in modeling nonlinear wave phenomena in plasmas [22,34], deep water surfaces [1,4,38], optical fibres [1,18,28], ferromagnetic materials [11,37], and Bose–Einstein condensates [26,27]. NLS equations with solutions decaying at infinity have been studied in detail [2–4,9,16]. After finding the Peregrine solutions , significant results on NLS equations with nonvanishing boundary conditions have been reported in Akhmediev et al. [5,6], Akhmediev and Korneev , Its et al. , Mihalache et al. , Tajiri and Watanabe , Zakharov and Gelash [35,36]. The direct and inverse scattering theory of the focusing NLS equation with nonvanishing boundary conditions has been studied systematically in Biondini and Kovačić , Demontis et al. . We shall frequently refer to Demontis et al.  for some of the direct scattering results.
In this article we consider the focusing 1 + 1 AKNS system
where v is a function of
with values in
, σ3 = diag(1, −1), and
is the potential. We assume that there exist two distinct 2 × 2 matrices
satisfying μ = |qr| = |ql| > 0 such that for some s ≥ 0 the integrability condition
holds. Condition (1.2) is usually assumed for s = 0, 1. The Lax pair equations whose compatibility condition is equivalent to the focusing NLS equation, are discussed in detail in Appendix B (cf. ).
Under condition (H0) and for
, there exist two fundamental eigensolutions
of (1.1) satisfying the asymptotic conditions
Under condition (H1) the fundamental eigensolutions can be defined for
. Their existence can easily be proven by iterating the Volterra integral equations 
We observe that the corresponding matrix groups are given by
appears in (1.6) only as the argument of even functions and hence the sign indeterminacy in defining λ(k) by means of a square root does not affect (1.6).
In this article we derive in a rigorous way the triangular representations
where for each
the integrability condition
holds. Triangular representations are well-known under vanishing boundary conditions [13,27–29] but have never been derived under integrability conditions of the form (1.9), with one notable exception. In Demontis et al.  the representations (1.8) have been derived under the conditions (H1) and
at the expense of replacing (1.9) by the quadratic integrability condition
After establishing the triangular representations of the fundamental eigensolutions, we introduce the conformal mapping λ(k) by (1.7) and distinguish between left and right versions of k ∈(−iμ, iμ) as boundary points of the analytic manifolds
in 1,1-correspondence with the complex half-planes
. We then go on to define the Jost functions and to derive their triangular representations. We also study in detail the conjugate transposition and conjugate non-transposition symmetries of the various quantities.
Once the triangular representations have been established, we go on studying the asymptotic behavior of the scattering coefficients as the spectral parameter k tends to ±iμ. In analogy with the case of the Schrödinger equation on the line [10,12,15], we shall make the distinction between the generic case where the corresponding Jost solutions are linearly independent, and the exceptional case where these solutions are proportional.
This paper is organized as follows. In Section 2 we establish the triangular representations of the fundamental eigensolutions. Their asymptotic behavior at either end of the real x-line will be the topic of Section 3. We then introduce the conformal mapping λ(k), define the Jost functions, and derive their triangular representations in Section 4. Section 5 is devoted to the conjugation symmetry properties of the various quantities. Next, in Section 6 we investigate the asymptotic behavior of the scattering coefficients near the endpoints of the branch cut k ∈[−iμ, iμ] and prove the integrability of the Fourier transforms of the reflection coefficients.
Finally, we discuss the contents of the various appendices. In Appendix A we discuss the Wiener algebra of constants plus Fourier transforms of L1-functions and invertibility within this algebra, well-known material treated at length in Gelfand et al. . In this appendix we introduce the notation of writing Zn×m as the set of n × m matrices with entries in Z. The time dependence of the scattering data is discussed in Appendix B. To avoid clogging notations in the main body of the paper, we do not indicate any time dependence unless it is absolutely necessary.
2. TRIANGULAR REPRESENTATIONS
In this section we derive the triangular representations of the fundamental eigensolutions in a rigorous way. We also relate the potential to the integral kernels appearing in the triangular representations.
Factoring out the asymptotic behavior of the fundamental eigensolutions, we write
Then, under condition (H1) and for
, we easily derive from (1.5) the Volterra integral equations
These equations can be written in the form
where the triangular representations (1.8) are putatively written as
We may thus convert the integral equations (2.3) into Volterra integral equations for the integral kernelsJ(x, x − α) and K(x, x + α). As in the vanishing case [3,13], we shall derive estimates for the integral kernels from the Volterra integral equations they satisfy.
Although at first sight the procedure explained in the past few lines may seem circular, below we shall in fact prove the existence of the integral kernels satisfying (1.9) by applying Gronwall’s inequality to the putative integral equations obtained by Fourier transforming the Volterra integral equations (2.3). By Fourier transformation of their unique solutions, we then arrive at the triangular representations for the fundamental eigensolutions satisfying (2.3), thus completing their proof.
The triangular representations (2.4) can be viewed as integral transforms of the type described by the following result.
Suppose the entries of F belong to
. Then the integral transform
allows the inversion formula
Let us first convert the sinc and sinc2 transforms into cosine transforms, where
Then (2.7) is true, while (2.8) implies
We now observe that
does not change when changing the sign of λ while keeping k invariant. Decomposing
into k-even and k-odd functions of
, we obtain
Applying Fourier cosine transform inversion and using (2.9) we obtain the inversion formula (2.6), as claimed.
Let us now derive the triangular representations of the fundamental eigensolutions following the procedure explained above.
Let condition (H1) be satisfied. Then there exist integral kernels J(x, y) and K(x, y) satisfying
such that the triangular representations
hold true. Moreover, if condition (Hs+1) is true for some s ≥ 0, then for
the integral kernels J(x, y) and K(x, y) satisfy the estimates
We prove the triangular representation (2.11a) first, then use a symmetry argument to derive (2.11b) within a few lines, and then establish the second part of the theorem involving condition (Hs+1) for s = 0, 1, 2, … by going through the modifications required in the estimates. The second part for noninteger s ≥ 0 then follows by an interpolation argument.
We shall derive L1-estimates for the inhomogeneous term Finh(α) and the J-dependent term Fh(α) and then apply Gronwall’s inequality to obtain an L1-estimate for J.
Let us compute the inhomogeneous term on the right-hand side, splitting it into k-even and k-odd parts. Using
, we get under condition (H2) for the k-even component of the inhomogeneous term
Similarly, under condition (H2) we obtain for the k-odd component of the inhomogeneous term divided by ik
Applying (2.9) to compute the Fourier cosine transform of Finh(α), we see that the terms containing factors of the form
cancel out. Using (2.8) we get
Consequently, under condition (H2) we get
Using an approximation argument, the estimate (2.14) is easily shown to hold under the more general condition (H1).
Let us now write the J-dependent term on the right-hand side of (2.3a) as a Fourier cosine transform. To do so, we use the trigonometric formulae
, as well as the trigonometric formula
. We thus get for the k-even component of the J-dependent term
For the k-odd component of the J-dependent term divided by ik we obtain
Applying (2.9) to compute the Fourier cosine transform of Fh(α), we see that the terms containing factors of the form
cancel out. Unfortunately, only half the terms containing factors
do. Using (2.8) we get
We now strip off the Fourier cosine transform using (2.8), directly in the terms containing a factor cos(λα) or
and indirectly after interchanging α and β in the terms containing a factor cos(λβ) or
. To avoid repeating nearly similar estimates, we write the cosine transform terms as the Fourier cosine transforms of various matrix functions of the form
where z − α ≤ z and z − α ≤ 2x − z − α for each α ≥ 0. In fact,
in the first three lines of (2.15),
in the fourth line of (2.15), and
in the fifth line of (2.15). Integrating (2.16a) and (2.16b) with respect to
we obtain the upper bounds
as well as
Using (2.14) and the various meanings of
, under condition (H2) we obtain for the J-dependent term
which is easily shown to hold under the more general condition (H1).
Applying Gronwall’s inequality  to the inequality
thus proving the triangular representation (2.11a).
The proof of the triangular representation (2.11b) is based on a simple parity symmetry argument. In fact, letting Q(#)(x) = Q(−x) we switch the roles of Qr and Ql by using
and obtain the following symmetry relations for the fundamental eigensolutions:
With the help of Gronwall’s inequality we then derive the final result for s = 0, 1, 2, ….
For noninteger s ≥ 0 we apply an interpolation argument  based on the Hölder estimate
where N ≤ s ≤ N + 1.
Let us now derive expressions to pass from the (1, 2)-elements of the integral kernel J(x, y) and K(x, y) to the potential Q(x). These expressions have been derived by different means in Demontis et al. [14, Eq. (3.5)] under the assumption that (H2) is valid and
Under condition (H1) we have
It suffices to extend the expression obtained in Demontis et al.  to general potentials satisfying (H1). Taking
in the expression
In this section we view (1.7) as a conformal mapping from a suitable k-manifold to a suitable λ-manifold and define the Jost functions. We also derive the triangular representations of the Jost functions. Finally, we introduce the scattering coefficients and the reflection coefficients and derive their representations as Fourier transforms.
4.1. Conformal Mapping
Let us now view
as the conformal mapping from the complex k-plane
cut along [−iμ, iμ] onto the complex λ plane
that satisfies λ ∼ k at infinity. Allowing each k ∈(−iμ, iμ) to have a left and a right copy to be put into 1, 1-correspondence with λ ∈(−μ, 0) and λ ∈(0, μ), respectively, we create diffeomorphisms between the analytic manifolds
and the open complex half-planes
and between the analytic manifolds with boundary
and the closed complex half-planes
. Doing it this way, many functions can be interchangeably viewed as functions of k and as functions of λ (Figure 1).
The following Fourier representation is true [24, 10.22.61]:
is the Bessel function of order one. Obviously, the left-hand side of (4.1) is the Fourier transform of a function in
and hence is continuous in
, is analytic in
, and vanishes as λ → ∞ from within the closed upper half complex λ-plane. For
we obtain by complex conjugation
4.2. Definition of Jost Functions
stand for the 2 × 2 matrix whose columns are the eigenvectors of Λr,l(k), i.e., letting
we define the Jost functions as ϕ(x, k) and ψ(x, k) for
The Jost functions in (4.4a) can also be defined for
and those in (4.4b) for
by computing the corresponding columns of Wr,l(k) for k in the complementary manifold. For
(and hence for λ ∈ (−∞, −μ] ∪ [μ, +∞) the Jost functions coincide when defined either way. For
the Jost matrices.
4.3. Definition of Scattering Coefficients
where S(k) and
are each other’s inverses, we obtain
are written in terms of the traditional a, b, c, and d functions [3, Ch. 2]. Consequently,
Using the identity
are the rows of Wr,l(k), we obtain
and similarly for the entries of
. Thus the scattering coefficients a(k) and c(k) are well-defined for
are well-defined for
, and the off-diagonal scattering coefficients for
, with the possible exception of k = ±iμ. The entries of s(k) and
may not be defined for k = ±iμ but they are when multiplied by
4.4. Triangular Representations of Jost Solutions
Using the triangular representations (2.11) and the Fourier representations (4.1) and (4.2), we get
Consequently, using that J1(w) = w + O(w3) as w → 0+, we get
Using the Wiener algebras defined in Appendix A, it is easily verified that
. The (1, 1)-element of (4.15a) and the (2, 2)-element of (4.15b) belong to
. The (2, 2)-element of (4.15a) and the (1, 1)-element of (4.15b) belong to
Let us now define the reflection coefficients
, we obtain the identities
where for the moment we leave open the existence of the reciprocals in (4.16) and (4.17). Hence, proving the reflection coefficients to belong to
is postponed to Section 6.
In this section we derive the matrix conjugate transpose symmetry properties and nontranspose conjugate symmetry properties for Jost functions and scattering and reflection coefficients. The dagger denotes the matrix conjugate transpose.
a. Conjugate transposition symmetry. For
the matrix functions
both satisfy the differential equation
as do the matrix functions
. We observe that
for k ∈[−iμ, iμ]. Thus for
the fundamental eigensolutions
are unitary matrices of determinant 1. We also get
where in (5.4a) the choice of the sign in defining λ from k does not matter. In (5.4b) this choice is to be made consistently. Using (2.4), (3.8), and (5.4a) we obtain for the integral kernels
Using (4.7) and (4.4) we obtain for the Jost matrices
as far as the second row of (5.6a) and the first row of (5.6b) are concerned and
as far as the first row of (5.6b) and the second row of (5.6b) are concerned. Using (4.5) we obtain for the matrix of scattering coefficients
Thus S(k) and
are unitary matrices if
. Since S(k) and
both have unit determinant, we get
provided the reciprocals in their definitions (4.16) exist.
b. Conjugation symmetry. Let
stand for the second Pauli matrix. Then it is easily verified that
both satisfy the differential equation (1.1). The same thing is true for the other fundamental eigensolution
. We thus obtain
It is well-known that in the scattering theory of the Schrödinger equation on the line with Faddeev class potential two cases can be distinguished [10,12,15]: the generic case where for k = 0 the two Jost functions are linearly independent, and the exceptional case where for k = 0 the two Jost functions are linearly dependent. The scattering theory in the exceptional case is more easily developed by strengthening the integrability condition on the potential (as done in Chadan and Sabatier , Deift and Trubowitz  and Faddeev ), though such strengthening can be avoided at the expense of more complicated mathematical arguments . For reflectionless potentials we are always in the exceptional case.
In the theory of the Schrödinger equation on the line with Faddeev class potential we can actually prove that there are no spectral singularities . In fact, for positive energy k2 the two Jost functions can be proven to be linearly independent. In the present situation we actually need to assume absence of spectral singularities. Indeed, it is well-known that a(k) = c(k) for
. Therefore, throughout this article we assume absence of spectral singularities:
There do not exist any
where a(k), c(k),
As a result, under this assumption the reflection coefficients ρ(k) and r(k) are well-defined for
and the reflection coefficients
are well-defined for
. Moreover, b(k),
, d(k), and
are well-defined for
. Their definitions for k = ±iμ are a different matter to be pursued presently.
it is clear that the Jost functions ϕ(x, iμ) and ψ(x, iμ) are linearly independent iff the Jost functions
are linearly independent. As in the Schrödinger case, we can therefore make a distinction between the following two cases:
the generic case: the Jost functions ϕ(x, iμ) and ψ(x, iμ) are linearly independent. OR: the Jost functions
are linearly independent.
the exceptional case: the Jost functions ϕ(x, iμ) and ψ(x, iμ) are linearly dependent. OR: the Jost functions
are linearly dependent.
Suppose condition (H1) is satisfied. Then we are in the generic case if and only if
exists and is nonzero. If this limit vanishes, we are in the exceptional case.
Since det(ϕ(k, x) ψ(k, x)) does not depend on
, we compute their determinants as x → ∞ and x → +∞ and obtain
thus proving again that a(k) = c(k). Since these determinants have the finite limit
As k → iμ from within
, we arrive at the desired conclusion.
We now observe that
Hence, in the exceptional case the (proportional) Jost functions ϕ(x, iμ) and ψ(x, iμ) are bounded in
and have finite nonzero limits as x → ± ∞.
Let us assume condition (H1) in the generic case and condition (H2) in the exceptional case, as well as absence of spectral singularities. Let us also assume that a(k) does not vanish as k → iμ. Then the reflection coefficients ρ(k),
, r(k), and
are Fourier transforms of functions in
. Moreover, there are only finitely many discrete eigenvalues.
In the absence of spectral singularities and in the generic case, the four reflection coefficients are all Fourier transforms of functions in
[cf. Appendix A]. Moreover, in this case there are at most finitely many discrete eigenvalues.
It remains to consider the exceptional case in detail. To do so, we strengthen the integrability assumption on the potential by assuming condition (H2). Since
can then easily be shown to be the Fourier transforms (in λ) of functions in
, respectively, we can then apply Taylor’s theorem and write
where in the exceptional case we must have a−1 = 0. We need to prove that, in the exceptional case, a0 ≠ 0 and b−1 = 0. Equations (5.7) and (5.8) imply
where a(k) = c(k). Substituting (6.2) in the (2, 2)-element of either equation we get
Substituting (6.2) [with λ* = −λ] in the (1, 2)-element of the left equation we get
The Ansatz b−1 = 0 and d−1 ≠ 0 leads to b0 = 0 and |a0| = 1, so that
, a contradiction. In the same way we arrive at a contradiction from the Ansatz b−1 ≠ 0 and d−1 = 0. We must therefore conclude that b−1 = d–1 = 0. Instead of (6.2b), we thus arrive at the identities
is a real number. Furthermore,
where the factors in either matrix product have unit determinant. Computing the two matrix products we get
This leads to two mutually exclusive possibilities:
a0 = 0 and b0d0 = 1. Since
with a(k) ≠ 0 for values of
approaching iμ, the absence of spectral singularities assumption implies that |b0| = |d0|. Consequently, there exists a phase
In the former case the reflection coefficients are Fourier transforms of functions in
, whereas in the latter case the reflection coefficients blow up as k → iμ.
Now observe that
is well-defined. Thus the identity
implies that in the exceptional case
where the proportionality constant is nonzero. In the same way we prove that
where the proportionality constant is nonzero as k = ±iμ. Since these two proportionality constants have product 1,
, we get
The proof of Theorem 6.2 forced us to consider the mutually exclusive versions of the exceptional case, denoted by (a) and (b). In the generic case and in the exceptional case (a) the reflection coefficients are Fourier transforms of L1-functions. Unfortunately this is no longer the case in the exceptional case (b), the so-called hyperexceptional case for want of a better term. At present we cannot exclude the occurrence of the hyperexceptional case, but we are not aware of any focusing potential leading to the hyperexceptional case either.
CONFLICTS OF INTEREST
The author declares no conflicts of interest.
The author is greatly indebted to Martin Klaus for discussions on 1 + 1 AKNS systems with nonvanishing boundary conditions and the occurrence of spectral singularities for piecewise constant potentials, and to Francesco Demontis for carefully reading the manuscript. The author has been partially supported by the Fondazione Banco di Sardegna in the framework of the research program Integro-Differential equations and non-local problems and by the Regione Autonoma della Sardegna in the framework of the research program Algorithms and models for imaging science, and by Istituto Nazionale della Alta Matematica, Gruppo Nazionale per la Fisica Matematica (INdAM-GNFM).
1. WIENER ALGEBRAS
By the (continuous) Wiener algebra
we mean the complex vector space of constants plus Fourier transforms of L1-functions
endowed with the norm |c| + ||h||1. Here we define the Fourier transform as follows:
. The invertible elements of the commutative Banach algebra
with unit element are exactly those
for which c ≠ 0 and
has the two closed subalgebras
consisting of those
such that h is supported on
, respectively. The invertible elements of
are exactly those
for which c ≠ 0 and
stand for the (nonunital) closed subalgebras of
consisting of those
for which c = 0, we obtain the direct sum decomposition
By Π± we now denote the (bounded) projection of
. Then Π+ and Π– are complementary projections. In fact,
for some p ∈ (1, +∞). These direct sum decompositions coupled by the Fourier transform can be schematically represented as follows:
Now observe that
acts as an isometric linear 1, 1-correspondence from
. If we define the norm of
, we obtain the direct sum decomposition
where the projection
is the restriction of an arbitrary
to the half-line
Throughout this article we denote the vector spaces of n × m matrices with entries in
, respectively. We write
for the vector spaces of n × m matrices with entries in
, respectively. Using a suitable (i.e., submultiplicative) matrix norm, we can turn all of these vector spaces into Banach spaces. It is then clear that
are noncommutative Banach algebras with unit element and
are (nonunital) noncommutative Banach algebras. The projections Π± can be extended in a natural way to matrices of Wiener algebra elements.
The following result is most easily proved using the Gelfand theory of commutative Banach algebras  but was known before to Wiener [32,33].
Theorem Appendix A.1.
If for some complex number h∞and some
the Fourier transform
and if h∞ ≠ 0, then there exists
1. TIME DEPENDENCE OF THE SCATTERING DATA
The focusing NLS equation
, arises as the compatibility condition of the Lax pair equations [2,3,16]
where v is a nonsingular 2 × 2 matrix function.
Using that Ψ, Φ, and v all satisfy the first order system (1.1), we can write Ψ = vC+ and Φ = vC−, where C± do not depend on
(but do depend on k and t). Since
TY - JOUR
AU - Cornelis van der Mee
PY - 2020
DA - 2020/12
TI - Focusing NLS Equations with Nonzero Boundary Conditions: Triangular Representations and Direct Scattering
JO - Journal of Nonlinear Mathematical Physics
SP - 68
EP - 89
VL - 28
IS - 1
SN - 1776-0852
UR - https://doi.org/10.2991/jnmp.k.200922.006
DO - https://doi.org/10.2991/jnmp.k.200922.006
ID - vanderMee2020