Journal of Nonlinear Mathematical Physics

Volume 28, Issue 1, March 2021, Pages 68 - 89

Focusing NLS Equations with Nonzero Boundary Conditions: Triangular Representations and Direct Scattering

Authors
Cornelis van der Mee*
Dip. Matematica e Informatica, Universit’a di Cagliari, Via Ospedale 72, Cagliari 09124, Italy
Corresponding Author
Cornelis van der Mee
Received 10 December 2019, Accepted 6 May 2020, Available Online 10 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.200922.006How to use a DOI?
Keywords
AKNS system, triangular representation
Abstract

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 𝕉[-iμ,iμ] , 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 ± of the branch cut, where we distinguish between the generic case and the exceptional case.

Copyright
© 2020 The Author. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

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 [24,9,16]. After finding the Peregrine solutions [25], significant results on NLS equations with nonvanishing boundary conditions have been reported in Akhmediev et al. [5,6], Akhmediev and Korneev [7], Its et al. [19], Mihalache et al. [23], Tajiri and Watanabe [29], 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 [8], Demontis et al. [14]. We shall frequently refer to Demontis et al. [14] for some of the direct scattering results.

In this article we consider the focusing 1 + 1 AKNS system

vx=(-ikσ3+Q)v, (1.1)
where v is a function of x𝕉 with values in 𝕉2 , σ3 = diag(1, −1), and Q=(0q-q*0) is the potential. We assume that there exist two distinct 2 × 2 matrices Qr=(0qr-qr*0) and Ql=(0ql-ql*0) satisfying μ = |qr| = |ql| > 0 such that for some s ≥ 0 the integrability condition
(Hs)0dy(1+|y|)s(Q(-y)-Ql+Q(y)-Qr)<+ (1.2)
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. [13]).

Under condition (H0) and for k𝕉(-iμ,iμ) , there exist two fundamental eigensolutions Φ˜(x,k) and Ψ˜(x,k) of (1.1) satisfying the asymptotic conditions

Φ˜(x,k)=exΛl(k)[I2+o(1)],x-, (1.3a)
Ψ˜(x,k)=exΛr(k)[I2+o(1)],x+, (1.3b)
where
Λl(k)=-ikσ3+Ql,Λr(k)=-ikσ3+Qr. (1.4)

Under condition (H1) the fundamental eigensolutions can be defined for k𝕉[-iμ,iμ] . Their existence can easily be proven by iterating the Volterra integral equations [14]

Φ˜(x,k)=exΛl(k)+-xdye(x-y)Λl(k)[Q(y)-Ql]Φ˜(y,k)eyΛl(k), (1.5a)
Ψ˜(x,k)=exΛr(k)-xdye-(y-x)Λr(k)[Q(y)-Qr]Ψ˜(y,k)eyΛr(k). (1.5b)

We observe that the corresponding matrix groups are given by

exΛr,l(k)=(cos(λx)I2+sin(λx)λQr,l)-iksin(λx)λσ3, (1.6)
where
λ(k)=k2+μ2 (1.7)
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

Φ˜(x,k)=exΛl(k)+-xdyJ(x,y)eyΛl(k), (1.8a)
Ψ˜(x,k)=exΛr(k)+xdyK(x,y)eyΛr(k), (1.8b)
where for each x𝕉 the integrability condition
-xdyJ(x,y)+xdyK(x,y)<+ (1.9)
holds. Triangular representations are well-known under vanishing boundary conditions [13,2729] but have never been derived under integrability conditions of the form (1.9), with one notable exception. In Demontis et al. [14] the representations (1.8) have been derived under the conditions (H1) and qxL1(𝕉) at the expense of replacing (1.9) by the quadratic integrability condition
-xdyJ(x,y)2+xdyK(x,y)2<+. (1.10)

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 ∈(−, ) as boundary points of the analytic manifolds k𝕂± in 1,1-correspondence with the complex half-planes λ𝔺± . 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 ±. 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 ∈[−, ] 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. [17]. 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

Φ˜(x,k)=M˜(x,k)exΛl(k),Ψ˜(x,k)=N˜(x,k)exΛr(k). (2.1)

Then, under condition (H1) and for k𝕉[-iμ,iμ] , we easily derive from (1.5) the Volterra integral equations

M˜(x,k)=I2+0dαeαΛl(k)[Q(x-α)-Ql]M˜(x-α,k)e-αΛl(k), (2.2a)
N˜(x,k)=I2-0dαe-αΛr(k)[Q(x+α)-Qr]N˜(x+α,k)eαΛr(k). (2.2b)

These equations can be written in the form

M˜(x,k)-I2=+0dαeαΛl(k)[Q(x-α)-Ql]e-αΛl(k)+0dαeαΛl(k)[Q(x-α)-Ql](M˜(x-α,k)-I2)e-αΛl(k), (2.3a)
N˜(x,k)-I2=-0dαe-αΛr(k)[Q(x+α)-Qr]eαΛr(k)-0dαe-αΛr(k)[Q(x+α)-Qr](N˜(x+α,k)-I2)eαΛr(k), (2.3b)
where the triangular representations (1.8) are putatively written as
M˜(x,k)=I2+0dαJ(x,x-α)e-αΛl(k), (2.4a)
N˜(x,k)=I2+0dαK(x,x+α)eαΛr(k). (2.4b)

We may thus convert the integral equations (2.3) into Volterra integral equations for the integral kernels J(x, xα) 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.

Proposition 2.1.

Suppose the entries of F belong to L1(𝕉+;(1+α)dα) . Then the integral transform

F^(k)=0dαF(α)e±αΛr,l(k)=0dαcos(λα)Fe(α)ik0dαcos(λα)Fo(α) (2.5)
allows the inversion formula
F(α)=4π20dλcos(λα)[F^(k)+F^(-k)2-F^(k)-F^(-k)2ikσ3Qr,l]. (2.6)

Moreover,

F(α)=Fe(α)Fo(α)σ3Qr,l. (2.7)

Let us first convert the sinc and sinc2 transforms into cosine transforms, where Ls1(𝕉+)=L1(𝕉+;(1+|y|)sdy) . For FL11(𝕉+) we have

0dysin(λy)λF(y)=[-sin(λy)λydzF(z)]0+0dycos(λy)ydzF(z)=0dycos(λy)ydzF(z). (2.8)

Furthermore, for FL21(𝕉+) we have

0dy(sin(λy)λ)2F(y) =[-(sin(λy)λ)2ydzF(z)]0+0dysin(2λy)λydzF(z)=[-sin(2λy)λydwwdzF(z)]0+20dycos(2λy)ydwwdzF(z)=0dycos(λy)y/2dwwdzF(z)=0dycos(λy)y/2dz(z-y2)F(z).

Next, write the first line of (2.5) as

F^(k)=0dαF(α)e±αΛr,l(k)=0dαcos(λα)Fe(α)ik0dαcos(λα)Fo(α),
where
0dαcos(λα)Fe(α)=0dαF(α)(cos(λα)I2±sin(λα)λQr,l),0dαcos(λα)Fo(α)=0dαsin(λα)λF(α)σ3.

Then (2.7) is true, while (2.8) implies Fe(α)=F(α)±αdβF(β)Qr,l and Fo(β)=αdβF(β)σ3 . Consequently,

F(α)=Fe(α)±Fo(α)σ3Qr,l. (2.9)

We now observe that F^(k) does not change when changing the sign of λ while keeping k invariant. Decomposing F^(k) into k-even and k-odd functions of k𝕉[-iμ,iμ] , we obtain

F^(k)+F^(-k)2=0dαcos(λα)Fe(α),F^(k)-F^(-k)2ik=0dαcos(λα)Fo(α).

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.

Theorem 2.1.

Let condition (H1) be satisfied. Then there exist integral kernels J(x, y) and K(x, y) satisfying

-xdyJ(x,y)+xdyK(x,y)<+,x𝕉, (2.10)
such that the triangular representations
Φ˜(x,k)=exΛl(k)+-xdyJ(x,y)eyΛl(k), (2.11a)
Ψ˜(x,k)=exΛr(k)+xdyK(x,y)eyΛr(k), (2.11b)
hold true. Moreover, if condition (Hs+1) is true for some s ≥ 0, then for x𝕉 the integral kernels J(x, y) and K(x, y) satisfy the estimates
-xdy(1+x-y)sJ(x,y)+xdy(1+y-x)sK(x,y)<+. (2.12)

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.

Let us now prove (2.11a). Substituting (2.4a) into (2.4b) we get

0dα^J(x,x-α^)e-α^Λl(k)=0dα^eα^Λl(k)[Q(x-α^)-Ql]e-α^Λl(k)+0dα^eα^Λl(k)[Q(x-α^)-Ql]0dβ^J(x-α^,x-α^-β^)e-(α^+β^)Λl(k)=0dα^Finh(α^)e-α^Λl(k)+0dα^Fh(α^)e-α^Λl(k).

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 k2λ2=1-μ2λ2 , we get under condition (H2) for the k-even component of the inhomogeneous term

0dα^[cos(λα^)I2+sin(λα^)λQl][Q(x-α^)-Ql][cos(λα^)I2-sin(λα^)λQl]+0dα^(1-μ2μ2)sin2(λα^)σ3[Q(x-α^)-Ql]σ3=120dαcos(λα)[Q(x-α2)-Ql]+140dαsin(λα)λ(Ql[Q(x-α2)-Ql]-[Q(x-α2)-Ql]Ql)+0dα(sin(λα)λ)2(μ2[Q(x-α)-Ql]-Ql[Q(x-α)-Ql]Ql).

Similarly, under condition (H2) we obtain for the k-odd component of the inhomogeneous term divided by ik

0dα^[-sin(λα^)λσ3][Q(x-α^)-Ql][cos(λα^)I2-sin(λα^)λQl]+0dα^[cos(λα^)I2+sin(λα^)λQl][Q(x-α^)-Ql][sin(λα^)λσ3]=120dαsin(λα)λ[Q(x-α2)-Ql]σ3+0dα(sin(λα)λ)2(σ3[Q(x-α)-Ql]Ql+Ql[Q(x-α)-Ql]σ3).

Applying (2.9) to compute the Fourier cosine transform of Finh(α), we see that the terms containing factors of the form (sin(λα)λ)2 cancel out. Using (2.8) we get

0dαcos(λα)Finh(α)=0dαcos(λα){12[Q(x-α2)-Ql]+14αdβ(Ql[Q(x-β2)-Ql]+[Q(x-β2)-Ql]Ql)}. (2.13)

Consequently, under condition (H2) we get

0dαFinh(α)-xdz(1+μ(x-z))Q(z)-Ql. (2.14)

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

cos(λ(α^+β^))cos(λα^)=12(cos(λα)+cos(λβ)),sin(λ(α^+β^))sin(λα^)=-12(cos(λα)-cos(λβ)),sin(λ(α^+β))λcos(λα^)=12(sin(λα)λ+sin(λβ)λ),cos(λ(α^+β))sin(λα^)λ=12(sin(λα)λ-sin(λβ)λ),
where α=2α^+β^ , β=β^ , 0dα^0dβ^=120dα0αdβ=120dββdα , as well as the trigonometric formula
sin(λ(α^+β))λsin(λα^)λ=(sin(λα)λ)2cos2(λβ)-cos2(λα)(sin(λβ)λ)2=(sin(λα)λ)2-(sin(λβ)λ)2,
where α=α^+12β^ , β=12β^ , 0dα^0dβ^=20dα0αdβ=20dββdα . We thus get for the k-even component of the J-dependent term
0dα^0dβ^[cos(λα^)I2+sin(λα^)λQl][Q(x-α^)-Ql]×J(x-α^,x-α^-β^)[cos(λ(α^+β^))I2-sin(λ(α^+β^))λQl]+0dα^0dβ^(1-μ2λ2)sin(λα^)sin(λ(α^+β^))σ3[Q(x-α^)-Ql]J(x-α^,x-α^-β^)σ3=140dα0αdβ(cos(λα)+cos(λβ))[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)-140dα0αdβ(cos(λα)-cos(λβ))σ3[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)σ3+140dα0αdβ[sin(λα)λ-sin(λβ)λ]Ql[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)-140dα0αdβ[sin(λα)λ+sin(λβ)λ][Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)Ql-20dα0αdβ[(sin(λα)λ)2-(sin(λβ)λ)2](+Ql[Q(x-α+β)-Ql]×J(x-α+β,x-α-β)Ql+μ2σ3[Q(x-α+β)-Ql]J(x-α+β,x-α-β)σ3).

For the k-odd component of the J-dependent term divided by ik we obtain

-0dα^0dβ^sin(λα^)λσ3[Q(x-α^)-Ql]J(x-α^,x-α^-β^)×[cos(λ(α^+β))I2-sin(λ(α^+β^))λQl]+0dα^0dβ^[cos(λα^)I2+sin(λα^)λQl]×[Q(x-α^)-Ql]J(x-α^,x-α^-β^)σ3sin(λ(α^+β^))λ=-140dα0αdβ(sin(λα)λ-sin(λβ)λ)σ3[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)+140dα0αdβ(sin(λα)λ+sin(λβ)λ)[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)σ3+20dα0αdβ[(sin(λα)λ)2-(sin(λβ)λ)2](+σ3[Q(x-α+β)-Ql]×J(x-α+β,x-α-β)Ql+Ql[Q(x-α+β)-Ql]J(x-α+β,x-α-β)σ3).

Applying (2.9) to compute the Fourier cosine transform of Fh(α), we see that the terms containing factors of the form (sin(λα)λ)2 cancel out. Unfortunately, only half the terms containing factors sin(λα)λ do. Using (2.8) we get

0dαcos(λα)Fh(α)=140dα0αdβ(cos(λα)+cos(λβ))[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)-140dα0αdβ(cos(λα)-cos(λβ))σ3[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)σ3+140dα0αdβ[sin(λα)λ-sin(λβ)λ](Ql[Q(x-α-β2)-Ql]J(x-α-β2,x-α+β2)-[Q(x-α-β2)-Ql]σ3J(x-α-β2,x-α+β2)σ3Ql). (2.15)

We now strip off the Fourier cosine transform using (2.8), directly in the terms containing a factor cos(λα) or sin(λα)λ and indirectly after interchanging α and β in the terms containing a factor cos(λβ) or sin(λβ)λ . To avoid repeating nearly similar estimates, we write the cosine transform terms as the Fourier cosine transforms of various matrix functions of the form

(0αdβ±αdβ)𝒬(x-|α-β|2)𝒥(x-|α-β|2,x-α+β2)=2x-α2xdz𝒬(z)𝒥(z,2x-z-α)±2-xdz𝒬(z)𝒥(z,z-α) (2.16a)
and
(αdγ0γdβ-αdγγdβ)𝒬(x-|γ-β|2)𝒥(x-|γ-β|2,x-γ+β2)=2-xdz𝒬(z)z-αmin(z,2x-z-α)dw𝒥(z,w), (2.16b)
where zα ≤ z and zα ≤ 2xzα for each α ≥ 0. In fact, 𝒬=Q-Ql and 𝒥=J in the first three lines of (2.15), 𝒬=Ql[Q-Ql] and 𝒥=J in the fourth line of (2.15), and 𝒬=Q-Ql and 𝒥=σ3Jσ3Ql in the fifth line of (2.15). Integrating (2.16a) and (2.16b) with respect to α𝕉+ we obtain the upper bounds
0dα2x-α2xdz𝒬(z)𝒥(z,2x-z-α)±2-xdz𝒬(z)𝒥(z,z-α)4-xdz𝒬(z)-zdw𝒥(z,w), (2.17a)
as well as
20dα-xdz𝒬(z)z-αmin(z,2x-z-α)dw𝒥(z,w)4-xdz(x-z)𝒬(z)-zdw𝒥(z,w). (2.17b)

Using (2.14) and the various meanings of 𝒬 and 𝒥 , under condition (H2) we obtain for the J-dependent term

0dαFh(α)2-xdz[1+μ(x-z)]Q(z)-Ql-zdwJ(z,w), (2.18)
which is easily shown to hold under the more general condition (H1).

Applying Gronwall’s inequality [14] to the inequality

-xdwJ(x,w)-xdz[1+μ(x-z)]Q(z)-Ql+2-xdz[1+μ(x-z)]Q(z)-Ql-zdwJ(z,w)
following from (2.14) and (2.18), we obtain
-xdwJ(x,w)[-xdz[1+μ(x-z)]Q(z)-Ql]×exp(2xdz[1+μ(x-z)]Q(z)-Ql), (2.19)
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 Qr,l(#)=Ql,r and obtain the following symmetry relations for the fundamental eigensolutions:

Ψ˜(#)(x,k)=σ3Φ˜(-x,-k)σ3,Φ˜(#)(x,k)=σ3Ψ˜(-x,-k)σ3.

We thus get the triangular representation

Ψ˜(k,x)=σ3Φ˜(#)(-x,-k)σ3=σ3e-xΛl(#)(-k)+--xdwσ3J(#)(-x,w)σ3σ3ewΛl(#)(-k)σ3=exΛr(k)+xdzσ3J(#)(-x,-z)σ3ezΛr(k),
so that
K(x,y)=σ3J(#)(-x,-y)σ3 (2.20)
has the integrability properties (2.10).

Let us now prove the second part of the theorem for s = 0, 1, 2, …. Under the hypothesis (Hs+1), we modify the estimates (2.14), (2.17a), and (2.17b), where s = 0, 1, 2, …. Instead of (2.14) we get

0(1+α)sFinh(α)120dα(1+α)sQ(x-α2)-Ql+μ20dα(1+α)sαdβQ(x-β2)-Ql=-xdz(1+2(x-z))sQ(z)-Ql+μs+1j=0s2j+1-xdz(x-z)j+1Q(z)-Ql.

Instead of (2.17a) we get

0dα(1+α)s2x-α2xdz𝒬(z)𝒥(z,2x-z-α)±2-xdz𝒬(z)𝒥(z,z-α)2-xdz𝒬(z)[2(x-z)dα(1+α)s𝒥(z,2x-z-α)+0dα(1+α)s𝒥(z,z-α)]=2-xdz𝒬(z)-zdw[(1+2(x-z)+z-w)s+(1+z-w)s]𝒥(z,w),
where (1+2(x-z)+z-w)s=j=0s(sj)2j(x-z)j(1+z-w)s-j . Instead of (2.17b) we get
20dα(1+α)s-xdz𝒬(z)z-αmin(z,2x-z-α)dw𝒥(z,w)=2-xdz𝒬(z)-zdwz-w2x-z-wdα(1+α)s𝒥(z,w)=2s+1j=0s2j+1-xdz(x-z)j+1𝒬(z)-zdw(1+z-w)s-j𝒥(z,w).

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 [34] based on the Hölder estimate

0dα(1+α)sF(α)[0dα(1+α)N+1F(α)]s-N×[0dα(1+α)NF(α)]N+1-s,
where NsN + 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 qxL1(𝕉) .

Theorem 2.2.

Under condition (H1) we have

J12(x,x)=12[q(x)-ql],K12(x,x)=-12[q(x)-qr]. (2.21)

It suffices to extend the expression obtained in Demontis et al. [14] to general potentials satisfying (H1). Taking α0+ in the expression

J(x,x-α)=Finh(α)+Fh(α),
we obtain using (2.13) and (2.15)
J(x,x)=12[Q(x)-Ql]+140dβ(Ql[Q(x-β2)-Ql]+[Q(x-β2)-Ql]Ql)+140dβ[Q(x-β2)-Ql]J(x-β2,x-β2)+140dβσ3[Q(x-β2)-Ql]J(x-β2,x-β2)σ3-140dβ0βdγ(Ql[Q(x-β-γ2)-Ql]J(x-β-γ2,x-β+γ2)-[Q(x-β-γ2)-Ql]σ3J(x-β-γ2,x-β+γ2)σ3Ql).

Using (2.19) we see that, under condition (H1), J(x,x)-12[Q(x)-Ql] is a continuous function of x𝕉 . Utilizing a continuity argument we easily extend (2.21) to arbitrary potentials satisfying (H1).

The proof for K(x, x) can in fact be obtained from the result for J(x, x) by using (2.20).

3. RELATING FUNDAMENTAL EIGENSOLUTIONS

To study the asymptotic behavior of the fundamental eigensolutions as x → ±∞, we write the Volterra integral equations (1.5) as follows [14, Eqs. (2.12)]:

Φ˜(x,k)=𝒢(x,0;k)+-xdy𝒢(x,y;k)[Q(y)-Qf(y)]Φ˜(y,k), (3.1a)
Ψ˜(x,k)=𝒢(x,0;k)-xdy𝒢(x,y;k)[Q(y)-Qf(y)]Ψ˜(y,k), (3.1b)
where
Qf(x)={Ql,x𝕉-,Qr,x𝕉+,
and
𝒢(x,y;k)={e(x-y)Λl(k),x,y𝕉-,e(x-y)Λr(k),x,y𝕉+,exΛl(k)e-yΛr(k),x0y,exΛr(k)e-yΛl(k),x0y,
is the evolution system associated with the first order system (1.1) associated with the piecewise constant potential Qf. Then [14, Eqs. (2.22) and (2.23)]
Φ˜(x,k)=Ψ˜(x,k)Bl(k),Ψ˜(x,k)=Φ˜(x,k)Br(k), (3.2)
where
Bl(k)=I2+-dy𝒢(0,y;k)[Q(y)-Qf(y)]Φ˜(y,k), (3.3a)
Br(k)=I2--dy𝒢(0,y;k)[Q(y)-Qf(y)]Ψ˜(y,k), (3.3b)
are each other’s inverses.

Letting V(x, t) be a square matrix solution of the AKNS system (1.1), we easily derive for V−1 the “inverse” AKNS system

[V-1]x=-V-1VxV-1=-V-1(-ikσ3+Q)VV-1=V-1(ikσ3-Q).

Consequently, in analogy with (1.5) we obtain the Volterra integral equations

Φ˜(x,k)-1=e-xΛl(k)--xdyΦ˜(y,k)-1[Q(y)-Ql]e-(x-y)Λl(k), (3.4a)
Ψ˜(x,k)-1=e-xΛr(k)+xdyΨ˜(y,k)-1[Q(y)-Qr]e(y-x)Λr(k). (3.4b)

We can also write the Volterra integral equations in the form

Φ˜(x,k)-1=𝒢(0,x;k)--xdyΦ˜(y,k)-1[Q(y)-Qf(y)]𝒢(y,x;k), (3.5a)
Ψ˜(x,k)-1=𝒢(0,x;k)+xdyΨ˜(y,k)-1[Q(y)-Qf(y)]𝒢(y,x;k), (3.5b)
in analogy with (3.1). Taking the limits as x → ±∞ and using (3.2) we get
Br(k)=I2--dyΦ˜(y,k)-1[Q(y)-Qf(y)]𝒢(y,0;k), (3.6a)
Bl(k)=I2+-dyΨ˜(y,k)-1[Q(y)-Qf(y)]𝒢(y,0;k). (3.6b)

Mimicking the proof of Theorem 2.1, we can derive the triangular representations

Φ˜(x,k)-1=e-xΛl(k)+-xdye-yΛl(k)J˜(y,x), (3.7a)
Ψ˜(x,k)-1=e-xΛr(k)+xdye-yΛr(k)K˜(y,x), (3.7b)
where
-xdyJ˜(y,x)+xdyK˜(y,x)<+,x𝕉.

Equations (3.7) can also be written in the form

M˜(x,k)-1=I2+0dαeαΛl(k)J˜(x-α,x), (3.8a)
N˜(x,k)-1=I2+0dαe-αΛr(k)K˜(x+α,x), (3.8b)
in analogy with (2.4).

4. JOST FUNCTIONS AND SCATTERING COEFFICIENTS

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

λ(k)=k2+μ2
as the conformal mapping from the complex k-plane 𝕂 cut along [−, ] onto the complex λ plane 𝔺 that satisfies λ ∼ k at infinity. Allowing each k ∈(−, ) 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 𝕂± and the open complex half-planes 𝔺± and between the analytic manifolds with boundary 𝕂±𝕂± 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).

Figure 1

The regions k𝕂± and λ𝔺± with manifold boundary. Note that 𝕂± have the common boundary (−∞, −μ]∪[μ, +∞) and 𝔺± have the real line as their common boundary.

The following Fourier representation is true [24, 10.22.61]:

0dteiλtJ1(μt)μt=-0dte-iλtJ1(μt)μt=iλ+k, (4.1)
where λ𝔺+𝕉 and k𝕂+𝕂+ . Here
J1(z)=k=0(-1)k(z/2)2k+1k!(k+1)!={z2[1+O(z2)],z0+,2πzcos(z-34π)[1+O(1z)],z+,
is the Bessel function of order one. Obviously, the left-hand side of (4.1) is the Fourier transform of a function in L1(𝕉+) and hence is continuous in λ𝔺+𝕉 , is analytic in λ𝔺+ , and vanishes as λ → ∞ from within the closed upper half complex λ-plane. For λ𝔺-𝕉 and k𝕂-𝕂- we obtain by complex conjugation
0dte-iλtJ1(μt)μt=-0dteiλtJ1(μt)μt=-iλ+k. (4.2)

4.2. Definition of Jost Functions

Letting

Wr,l(k)=(wr,l(1)(k)wr,l(2)(k))=I2-iλ+kσ3Qr,l
stand for the 2 × 2 matrix whose columns are the eigenvectors of Λr,l(k), i.e., letting
exΛr,l(k)Wr,l(k)=Wr,l(k)e-iλxσ3, (4.3)
we define the Jost functions as ϕ(x, k) and ψ(x, k) for k𝕂+𝕂+ and ψ¯(x,k) and ϕ¯(x,k) for k𝕂-𝕂- as follows:
ϕ(x,k)=Φ˜(x,k)wl(1)(k),ψ(x,k)=Ψ˜(k,x)wr(2)(k),k𝕂+𝕂+, (4.4a)
ψ¯(x,k)=Ψ˜(x,k)wr(1)(k),ϕ¯(x,k)=Φ˜(x,k)wl(2)(k),k𝕂-𝕂-. (4.4b)

The Jost functions in (4.4a) can also be defined for k𝕂-𝕂- and those in (4.4b) for k𝕂+𝕂+ by computing the corresponding columns of Wr,l(k) for k in the complementary manifold. For 0k𝕉 (and hence for λ ∈ (−∞, −μ] ∪ [μ, +∞) the Jost functions coincide when defined either way. For k𝕂+𝕂- we call

Φ(x,k)=(ϕ(x,k)ϕ¯(x,k)),Ψ(x,k)=(ψ¯(x,k)ψ(x,k)),
the Jost matrices.

4.3. Definition of Scattering Coefficients

Putting

S(k)=Wr(k)-1Bl(k)Wl(k),S¯(k)=Wl(k)-1Br(k)Wr(k),
where S(k) and S¯(k) are each other’s inverses, we obtain
Φ(x,k)=Ψ(x,k)S(k),Ψ(x,k)=Φ(x,k)S¯(k), (4.5)
where
S(k)=(a(k)b¯(k)b(k)a¯(k)),S¯(k)=(c¯(k)d(k)d¯(k)c(k)),
are written in terms of the traditional a, b, c, and d functions [3, Ch. 2]. Consequently,
2λλ+kS(k)=[2λλ+kΨ(x,k)-1]Φ(x,k), (4.6a)
2λλ+kS¯(k)=[2λλ+kΦ(x,k)-1]Ψ(x,k). (4.6b)

Using the identity

2λλ+kWr,l(k)-1=σ3Wr,l(k)σ3=σ3(wr,l[1](k)wr,l[2](k))σ3, (4.7)
where wr,l[1](k) and wr,l[2](k) are the rows of Wr,l(k), we obtain
2λλ+ka(k)=wr[1](k)σ3Bl(k)wl(1)(k),2λλ+kb(k)=-wr[2](k)σ3Bl(k)wl(1)(k),2λλ+kb¯(k)=wr[1](k)σ3Bl(k)wl(2)(k),2λλ+ka¯(k)=-wr[2](k)σ3Bl(k)wl(2)(k),
and similarly for the entries of S¯(k) . Thus the scattering coefficients a(k) and c(k) are well-defined for k𝕂+𝕂+ and for k𝕂- , a¯(k) and c¯(k) are well-defined for k𝕂-𝕂- and for k𝕂+ , and the off-diagonal scattering coefficients for k𝕉𝕂+𝕂- , with the possible exception of k = ±. The entries of s(k) and S¯(k) may not be defined for k = ± but they are when multiplied by 2λλ+k .

4.4. Triangular Representations of Jost Solutions

Using the triangular representations (2.11) and the Fourier representations (4.1) and (4.2), we get

eiλxϕ(x,k)=wl(1)(k)+0dαeiλαJ(x,x-α)wl(1)(k)=(10)+0dαeiλαJ(x,x-α), (4.8a)
e-iλxψ(x,k)=wr(2)(k)+0dαeiλαK(x,x+α)wr(2)(k)=(01)+0dαeiλαK(x,x+α), (4.8b)
for k𝕂+𝕂+ , and
eiλxψ¯(x,k)=wr(1)(k)+0dαe-iλαK(x,x+α)wr(1)(k)=(10)+0dαe-iλαK¯(x,x+α), (4.8c)
e-iλxϕ¯(x,k)=wl(2)(k)+0dαe-iλαJ(x,x-α)wl(2)(k)=(01)+0dαe-iλαJ¯(x,x-α), (4.8d)
for k𝕂-𝕂- , where
K¯(x,y)=K(x,y)(10)+J1(μ[y-x])μ[y-x]qr*(01)+xydzJ1(μ[y-z])μ[y-z]qr*K(x,z)(01), (4.9a)
K(x,y)=K(x,y)(01)-J1(μ[y-x])μ[y-x]qr(10)-xydzJ1(μ[y-z])μ[y-z]qrK(x,z)(10), (4.9b)
J(x,y)=J(x,y)(10)-J1(μ[x-y])μ[x-y]ql*(01)-yxdzJ1(μ[z-y])μ[z-y]ql*J(x,z)(01), (4.9c)
J¯(x,y)=J(x,y)(01)+J1(μ[x-y])μ[x-y]ql(10)+yxdzJ1(μ[z-y])μ[z-y]qlJ(x,z)(10). (4.9d)

Consequently, using that J1(w) = w + O(w3) as w → 0+, we get

(K¯(x,x)K(x,x))=K(x,x)-Qr, (4.10a)
(J(x,x)J¯(x,x))=J(x,x)+Ql. (4.10b)

Writing (4.8a) and (4.8b) as Fourier transforms for k𝕂- and using (4.2) we get

eiλxϕ(x,k)=(10)-0dαe-iλαJ1(μα)μαql*(01)+0dαeiλαJ(x,x-α)(10)--dαe-iλα0dβJ1(μ[α+β])μ[α+β]ql*J(x,x-β)(01), (4.11a)
e-iλxψ(x,k)=(01)-0dαe-iλαJ1(μα)μαql(10)+0dαeiλαK(x,x+α)(01)--dαe-iλα0dβJ1(μ[α+β])μ[α+β]qrK(x,x+β)qr(10). (4.11b)

On the other hand, writing (4.8c) and (4.8d) as Fourier transforms for k𝕂+ and using (4.1) we get

eiλxψ¯(x,k)=(10)+0dαeiλαJ1(μα)μαqr*(01)+0dαe-iλαK(x,x+α)(10)+-dαeiλα0dβJ1(μ[α+β])μ[α+β]qr*K(x,x+β)(01), (4.11c)
e-iλxϕ¯(x,k)=(01)+0dαeiλαJ1(μα)μαql(10)+0dαe-iλαJ(x,x-α)(01)+-dαe-iλα0dβJ1(μ[α+β])μ[α+β]qlJ(x,x-β)(10). (4.11d)

Let us now apply the projections Π+ and Π defined by (A.1) to (4.11). Applying the projections Π+ to (4.11a) and (4.11b) and Π to (4.11c) and (4.11d), we obtain with the help of (A.1)

Π+[eiλxϕ(x,k)]=0dαeiλα{J(x,x-α)(10)-0dβJ1(μ[β-α])μ[β-α]J(x,x-β)(01)ql*}, (4.12a)
Π+[e-iλxψ(x,k)]=0dαeiλα{K(x,x+α)(01)-0dβJ1(μ[β-α])μ[β-α]K(x,x+β)(10)qr}, (4.12b)
Π-[eiλxψ¯(x,k)]=0dαe-iλα{K(x,x+α)(10)+0dβJ1(μ[β-α])μ[β-α]K(x,x+β)(01)qr*}, (4.12c)
Π-[e-iλxϕ¯(x,k)]=0dαe-iλα{J(x,x-α)(01)+0dβJ1(μ[β-α])μ[β-α]J(x,x-β)(10)ql}. (4.12d)

In a similar way we get by applying the projections Π- to (4.11a) and (4.11b) and Π+ to (4.11c) and (4.11d)

Π-[eiλxϕ(x,k)]=-0dαe-iλα{J1(μα)μαql*(01)+0dβJ1(μ[α+β])μ[α+β]J(x,x-β)(01)ql*}, (4.13a)
Π-[e-iλxψ(x,k)]=-0dαe-iλα{J1(μα)μαqr(10)+0dβJ1(μ[α+β])μ[α+β]K(x,x+β)(10)qr}, (4.13b)
Π+[eiλxψ¯(x,k)]=0dαeiλα{J1(μα)μαqr*(01)+0dβJ1(μ[α+β])μ[α+β]K(x,x+β)(01)qr*}, (4.13c)
Π+[e-iλxϕ¯(x,k)]=0dαeiλα{J1(μα)μαql(10)+0dβJ1(μ[α+β])μ[α+β]J(x,x-β)(10)ql}. (4.13d)

4.5. Scattering Coefficients as Fourier Transforms

Writing

Φ(x,k)-1=(ϕ¯(x,k)ϕ(x,k)),Ψ(x,k)-1=(ψ(x,k)ψ¯(x,k))
for 0k𝕉 , using the identity (4.7), and the triangular representations (3.8) we obtain
2λλ+ke-iλxϕ¯(x,k)=wl[1](k)σ3+0dαe-iλαwl[1](k)σ3J˜(x-α,x), (4.14a)
2λλ+keiλxψ¯(x,k)=-wr[2](k)σ3-0dαe-iλαwr[2](k)σ3K˜(x+α,x), (4.14b)
2λλ+ke-iλxψ(x,k)=wr[1](k)σ3+0dαeiλαwr[1](k)σ3K˜(x+α,x), (4.14c)
2λλ+keiλxϕ(x,k)=-wl[2](k)σ3-0dαeiλαwl[2](k)σ3J˜(x-α,x), (4.14d)
where k𝕂-𝕂- in (4.14a) and (4.14b) and k𝕂+𝕂+ in (4.14c) and (4.14d).

Using the Wiener algebras defined in Appendix A, it is easily verified that

2λλ+ke-iλxσ3S(k)eiλxσ3=[σ3Wr(k)σ3+0dαeiλασ3σ3Wr(k)σ3K˜(x+α,x)]×[Wl(k)+0dβJ(x,x-β)Wl(k)eiλβσ3], (4.15a)
2λλ+ke-iλxσ3S¯(k)eiλxσ3=[σ3Wl(k)σ3+0dαe-iλασ3σ3Wl(k)σ3J˜(x-α,x)]×[Wr(k)+0dβK(x,x+β)Wr(k)e-iλβσ3], (4.15b)
belong to 𝒲2×2 . The (1, 1)-element of (4.15a) and the (2, 2)-element of (4.15b) belong to 𝒲+ . The (2, 2)-element of (4.15a) and the (1, 1)-element of (4.15b) belong to 𝒲- . Further, 2λλ+ka(k)-1 and 2λλ+kc(k)-1 belong to 𝒲+ , 2λλ+ka¯(k)-1 and 2λλ+kc¯(k)-1 to 𝒲- , and 2λλ+kb(k) , 2λλ+kb¯(k) , 2λλ+kd(k) , and 2λλ+kd¯(k) to 𝒲 .

Let us now define the reflection coefficients

ρ(k)=b(k)a(k)-1,r(k)=d(k)c(k)-1,k𝕂+, (4.16a)
ρ¯(k)=b¯(k)a¯(k)-1,r¯(k)=d¯(k)c¯(k)-1,k𝕂-. (4.16b)

Using that S(k)S¯(k)=I2=S¯(k)S(k) , we obtain the identities

ρ(k)=-c(k)-1d¯(k),r(k)=-a(k)-1b¯(k), (4.17a)
ρ¯(k)=-c¯(k)-1d(k),r¯(k)=-a¯(k)-1b(k), (4.17b)
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 𝒲0 is postponed to Section 6.

5. SYMMETRIES

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 k𝕉[-iμ,iμ] the matrix functions Ψ˜(x,k)-1 and Ψ˜(x,k*) both satisfy the differential equation

Vx=V(x,t)(ikσ3-Q),
as do the matrix functions Φ˜(x,k)-1 and Φ˜(x,k*) . Thus,
Φ˜(x,k*)=Φ˜(x,k)-1,Ψ˜(x,k*)=Ψ˜(x,k)-1, (5.1)
where k𝕉[-iμ,iμ] . We observe that k*=k for k𝕉 and k*=-k for k ∈[−, ]. Thus for k𝕉 the fundamental eigensolutions Φ˜(x,k) and Ψ˜(x,k) are unitary matrices of determinant 1. We also get
M˜(x,k*)=M˜(x,k)-1,N˜(x,k*)=N˜(x,k)-1, (5.2)
where k𝕉[-iμ,iμ] . Using (3.2) we also obtain
Bl(k*)=Br(k)=Bl(k)-1,Br(k*)=Bl(k)=Br(k)-1, (5.3)
where k𝕉[-iμ,iμ] .

Next, we easily derive the identities

Λr,l(k*)=-Λr,l(k), (5.4a)
Wr,l(k*)=σ3Wr,l(k)σ3=2λλ+kWr,l(k)-1, (5.4b)
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
J(x,x-α)=J˜(x-α,x),K(x,x+α)=K˜(x+α,x), (5.5)
where α𝕉+ .

Using (4.7) and (4.4) we obtain for the Jost matrices

Φ(x,k*)=2λλ+kΦ(x,k)-1, (5.6a)
Ψ(x,k*)=2λλ+kΨ(x,k)-1, (5.6b)
where k𝕂+𝕂+ as far as the second row of (5.6a) and the first row of (5.6b) are concerned and k𝕂-𝕂- 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
S(k*)=S(k)-1=S¯(k),S¯(k*)=S¯(k)=S(k). (5.7)

Thus S(k) and S¯(k) are unitary matrices if k𝕉 . Since S(k) and S¯(k) both have unit determinant, we get

a(k*)*=a¯(k),c(k*)*=c¯(k), (5.8a)
b(k*)*=-b¯(k),d(k*)*=-d¯(k), (5.8b)
where -iμk𝕂-𝕂- in (5.8a) and -iμk𝕂- in (5.8b). Equations (4.16) imply that the reflection coefficients satisfy the symmetry relations
ρ¯(k)=-ρ(k*)*,r¯(k)=-r(k*)*, (5.9)
provided the reciprocals in their definitions (4.16) exist.

b. Conjugation symmetry. Let σ2=(0-ii0) stand for the second Pauli matrix. Then it is easily verified that σ2Ψ˜(x,k*)*σ2 and Ψ˜(x,k) both satisfy the differential equation (1.1). The same thing is true for the other fundamental eigensolution Φ˜ . We thus obtain

σ2Φ˜(x,k*)*σ2=Φ˜(x,k),σ2Ψ˜(x,k*)*σ2=Ψ˜(x,k), (5.10)
where k𝕉[-iμ,iμ] . Furthermore,
σ2M˜(x,k*)*σ2=M˜(x,k),σ2N˜(x,k*)*σ2=N˜(x,k). (5.11)
where k𝕉[-iμ,iμ] . Moreover,
σ2Br,l(k*)*σ2=Br,l(k), (5.12)
where k𝕉[-iμ,iμ] . Using (5.11) and (2.4) we get
σ2J(x,x-α)*σ2=J(x,x-α),σ2K(x,x+α)*σ2=K(x,x+α), (5.13)
where α𝕉+ .

Next, we easily derive the identities

Λr,l(k*)*=σ2Λr,l(k)σ2, (5.14a)
Wr,l(k*)*=σ2Wr,l(k)σ2, (5.14b)
where in (5.14a) the choice of the sign in defining λ from k does not matter. In (5.14b) this choice is to be made consistently. Using (5.10) and (5.14b) we obtain for the Jost matrices
Φ(x,k*)*=σ2Φ(x,k)σ2,Ψ(x,k*)*=σ2Ψ(x,k)σ2. (5.15)

Consequently,

S(k*)*=σ2S(k)σ2,S¯(k*)*=σ2S¯(k)σ2. (5.16)

We immediately recover (5.8).

6. GENERIC AND EXCEPTIONAL CASES

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 [10], Deift and Trubowitz [12] and Faddeev [15]), though such strengthening can be avoided at the expense of more complicated mathematical arguments [20]. 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 [12]. 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 0λ𝔺+𝕉 and a¯(k)=c¯(k) for 0λ𝔺-𝕉 . Therefore, throughout this article we assume absence of spectral singularities:

There do not exist any 0λ(k)𝕉 where a(k), c(k), a¯(k) , and c¯(k) vanish.

As a result, under this assumption the reflection coefficients ρ(k) and r(k) are well-defined for iμk𝕂+𝕂+ and the reflection coefficients ρ¯(k) and r¯(k) are well-defined for -iμk𝕂-𝕂- . Moreover, b(k), b¯(k) , d(k), and d¯(k) are well-defined for k𝕂+𝕂- . Their definitions for k = ± are a different matter to be pursued presently.

Under condition (H1) we can define

ϕ(x,iμ)=Φ˜(x,iμ)(1-ql*/μ),ψ(x,iμ)=Ψ˜(x,iμ)(-qr/μ1),ψ¯(x,-iμ)=Ψ˜(x,-iμ)(1-qr*/μ),ϕ¯(x,-iμ)=Φ˜(x,-iμ)(-ql/μ1).

Since [see (5.15)]

(ϕup(x,iμ)ψup(x,iμ)ϕdn(x,iμ)ψdn(x,iμ))*=σ2(ψ¯up(x,-iμ)ϕ¯up(x,-iμ)ψ¯dn(x,-iμ)ϕ¯dn(x,-iμ))σ2,
it is clear that the Jost functions ϕ(x, ) and ψ(x, ) are linearly independent iff the Jost functions ψ¯(x,-iμ) and ϕ¯(x,-iμ) are linearly independent. As in the Schrödinger case, we can therefore make a distinction between the following two cases:
  1. (a)

    the generic case: the Jost functions ϕ(x, ) and ψ(x, ) are linearly independent. OR: the Jost functions ϕ¯(x,-iμ) and ψ¯(x,-iμ) are linearly independent.

  2. (b)

    the exceptional case: the Jost functions ϕ(x, ) and ψ(x, ) are linearly dependent. OR: the Jost functions ϕ¯(x,-iμ) and ψ¯(x,-iμ) are linearly dependent.

Theorem 6.1.

Suppose condition (H1) is satisfied. Then we are in the generic case if and only if

limkiμk𝕂+𝕂+2λλ+ka(k)
exists and is nonzero. If this limit vanishes, we are in the exceptional case.

Using (4.3) and (4.4) we get for iμk𝕂+𝕂+

ϕ(x,k){e-iλx(1-iλ+kql*),x-,e-iλx(a(k)-iλ+kqre2iλxb(k),e2iλxb(k)-iλ+kqr*a(k)),x+,ψ(x,k)={eiλx(e-2iλxd(k)-iλ+kqlc(k)c(k)-iλ+kql*e-2iλxd(k)),x-,eiλx(-iλ+kqr1),x+.

Since det(ϕ(k, x) ψ(k, x)) does not depend on x𝕉 , we compute their determinants as x → ∞ and x → +∞ and obtain

2λλ+kc(k)=2λλ+ka(k),
thus proving again that a(k) = c(k). Since these determinants have the finite limit
det(Φ˜(x,iμ)(1-ql*/μ)Ψ˜(x,iμ)(-qr/μ1))

As k from within 𝕂+𝕂+ , we arrive at the desired conclusion.

We now observe that

ϕ(x,iμ)[I2+xQl+μxσ3](1-ql*/μ)=(1-ql*/μ),x-, (6.1a)
ψ(x,iμ)[I2+xQr+μxσ3](-qr/μ1)=(-qr/μ1),x+. (6.1b)

Hence, in the exceptional case the (proportional) Jost functions ϕ(x, ) and ψ(x, ) are bounded in x𝕉 and have finite nonzero limits as x → ± ∞.

Theorem 6.2.

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 kiμ. Then the reflection coefficients ρ(k), ρ¯(k) , r(k), and r¯(k) are Fourier transforms of functions in L1(𝕉) . 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 L1(𝕉) [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 2λλ+ka(k) and 2λλ+kb(k) can then easily be shown to be the Fourier transforms (in λ) of functions in L1(𝕉+;(1+α)dα) and L1(𝕉;(1+|α|)dα) , respectively, we can then apply Taylor’s theorem and write

a(k)=c(k)=a-1λ+a0+o(1),kiμin𝕂+𝕂+, (6.2a)
b(k)=b-1λ+b0+o(1),kiμin𝕂+, (6.2b)
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
(a(k)-b(k*)*b(k)a(k*)*)(c(k*)*d(k)-d(k*)*c(k))=I2=(c(k*)*d(k)-d(k*)*c(k))(a(k)-b(k*)*b(k)a(k*)*),
where a(k) = c(k). Substituting (6.2) in the (2, 2)-element of either equation we get
b-1d-1=0,b-1d0+b0d-1=0,|a0|2+b0d0=|a0|2+b0*d0*=1.

Substituting (6.2) [with λ* = −λ] in the (1, 2)-element of the left equation we get

a0(d-1+b-1*)=0,a0d0=a0b0*.

The Ansatz b−1 = 0 and d−1 ≠ 0 leads to b0 = 0 and |a0| = 1, so that d-1+b-1*=0 , 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

b(k)=b0+o(1),d(k)=d0+o(1), (6.3)
where b0d0=b0*d0*=1-|a0|2 is a real number. Furthermore,
(a0-b0*b0a0*)(a0*d0-d0*a0)=I2=(a0*d0-d0*a0)(a0-b0*b0a0*),
where the factors in either matrix product have unit determinant. Computing the two matrix products we get
(|a0|2+b0*d0*a0(d0-b0*)a0*(b0-d0*)|a0|2+b0d0)=I2=(|a0|2+b0d0a0*(d0-b0*)a0(b0-d0*)|a0|2+b0*d0*).

This leads to two mutually exclusive possibilities:

  1. (a)

    a00 , d0=b0* , and |a0|2+|b0|2=|a0|2+|d0|2=1 .

  2. (b)

    a0 = 0 and b0d0 = 1. Since a(k)[d(k)+b¯(k)]=0 with a(k) ≠ 0 for values of k𝕂+ approaching , the absence of spectral singularities assumption implies that |b0| = |d0|. Consequently, there exists a phase θ𝕉 such that b0=d0*=eiθ .

In the former case the reflection coefficients are Fourier transforms of functions in L1(𝕉) , whereas in the latter case the reflection coefficients blow up as k.

Now observe that

ψ¯(x,iμ)=Ψ˜(x,iμ)(1-qr*/μ)=-qr*μΨ˜(x,iμ)(-qr/μ1)=-qr*μψ(x,iμ)
is well-defined. Thus the identity ϕ=ψ¯a+ψb implies that in the exceptional case
ϕ(x,iμ)=(b0-qr*μa0)ψ(x,iμ),
where the proportionality constant is nonzero. In the same way we prove that
ψ(x,iμ)=(d0-qlμa0)ϕ(x,iμ),
where the proportionality constant is nonzero as k = ±. Since these two proportionality constants have product 1, d0=b0* , and qlqr*=μ2ei(θl-θr) , we get
(b0-e-iθra0)(b0*-eiθla0)=1.

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.

ACKNOWLEDGMENTS

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).

APPENDIX A

1. WIENER ALGEBRAS

By the (continuous) Wiener algebra 𝒲 we mean the complex vector space of constants plus Fourier transforms of L1-functions

𝒲={c+h^:c𝔺,hL1(𝕉)}
endowed with the norm |c| + ||h||1. Here we define the Fourier transform as follows: (𝒡h)(λ)=h^(λ)=-dyeiλyh(y) . The invertible elements of the commutative Banach algebra 𝒲 with unit element are exactly those c+h^𝒲 for which c ≠ 0 and c+h^(λ)0 for each λ𝕉 [17].

The algebra 𝒲 has the two closed subalgebras 𝒲+ and 𝒲- consisting of those c+h^ such that h is supported on 𝕉+ and 𝕉- , respectively. The invertible elements of 𝒲± are exactly those c+h^𝒲± for which c ≠ 0 and c+h^(λ)0 for each λ𝔺±𝕉 [17]. Letting 𝒲0± and 𝒲0 stand for the (nonunital) closed subalgebras of 𝒲± and 𝒲 consisting of those c+h^ for which c = 0, we obtain the direct sum decomposition

𝒲=𝔺𝒲0+𝒲0-,𝒲0=𝒲0+𝒲0-.

By Π± we now denote the (bounded) projection of 𝒲 onto 𝒲0± along 𝔺𝒲0 . Then Π+ and Π are complementary projections. In fact,

(Π±f)(λ)=12πi-dζf(ζ)ζ-(λ±i0+), (A.1)
where f𝒲0Lp(𝕉) for some p ∈ (1, +∞). These direct sum decompositions coupled by the Fourier transform can be schematically represented as follows:
L1(𝕉)=L1(𝕉-)L1(𝕉+)𝒡𝒡𝒡𝒲0=𝒲0-𝒲0+

Now observe that 𝒡 acts as an isometric linear 1, 1-correspondence from L1(𝕉) onto 𝒲0 . If we define the norm of 𝔺L1(𝕉) as c+h=|c|+h1 , we obtain the direct sum decomposition

L1(𝕉)=L1(𝕉+)L1(𝕉-),
where the projection 𝒡-1Π±𝒡 is the restriction of an arbitrary hL1(𝕉) to the half-line 𝕉± .

Throughout this article we denote the vector spaces of n × m matrices with entries in 𝒲 , 𝒲± , and 𝒲0± by 𝒲n×m , 𝒲±n×m , and 𝒲0±n×m , respectively. We write L1(𝕉)n×m and L1(𝕉±)n×m for the vector spaces of n × m matrices with entries in L1(𝕉) and L1(𝕉±) , 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 𝒲n×n and 𝒲±n×n are noncommutative Banach algebras with unit element and 𝒲0±n×n 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 [17] but was known before to Wiener [32,33].

Theorem Appendix A.1.

If for some complex number h and some hL1(𝕉) the Fourier transform h+-dzeiλzh(z)0 for every λ𝕉 and if h ≠ 0, then there exists kL1(𝕉) such that

1h+-dzeiλzh(z)=1h+-dzeiλzk(z)
for every λ𝕉 .

APPENDIX B

1. TIME DEPENDENCE OF THE SCATTERING DATA

The focusing NLS equation

iσ3Qt=Qxx-2Q3, (B.1)
where Q=(0Q-Q*0) , arises as the compatibility condition of the Lax pair equations [2,3,16]
vx=(-ikσ3+Q)v,vt=(2ik2σ3+iσ3Q2-2kQ-iσ3Qx)v, (B.2)
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 x𝕉 (but do depend on k and t). Since

Ψt=vtC+-1-vC+-1[C+]tC+-1=(2ik2σ3+iσ3Q2-2kQ-iσ3Qx)vC+-1-vC+-1[C+]tC+-1=(2ik2σ3+iσ3Q2-2kQ-iσ3Qx)Ψ-Ψ[C+]tC+-1,
we obtain
[C+]tC+-1=Ψ-1(2ik2σ3+iσ3Q2-2kQ-iσ3Qx)Ψ-Ψ-1Ψt, (B.3a)
and similarly
[C-]tC--1=Φ-1(2ik2σ3+iσ3Q2-2kQ-iσ3Qx)Φ-Φ-1Φt. (B.3b)

Using (B.1) we observe that

[Qr,l]t=2iσ3Qr,l(t)3=-2iμ2σ3Qr,l(t)=2iμ2Qr,l(t)σ3=iμ2{Qr,l(t)σ3-σ3Qr,l(t)}, (B.4)
so that
Qr,l(t)=e-iμ2tσ3Qr,l(0)eiμ2tσ3
and
[Wr,l(k)]t=-iλ+kσ3[Qr,l]t=-2μ2λ+kQr,l(t).

Since the left-hand sides of (B.3) do not depend on x𝕉 , we can take the limits of the right-hand sides as x → ±∞ and obtain

[C+]tC+-1=eiλxσ3Wr(k)-1[(2ik2σ3-iμ2σ3-2kQr)Wr(k)-[Wr(k)]t]e-iλxσ3=eiλxσ3Wr(k)-1[(2iλk-iμ2)Wr(k)σ3]e-iλxσ3=(2iλk-iμ2)σ3,[C-]tC--1=eiλxσ3Wl(k)-1[(2ik2σ3-iμ2σ3-2kQl)Wl(k)-[Wl(k)]t]e-iλxσ3=eiλxσ3Wl(k)-1[(2iλk-iμ2)Wl(k)σ3]e-iλxσ3=(2iλk-iμ2)σ3.

Next, differentiating S = Ψ−1Φ with respect to t and writing the second identity in (B.2) as vt = (…)v, we obtain

St=Φ-1Φt-Ψ-1ΨtΨ-1Φ=SΦ-1Φt-Ψ-1ΨtS=S(Φ-1()Φ-[C-]tC--1)-(Ψ-1()Ψ-[C+]tC+-1)S=-S[C-]tC--1+[C+]tC+-1S=(2iλk-iμ2)(σ3S-Sσ3).

Consequently,

S(k,t)=e(2iλk-iμ2)tσ3S(k,0)e-(2iλk-iμ2)tσ3.

As a result, the diagonal scattering coefficients a(k) and a¯(k) are time independent, whereas

b(k,t)=e-(4iλk-2iμ2)tb(k,0),b¯(k,t)=e(4iλk-2iμ2)tb¯(k,0).

Using (4.17) we see that the reflection coefficients have the time evolution

ρ(k,t)=e-(4iλk-2iμ2)tρ(k,0),ρ¯(k,t)=e(4iλk-2iμ2)tρ¯(k,0),r(k,t)=e(4iλk-2iμ2)tr(k,0),r¯(k,t)=e-(4iλk-2iμ2)tr¯(k,0).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 1
Pages
68 - 89
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.200922.006How to use a DOI?
Copyright
© 2020 The Author. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

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
ER  -