Reflectionless Solutions for Square Matrix NLS
with Vanishing Boundary Conditions
Francesco Demontis
∗, Cornelis van der Mee
∗Abstract
In this article we derive the reflectionless solutions of the 2 + 2 matrix NLS equation with vanishing boundary conditions and four different symmetries by using the matrix triplet method of represent-ing the Marchenko integral kernel in separated form. Apart from usrepresent-ing the Marchenko method, these solutions are also verified by direct sub-stitution in the 2 + 2 NLS equation.
1
Introduction
In this paper we study the matrix nonlinear Schr¨odinger equation
iQt+ Qxx− 2QRQ = 02×2, (1.1)
where Q(x, t) is a 2 × 2 complex matrix valued function and R = ΣQ†Ω for fixed matrices Σ, Ω ∈ {±I2, ±σ3}. Here the dagger indicates the complex
conjugate transpose and σ1 = (0 11 0), σ2 = (0 −ii 0 ), and σ3 = (1 00 −1) are the
Pauli matrices. These choices of Σ and Ω lead to four physically relevant cases (see [22] for more details on this fact): case 1 (the defocusing case): Σ = Ω = I2, case 2 (the focusing case): Σ = −Ω = I2, case 3: Σ = Ω = σ3,
and case 4: Σ = −Ω = σ3, where we note that a simultaneous sign change of
Σ and Ω does not create new cases. In addition to these four symmetries, we also discuss the special case where Q(x, t) is a complex symmetric matrix.
∗Dipartimento di Matematica e Informatica, Universit`a di Cagliari, Viale Merello 92,
Equation (1.1) represents the natural generalization of the matrix NLS equation, i.e.,
iQt+ Qxx− 2µQQ†Q = 02×2, µ = ±1 , (1.2)
where µ = −1 is the focusing case corresponding to case 2 and µ = 1 is the defocusing case equivalent to case 1. Equation (1.2) plays an important role in various applicative contexts and, in particular, in the description of Bose-Einstein condensates [18, 19]. The choice µ = 1 (defocusing case) corresponds to repulsive interatomic interactions and antiferromagnetic spin-exchange interactions, while the choice µ = −1 (focusing case) takes into account attractive interatomic interactions and ferromagnetic spin-exchange interactions.
The cases 3 and 4 of (1.1) discussed above arise in a natural way in var-ious models of nonlinear optics [15, 16, 20] and four-component fermionic condensates [17, 26, 12, 25]. More specifically, in [15, 16] the nonlinear dy-namics of the energy transfer process between the fundamental and second harmonic fields in the presence of the phase matched direct current field has been studied in detail; the arising spatio-temporal phenomena can be mod-eled by cases 3 and 4 of equation (1.1). On the other hand, in [20] cases 3 and 4 of equation (1.1) have been used to model various multicolor optical spatio-temporal solitary waves created by the interaction of light at a central frequency with two sideband waves, both through cross-phase modulation and parametric four-wave mixing of opposite signs. Moreover, cases 3 and 4 of (1.1) appear in the context of fermionic condensates of ultracold atoms and, in particular, such cases allow to investigate four component spin-32 cold atomic systems under special conditions (see [17, 26, 12, 25]).
Taking into account the important applicative contexts where (1.1) ap-pears, in [22] the Inverse Scattering Transform (IST) for (1.1) has been de-veloped as a tool to solve the associated initial value problem with vanishing boundary conditions, i.e. Q(x, t) → 02×2 as x → ±∞ for each fixed t, which
we assume from now on. While the IST for the cases 1 and 2 is well estab-lished [1, 13, 2], both with zero and nonzero boundary conditions, the IST and soliton solutions corresponding to the reductions 3 and 4 are novel [22], apart from the profound analysis of the one-soliton solutions in cases 2–4 given in [22]. We use the results on the IST obtained in [22, 21] to derive a general reflectionless solution formula for (1.1). In fact, in [22] the soliton solutions of (1.1) have been derived by solving the Riemann-Hilbert prob-lem, also in the one-soliton case. Some soliton solutions for equation (1.2)
have been derived by Ieda et al. [23] by solving the Marchenko equation with nonvanishing boundary conditions. In the present paper we formulate the inverse scattering problem in terms of the Marchenko integral equations, but unlike in [23] we find an explicit reflectionless solution formula by us-ing the so-called matrix triplet method (see, for example, [4, 11, 24, 7, 8] for more details on this method). As a consequence of applying the matrix triplet method, we get an explicit reflectionless solution formula comprising not only the soliton solutions already known in the literature but also a new class of explicit soliton solutions such as the two soliton solutions for cases 3 and 4 as well as the double poles soliton solutions for the same cases. It is also important to remark that in this article we verify by algebraic means that our solution formula satisfies (1.1).
Let us briefly discuss the contents of this article. In Section 2 we summa-rize the well established direct scattering theory of the 2 + 2 AKNS system [1, 13, 2] and discuss the symmetries of Jost solutions and scattering coeffi-cients in all four cases. In Section 3 we derive the Marchenko integral equa-tions and discuss how to compute the potential from the scattering data. In Section 4 we compute the reflectionless solutions by using the matrix triplet method. In this way we generalize the soliton solutions obtained in the fo-cusing case [11] to all four cases, where we take into account both the adjoint symmetry underlying the cases 1–4 and the assumption that Q(x, t) is a com-plex symmetric matrix. In Section 5 we substitute the reflectionless solution formula obtained back into the matrix NLS equation (1.1) and verify that it really is a solution. Section 6 is devoted to some illustrative examples, where we recover the well-known one-soliton solutions [22] by other means. The derivation of the Volterra integral equations for the Fourier transforms of the Jost solutions is given in Appendix A.
2
Direct scattering
In this section we describe the direct scattering theory of the 2 × 2 AKNS system, where the potential Q(x, t) satisfies the symmetry relation underly-ing the case 1–4 under consideration. At each occasion we point out when the second symmetry of having Q(x, t) complex symmetric is being used.
2.1
Jost solutions
It is well-known that the matrix NLS equation is equivalent to the compati-bility condition of a Lax pair for the potential matrix Q(x, t), where the first equation in the Lax pair is the so-called matrix Zakharov-Shabat (ZS) or Ablowitz-Kaup-Newell-Segur (AKNS) system. Specifically, Eq. (1.1) admits the Lax pair
ϕx = U ϕ, ϕt= Vϕ, (2.1)
where
U (x, t, k) = −ikσ3+ Q(x, t), V(x, t, k) = −2ik2σ3+ 2kQ + iσ3Qx−iσ3Q2.
Here Q(x, t) = 02×2 Q(x,t)
R(x,t) 02×2
, σ3 = I2⊕ (−I2), and R = ΣQ†Ω.
As in [22], for (x, t, k) ∈ R3 we define the Jost eigenfunctions as follows:
Φ(x, t, k) = φ(x, t, k) φ(x, t, k) = e−iθ(x,t,k)σ3[I 4 + o(1)], x → −∞, (2.2a) Ψ(x, t, k) = ψ(x, t, k) ψ(x, t, k) = e−iθ(x,t,k)σ3[I 4+ o(1)], x → +∞, (2.2b) where θ(x, t, k) = k(x + 2kt) = kx + 2k2t, (2.3) and φ(x, t, k) and φ(x, t, k) (respectively, ψ(x, t, k) and ψ(x, t, k)) are 4 × 2 matrices, called Jost solutions, comprising the first two and last two columns of the 4 × 4 matrix solutions Φ(x, t, k) (respectively, Ψ(x, t, k)). Taking out the asymptotic parts we obtain the modified eigenfunctions
M (x, t, k) M (x, t, k) = Φ(x, t, k)eiθ(x,t,k)σ3 ∼ I
4, x → −∞, (2.4a)
N (x, t, k) N (x, t, k) = Ψ(x, t, k)eiθ(x,t,k)σ3 ∼ I
4, x → +∞. (2.4b)
Assuming that Q(x, t) has its entries in L1(R, dx) for each t ∈ R, we can
prove the existence of the Jost eigenfunctions as well as their k-analyticity properties from the Volterra integral equations
M (x, t, k) M (x, t, k) = I4+
Z x
−∞
dy e−ik(x−y)σ3Q(y, t)×
× M (y, t, k) M (y, t, k) e−ik(y−x)σ3, (2.5a)
N (x, t, k) N (x, t, k) = I4−
Z ∞
x
dy eik(y−x)σ3Q(y, t)×
Using substitution of the so-called triangular representations M (x, t, k) M (x, t, k) = I4+ Z x −∞ dy L(x, y, t) L(x, y, t) eik(x−y)σ3, (2.6a) N (x, t, k) N (x, t, k) = I4+ Z ∞ x dy K(x, y, t) K(x, y, t) e−ik(y−x)σ3, (2.6b) into the Volterra integral equations (2.5) and stripping off the Fourier trans-forms, we get coupled Volterra integral equations for the kernel functions L(x, y, t) and L(x, y, t) and for the kernel functions K(x, y, t) and K(x, y, t) for each (x, t) ∈ R2 which can easily be proven to be uniquely solvable in an
L1-setting [cf. [2, 24, 6]; also Appendix A]. For each t ∈ R we thus obtain
for the kernel functions sup x∈R Z x −∞ dy kL(x, y, t)k + kL(x, y, t)k < +∞, (2.7a) sup x∈R Z ∞ x dy kK(x, y, t)k + kK(x, y, t)k < +∞. (2.7b) As an ancillary result [see (A.1) and (A.2)], we obtain the inversion for-mulas Q(x, t) = −2Kup (x, x, t), R(x, t) = −2Kdn(x, x, t), (2.8a) Q(x, t) = 2Lup(x, x, t), R(x, t) = 2Ldn (x, x, t), (2.8b) Z ∞ x dz (QR)(z, t) = 2Kup(x, x, t), Z ∞ x dz (RQ)(z, t) = 2Kdn (x, x, t), (2.8c) Z x −∞ dz (QR)(z, t) = 2Lup (x, x, t), Z x −∞ dz (RQ)(z, t) = 2Ldn(x, x, t). (2.8d) If the entries of Q(x, t) and Qx(x, t) belong to L1(R, dx) for each t ∈ R, we
can derive (2.8) alternatively by deriving the analog of [22, Eqs. (60)]. For later use we present, for (x, t, k) ∈ R3, the triangular representations
˘M (x, t, k) ˘ M (x, t, k) ! = M (x, t, k) M (x, t, k)−1 = I4+ Z x −∞ dy e−ik(x−y)σ3 ˘L(x, y, t) ˘ L(x, y, t) ! , (2.9a)
˘N (x, t, k) ˘ N (x, t, k) ! = N (x, t, k) N (x, t, k)−1 = I4+ Z ∞ x dy eik(y−x)σ3 ˘K(x, y, t) ˘ K(x, y, t) ! , (2.9b)
where for each t ∈ R sup x∈R Z x −∞ dy k˘L(x, y, t)k + k ˘L(x, y, t)k < +∞, (2.10a) sup x∈R Z ∞ x dy k ˘K(x, y, t)k + kK(x, y, t)k˘ < +∞. (2.10b) Equations (2.9) can be derived from Volterra integral equations for the in-verted Jost eigenfunctions by using the methods explained in Appendix A.
It is easily verified [22] that the 4 × 4 potential Q satisfies the symmetry relations Q†= −Ξ−1QΞ, Ξ = Ω −1 0 2×2 02×2 −Σ , (2.11)
Equation (2.11) implies the symmetry relations for the Jost eigenfunctions Φ−1(x, t, k) = ΞΦ†(x, t, k∗)Ξ−1, (2.12a) Ψ−1(x, t, k) = ΞΨ†(x, t, k∗)Ξ−1, (2.12b)
where we note that θ(x, t, k∗)∗ = θ(x, t, k).
Also assuming Q(x, t) to be a complex symmetric matrix and putting
F = 02×2 −I2 I2 02×2 , cases 1 and 3, 02×2 I2 −I2 02×2 , cases 2 and 4, we obtain the symmetry relations [22]
ΦT(x, t, k)F Φ(x, t, k) = F , (2.13a) ΨT(x, t, k)F Ψ(x, t, k) = F , (2.13b)
2.2
Scattering coefficients
Since U has zero trace and hence det Φ(x, t, k) = det Ψ(x, t, k) = 1 for (x, t, k) ∈ R3, there exists a 4 × 4 scattering matrix S(k) independent of
(x, t) such that
Φ(x, t, k) = Ψ(x, t, k)S(k). (2.14) Thus we can write in terms of 2 × 2 blocks
S(k) =a(k) b(k) b(k) a(k) , S(k)−1 =c(k) d(k) d(k) c(k) , (2.15) where det S(k) = 1 for every k ∈ R. The four diagonal blocks in S(k) and S(k)−1 tend to I2 and the four off-diagonal blocks vanish as k → ±∞. Using
(2.2), (2.6), (2.9), and (2.14) we obtain for the four diagonal blocks a = ˘N M , a =N M , c =˘ M N , and c = ˘˘ M N . Hence, by virtue of (2.6) and (2.9), there exist matrix functions ˆa, ˆa, ˆc, and ˆc such that
a(k) = I2+
Z ∞
0
dy eikyˆa(y), a(k) = I2+
Z ∞ 0 dy e−ikyˆa(y), c(k) = I2+ Z ∞ 0 dy e−ikyˆc(y), c(k) = I2+ Z ∞ 0 dy eikyˆc(y), where Z ∞ 0
dy kˆa(y)k + kˆa(y)k + kˆc(y)k + kˆc(y)k < +∞.
In the same way, using e2iθb = N M , e˘ −2iθb = ˘N M , e2iθd = ˘M N , and
e−2iθd =M N , we prove the existence of matrix functions ˆ˘ b, ˆb, ˆd, and ˆd such that
e4ik2tb(k) = Z ∞
−∞
dy e−ikyˆb(y, t), e−4ik2tb(k) = Z ∞ −∞ dy eikyˆb(y, t), e4ik2td(k) = Z ∞ −∞
dy e−ikyˆd(y, t), e−4ik2td(k) = Z ∞
−∞
dy eikyd(y, t),ˆ
where for each t ∈ R Z ∞
−∞
dy
kˆb(y, t)k + kˆb(y, t)k + kˆd(y, t)k + k ˆd(y, t)k
Using (2.12) we obtain the symmetry relations
S(k)−1 = ΞS(k)†Ξ−1 = F−1S(k)TF , k ∈ R, (2.16) where the transposition symmetry requires Q to be a complex symmetric ma-trix. These symmetry relations can obviously be transformed into symmetry relations for their separate 2 × 2 entries.
Using SS−1 = S−1S = I4 in (2.15), it is easily verified that det a(k) =
det c(k) and det a(k) = det c(k). In case 1 (defocusing case) these determi-nants are positive numbers for each k ∈ R. In this article we assume that in cases 2-4 these determinants are nonzero for any k ∈ R as well. In other words, we assume the absence of spectral singularities. We then define the (matrix) reflection coefficients as follows:
ρ(k) = b(k)a(k)−1, ρ(k)= b(k)a(k)−1, (2.17a) r(k) = d(k)c(k)−1, r(k)= d(k)c(k)−1. (2.17b) Under the absence of spectral singularities, there exist matrix functions ˆρ, ˆρ, ˆ
r, and ˆr such that e4ik2tρ(k) =
Z ∞
−∞
dy e−ikyρ(y, t),ˆ e−4ik2tρ(k) = Z ∞
−∞
dy eikyˆρ(y, t), (2.18a)
e4ik2tr(k) = Z ∞
−∞
dy e−ikyˆr(y, t), e−4ik2tr(k) = Z ∞
−∞
dy eikyˆr(y, t), (2.18b)
where for each t ∈ R Z ∞
−∞
dy k ˆρ(y, t)k + kˆρ(y, t)k + kˆr(y, t)k + kˆr(y, t)k < +∞. Using (2.16), (2.17), and SS−1 = S−1S = I4, we obtain
ρ†= ΩρΣ, ρ† = Σ−1ρΩ−1, r†= ΩrΣ, r†= Σ−1rΩ−1, (2.19a) ρT = ρ, ρT = ρ, rT = r, rT = r, (2.19b) where we have not written the k-dependence and (2.19b) requires Q(x, t) to be complex symmetric. Equations (2.18) and (2.19a) imply that for a.e. (y, t) ∈ R2
ˆ
ρ(y; t)† = Ωˆρ(y; t)Σ, ˆρ(y; t)†= Σ−1ρ(y; t)Ωˆ −1, (2.20a) ˆ
r(y; t)† = Ωˆr(y; t)Σ, r(y; t)ˆ †= Σ−1ˆr(y; t)Ω−1. (2.20b) Using (2.18) and (2.19b), it is clear that ˆρ, ˆρ, ˆr, and ˆr are complex symmetric matrices whenever Q(x, t) is a complex symmetric matrix.
3
Marchenko equations
In this section we derive the Marchenko integral equations from Riemann-Hilbert problems for the Jost solutions. We only give full details if the discrete eigenvalues are algebraically and geometrically simple.
3.1
Riemann-Hilbert problems
Assuming the absence of spectral singularities and the definitions (2.17) we write the 4 × 2 columns of the identities Φ = ΨS and Ψ = ΦS−1 in the form
ψ(x, t, k) = φ(x, t, k)a(k)−1− ψ(x, t, k)ρ(k), (3.1a) ψ(x, t, k) = φ(x, t, k)a(k)−1− ψ(x, t, k)ρ(k), (3.1b) φ(x, t, k) = ψ(x, t, k)c(k)−1− φ(x, t, k)r(k), (3.1c) φ(x, t, k) = ψ(x, t, k)c(k)−1− φ(x, t, k)r(k). (3.1d) Replacing Jost functions by modified Jost functions and rearranging terms we obtain M (x, t, k)a(k)−1− I2 02×2 = N (x, t, k)− I2 02×2 + e2iθ(x,t,k)N (x, t, k)ρ(k), (3.2a) M (x, t, k)a(k)−1−02×2 I2 = N (x, t, k)−02×2 I2 + e−2iθ(x,t,k)N (x, t, k)ρ(k), (3.2b) N (x, t, k)c(k)−1− I2 02×2 = M (x, t, k)− I2 02×2 + e2iθ(x,t,k)M (x, t, k)r(k), (3.2c) N (x, t, k)c(k)−1−02×2 I2 = M (x, t, k)−02×2 I2 + e−2iθ(x,t,k)M (x, t, k)r(k). (3.2d) Thus either side of each of (3.2) is the Fourier transform of a 4 × 2 matrix function with entries in L1(R).
-functions supported on R, R+, and R−, we get the commutative diagram
L1(R) = L1(R+) ⊕ L1(R−)
↓ ↓ ↓
W = W+ ⊕ W−
where the downarrows represent the Fourier transform F defined by (F f )(k) =
Z ∞
−∞
dy eikyf (y)
and the norms on W, W+, and W− are defined in such a way that the
downarrows become isometries. Then the complementary projections Π± of
W onto W± along W∓ are given by the Plemelj formulas
(Π±f )(k) = ±1 2πi Z ∞ −∞ dξ f (ξ) ξ − (k ± i0+), f ∈ W. (3.3)
Applying the projection Π−to (3.2a) and (3.2d) and the projection Π+to
(3.2b) and (3.2c), we arrive at 4×2 matrix singular integral equations (usually called Riemann-Hilbert problems in the integrable systems literature) which couple all four functions N , N , M , and M . To arrive at two equations coupling N and N and two equations coupling M and M , we introduce the so-called norming constants.
If all of the poles ikj in the upper half-plane C+ are simple, letting τj
and ˘τj be the residues of a(k)−1 and c(k)−1 at k = ikj and τj and ˘τj the
residues of a(k)−1 and c(k)−1 at k = −ik∗j, we define the norming constants Cj, Cj, Dj, and Dj by rewriting the residues of the left-hand sides of (3.2)
as follows: M (x, t, ikj)τj = e2iθ(x,t,ikj)N (x, t, ikj)Cj, (3.4a) M (x, t, −ikj∗)τj = e−2iθ(x,t,−ik ∗ j)N (x, t, −ik∗ j)Cj, (3.4b) N (x, t, −ikj∗)˘τj = e2iθ(x,t,−ik ∗ j)M (x, t, −ik∗ j)Dj, (3.4c) N (x, t, ikj)˘τj = e−2iθ(x,t,ikj)M (x, t, ikj)Dj. (3.4d) As a result, we get X j e2iθ(x,t,ikj)N (x, t, ik j)Cj k − ikj = N (x, t, k) − I2 02×2 − Z ∞ −∞ dξ 2πi e2iθ(x,t,ξ)N (x, t, ξ)ρ(ξ) ξ − k + i0+ , (3.5a)
X j e−2iθ(x,t,−ikj∗)N (x, t, −ik∗ j)Cj k + ikj∗ = N (x, t, k) − 02×2 I2 + Z ∞ −∞ dξ 2πi e−2iθ(x,t,ξ)N (x, t, ξ)ρ(ξ) ξ − k − i0+ , (3.5b) X j e2iθ(x,t,−ik∗j)M (x, t, −ik∗ j)Dj k + ik∗ j = M (x, t, k) − I2 02×2 + Z ∞ −∞ dξ 2πi e2iθ(x,t,ξ)M (x, t, ξ)r(ξ) ξ − k − i0+ , (3.5c) X j e−2iθ(x,t,ikj)M (x, t, ik j)Dj k − ikj = M (x, t, k) −02×2 I2 − Z ∞ −∞ dξ 2πi e−2iθ(x,t,ξ)M (x, t, ξ)r(ξ) ξ − k + i0+ . (3.5d)
Substituting k = ikj in (3.5b) and (3.5c) and k = −ik∗j in (3.5a) and (3.5d),
we obtain linear systems for N (x, t, ikj) and M (x, t, ikj) and for N (x, t, −ik∗j)
and M (x, t, −ik∗j), respectively, which enable us to compute the modified Jost solutions. These linear constraints are equivalent to the usual analyticity requirements on the modified Jost solutions.
If there is a pole of order mj at ikj, we define the corresponding
norm-ing constants Cj,s and Dj,s (s = 0, 1, . . . , mj − 1) in such a way that the
expressions M (x, t, k)a(k)−1− e2iθ(x,t,k)N (x, t, k) mj−1 X s=0 Cj,s (k − ikj)s+1 , N (x, t, k)c(k)−1− e−2iθ(x,t,k)M (x, t, k) mj−1 X s=0 Dj,s (k − ikj)s+1 ,
are analytic in a neighbourhood of k = ikj. In a similar way we define the
a way that the expressions M (x, t, k)a(k)−1− e2iθ(x,t,k)N (x, t, k) mj−1 X s=0 Cj,s (k + ik∗ j)s+1 , N (x, t, k)c(k)−1− e−2iθ(x,t,k)M (x, t, k) mj−1 X s=0 Dj,s (k + ik∗j)s+1,
are analytic in a neighborhood of k = −ikj∗. The resulting generalizations of (3.5) can then be used to derive linear constraints involving the values and k-derivatives of the modified Jost solutions up to the order mj− 1 which are
equivalent to the usual analyticity properties of the modified Jost solutions ([22, Appendix B] for double poles).
3.2
Derivation of Marchenko equations
The Marchenko integral equations are equivalent to the Riemann-Hilbert problems (3.5) (and their multipole analogs). Replacing the modified Jost solutions M , M , N , and N by their corresponding kernel functions L, L, K, and K by using the triangular representations and stripping off the Fourier transform, the Marchenko integral equations appear as a result. Instead of considering the Riemann-Hilbert problems in W−4×2, W+4×2, W+4×2, and
W4×2
− , we study the Marchenko equations in L1(−∞, x)4×2, L1(x, +∞)4×2,
L1(x, +∞)4×2, and L1(−∞, x)4×2, respectively. Equations (2.6) imply K(x, y, t) = 1 2π Z ∞ −∞ dk eik(y−x) N (x, t, k) − I2 02×2 , (3.6a) K(x, y, t) = 1 2π Z ∞ −∞ dk eik(x−y) N (x, t, k) −02×2 I2 , (3.6b) L(x, y, t) = 1 2π Z ∞ −∞ dk eik(y−x) M (x, t, k) − I2 02×2 , (3.6c) L(x, y, t) = 1 2π Z ∞ −∞ dk eik(x−y) M (x, t, k) −02×2 I2 . (3.6d) Applying 2π1 R−∞∞ dk eik(y−x) to (3.5a) and (3.5c) and 1
2π
R∞
−∞dk e
ik(x−y) to
ex-ample, [7] for more details on the following equations) K(x, y, t) + 02×2 Fr(x + y, t) + Z ∞ x dz K(x, z, t)Fr(z + y, t) = 04×2, (3.7a) K(x, y, t) +Fr(x + y, t) 02×2 + Z ∞ x dz K(x, z, t)Fr(z + y, t) = 04×2, (3.7b) L(x, y, t) + 02×2 Fl(x + y, t) + Z x −∞ dz L(x, z, t)Fl(z + y, t) = 04×2, (3.7c) L(x, y, t) +Fl(x + y, t) 02×2 + Z x −∞ dz L(x, z, t)Fl(z + y, t) = 04×2, (3.7d)
where the Marchenko kernels are given by Fr(w, t) = ˆρ(w, t) − i X j e−kjwe−4ik2jtC j, (3.8a) Fr(w, t) = ˆρ(w, t) + i X j e−k∗jwe4ikj∗2tC j, (3.8b) Fl(w, t) = ˆr(w, t) + i X j ek∗jwe−4ik ∗ j 2t Dj, (3.8c) Fl(w, t) = ˆr(w, t) − i X j ekjwe4ikj2tD j. (3.8d)
The Marchenko integral kernels Fr, Fr, Fl, and Flare complex symmetric
matrices if Q is a complex symmetric matrix. Moreover, in this case
Fr(w, t)† = ΩFr(w, t)Σ, Fr(w, t)†= Σ−1Fr(w, t)Ω−1, (3.9a)
Fl(w, t)† = ΩFl(w, t)Σ, Fl(w, t)†= Σ−1Fl(w, t)Ω−1, (3.9b)
since the Marchenko integral kernels satisfy the same conjugation and trans-position properties as the corresponding Fourier transposed reflection coeffi-cients [10, 8]. Thus, for complex symmetric potentials Q(x, t) the norming constants Cj, Cj, Dj, and Dj are complex symmetric matrices satisfying
Cj†= ΩCjΣ, D † j = Σ
−1
DjΩ−1.
Similar properties hold for the generalized norming constants in the case of multiple poles.
The principle of generalizing the symmetries of the reflection coefficients to identical symmetries for the corresponding Marchenko integral kernels tells us that the four Marchenko integral kernels are complex symmetric 2 × 2 matrices. A rigorous proof of the above principle can be found in [10, 8].
4
Matrix triplet method
In this section we employ the matrix triplet method to express the reflec-tionless NLS solutions in a triplet of three matrices. The impact on these triplets of having a complex symmetric potential Q(x, t) is discussed at the end of this section.
4.1
Marchenko kernels and kernel functions
Let us assume that the reflection coefficients ρ, ρ, r, and r vanish identically. Then the Marchenko integral kernels have separated variables. We can then introduce the four matrix triplets (Ar, Br, Cr), (Ar, Br, Cr), (Al, Bl, Cl), and
(Al, Bl, Cl) such that Fr(w, t) = Cre−wAre−4itA 2 rB r, (4.1a) Fr(w, t) = Cre−wAre4itA 2 rB r, (4.1b) Fl(w, t) = ClewAle−4itA 2 lB l, (4.1c) Fl(w, t) = ClewAle4itA 2 lB l. (4.1d)
The four Marchenko kernels are complex symmetric 2 × 2 matrices and each triplet is such that the dimension of the matrices Ar,l (Ar,l), Cr,l (Cr,l) and
Br,l (Br,l) are, respectively, p × p, 2 × p and p × 2. If the triplet (Ar, Br, Cr)
is minimal in the sense of representing Fr(w, t) using matrices of minimal
sizes, then there exists a complex symmetric invertible matrix Sr such that
SrAr = ATrSr, SrBr = CrT, and CrSr−1 = BrT (cf. [5]). Analogous properties
hold for the other three matrix triplets when minimal. Equations (3.7) imply K(x, y, t) = −Wr(x, t)e−4itA 2 re−yArB r, (4.2a) K(x, y, t) = −Wr(x, t)e4itA 2 re−yArB r, (4.2b) L(x, y, t) = −Wl(x, t)e−4itA 2 leyAlB l, (4.2c) L(x, y, t) = −Wl(x, t)e4itA 2 leyAlB l, (4.2d)
where Wr(x, t) = 02×p Cre−xAr + Z ∞ x dz K(x, z, t)Cre−zAr, (4.3a) Wr(x, t) = Cr e−xAr 02×p + Z ∞ x dz K(x, z, t)Cre−zAr, (4.3b) Wl(x, t) = 02×p ClexAl + Z x −∞ dz L(x, z, t)ClezAl, (4.3c) Wl(x, t) = ClexAl 02×2 + Z x −∞ dz L(x, z, t)ClezAl. (4.3d)
Introducing the unique solutions Qr = Z ∞ 0 dz e−zArB rCre−zAr, Nr = Z ∞ 0 dz e−zArB rCre−zAr, (4.4a) Ql = Z 0 −∞ dz ezAlB lClezAl, Nl = Z 0 −∞ dz ezAlB lClezAl, (4.4b)
of the Sylvester equations
ArQr+ QrAr = BrCr, ArNr+ NrAr = BrCr, (4.5a)
AlQl+ QlAl = BlCl, AlNl+ NlAl = BlCl, (4.5b)
and substituing equations (4.1) and (4.2) in (3.7) we obtain the linear systems
Wr(x, t) Wr(x, t) Ip Kr(x, t) Jr(x, t) Ip =Cre −xAr 0 2×p 02×p Cre−xAr , (4.6a) Wl(x, t) Wl(x, t) Ip Kl(x, t) Jl(x, t) Ip =Cle xAl 0 2×p 02×p ClexAl , (4.6b) where Jr(x, t) = e−4itA 2 re−xArN re−xAr, Kr(x, t) = e4itA 2 re−xArQ re−xAr, Jl(x, t) = e−4itA 2 lexAlQ lexAl, Kl(x, t) = e4itA 2 lexAlN lexAl.
From the systems (4.6a) and (4.6b) we easily get Wr(x, t)Γr(x, t) = Cre−xAr −Cre−2xAre−4itA 2 rN re−xAr (4.7a) Wr(x, t)Γr(x, t) =−Cr e−2xAre4itA2rQ re−xAr Cre−xAr (4.7b) Wl(x, t)Γl(x, t) = −Cle2xAle4itA 2 lNlexAl Cle−2xAl (4.7c) Wl(x, t)Γl(x, t) = C le−xAl −Cle2xAle−4itA 2 lQ lexAl . (4.7d) where Γr(x, t) = Ip− e4itA 2 re−xArQ re−4itA 2 re−2xArN re−xAr, (4.8a) Γr(x, t) = Ip− e−4itA 2 re−xArN re4itA 2 re−2xArQ re−xAr, (4.8b) Γl(x, t) = Ip− e4itA 2 lexAlN le−4itA 2 le2xAlQ lexAl, (4.8c) Γl(x, t) = Ip− e−4itA 2 lexAlQ le4itA 2 le2xAlN lexAl, (4.8d)
Then, from equations (4.2) we arrive at the following Kup(x, y, t) = Cre−2xAre4itA 2 rQ re−xArΓr(x, t)−1e−4itA 2 re−yArB r, (4.9a) Kdn(x, y, t) = −Cre−xArΓr(x, t)−1e−4itA 2 re−yArB r, (4.9b) Kup (x, y, t) = −Cre−xArΓr(x, t)−1e4itA 2 re−yArB r, (4.9c) Kdn (x, y, t) = Cre−2xAre−4itA 2 rN re−xArΓr(x, t)−1e4itA 2 re−yArB r, (4.9d)
provided the matrix inverses exist. Proceeding in a very similar way we get as well Lup (x, y, t) = Cle2xAle4itA 2 lN lexAlΓl(x, t)−1e−4itA 2 leyAlB l, (4.10a) Ldn (x, y, t) = −ClexAlΓl(x, t)−1e−4itA 2 leyAlB l, (4.10b) Lup(x, y, t) = −ClexAlΓl(x, t)−1e4itA 2 leyAlB l, (4.10c) Ldn(x, y, t) = Cle2xAle−4itA 2 lQ lexAlΓl(x, t)−1e4itA 2 leyAlBl, (4.10d)
provided the matrix inverses exist. In fact, applying the Sherman-Morrison formula [14] to Γr(x, t) = I2−Kr(x, t)Jr(x, t), Γr(x, t) = I2−Jr(x, t)Kr(x, t),
Γl(x, t) = I2− Kl(x, t)Jl(x, t), and Γl(x, t) = I2− Jl(x, t)Kl(x, t), we obtain
det Γr(x, t) = det Γr(x, t), det Γl(x, t) = det Γl(x, t).
Matrix triplets where the Sylvester equations (4.5) are uniquely solvable, but Ar, Ar, Al, or Al may have some eigenvalues with nonpositive real part,
can always be transformed into matrix triplets of the above kind without changing the Marchenko kernels [3].
In case 1 (defocusing case) there do not exist any soliton solutions [22]. In the other three cases we replace the overlined quantities by complex conjugate transposes or ordinary transposes using the symmetry relations. In fact [5], if the representations (4.1) are minimal in the sense that Ar, Ar, Al, and
Al have minimal matrix order among all matrix triplets leading to the same
Marchenko integral kernels, there exist unique nonsingular matrices Sr and
Sl such that Ar = Sr−1A † rSr, Br = Sr−1C † rΣ −1 , Cr = Ω−1Br†Sr, (4.11a) Al = Sl−1A † lSl, Bl= Sl−1C † lΣ −1 , Cl = Ω−1B † lSl. (4.11b)
Then the Lyapunov solutions in (4.4) satisfy
Qr = Sr−1Qr,Σ, Nr = Nr,ΩSr, Ql = Sl−1Ql,Σ, Nl = Nl,ΩSl, (4.12)
where Qr,Σ, Nr,Ω, Ql,Σ, and Nl,Ω are the hermitian matrices given by
Qr,Σ = Z ∞ 0 dz e−zA†rC† rΣ −1 Cre−zAr, (4.13a) Nr,Ω = Z ∞ 0 dz e−zArB rΩ−1Br†e −zA†r, (4.13b) Ql,Σ= Z ∞ 0 dz e−zA†lC† lΣ −1C le−zAl, (4.13c) Nl,Ω= Z ∞ 0 dz e−zAlB lΩ−1B † le −zA†l. (4.13d)
It is readily verified that the expressions in (4.13) are the unique solutions of the respective Lyapunov equations
A†rQr,Σ+ Qr,ΣAr = Cr†Σ −1 Cr, ArNr,Ω+ Nr,ΩA†r = BrΩ−1Br†, (4.14a) A†lQl,Σ+ Ql,ΣAl = C † lΣ −1C l, AlNl,Ω+ Nl,ΩA † l = BlΩ −1B† l. (4.14b)
Using (4.9), (4.10), (4.11), and (4.12) we obtain Kup(x, y, t) = Ω−1Br† h e−4itA†r 2 e2xA†r − Q r,Σe−4itA 2 re−2xArN r,Ω i−1 × × Qr,Σe−4itA 2 re−(x+y)ArB r, (4.15a) Kdn(x, y, t) = −Cr h e4itA2re2xAr − N r,Ωe4itA † r 2 e−2xA†rQ r,Σ i−1 × × e−(y−x)ArB r, (4.15b) Kup (x, y, t) = −Ω−1Br†he−4itA†r 2 e2xA†r − Q r,Σe−4itA 2 re−2xArN r,Ω i−1 × × e−(y−x)A†rC† rΣ −1 , (4.15c) Kdn (x, y, t) = Cr h e4itA2re2xAr − N r,Ωe4itA † r 2 e−2xA†rQ r,Σ i−1 × × Nr,Ωe4itA † r 2 e−(x+y)A†rC† rΣ −1 , (4.15d) as well as Lup (x, y, t) = Cl h e−4itA2le−2xAl− N l,Ωe−4itA † l 2 e2xA†lQ l,Σ i−1 × × Nl,Ωe−4itA † l 2 e(x+y)A†lC† lΣ −1 , (4.16a) Ldn (x, y, t) = −Ω−1Bl†he4itA†l 2 e−2xA†l − Q l,Σe4itA 2 le2xAlN l,Ω i−1 × × e−(x−y)A†lC† lΣ −1 , (4.16b) Lup(x, y, t) = −Cl h e−4itA2le−2xAl− N l,Ωe−4itA † l 2 e2xA†lQ l,Σ i−1 × × e−(x−y)AlB l, (4.16c) Ldn(x, y, t) = Ω−1Bl†he4itA†l 2 e−2xA†l − Q l,Σe4itA 2 le2xAlN l,Ω i−1 × × Ql,Σe4itA 2 le(x+y)AlB l. (4.16d)
We observe that the expressions (4.15) and (4.16) do not require the mini-mality of any matrix triplets.
Let us now use (2.8) in (4.15) and (4.16). We get Q(x, t) = 2Ω−1Br† h e2xA†re−4itA†r 2 − Qr,Σe−2xAre−4itA 2 rN r,Ω i−1 Cr†Σ−1, (4.17a) Q(x, t) = −2Cl h e−2xAle−4itA2l − N l,Ωe2xA † le−4itA † l 2 Ql,Σ i−1 Bl, (4.17b) R(x, t) = 2Cr h e2xAre4itA2r − N r,Ωe−2xA † re4itA † r 2 Qr,Σ i−1 Br, (4.17c) R(x, t) = −2Ω−1Bl†he−2xA†le4itA † l 2 − Ql,Σe2xAle4itA 2 lN l,Ω i−1 Cl†Σ−1. (4.17d)
Observe that the symmetry relation R = ΣQ†Ω is satisfied.
4.2
Jost solutions and scattering coefficients
The modified Jost solutions are as follows: Nup(x, t, k) = I2− iΩ−1Br† h e−4itA†r 2 e2xA†r − Q r,Σe−4itA 2 re−2xArN r,Ω i−1 × × Qr,Σe−4itA 2 re−2xAr(kI p− iAr)−1Br, (4.18a) Ndn(x, t, k) = iCr h e4itA2re2xAr − N r,Ωe4itA † r 2 e−2xA†rQ r,Σ i−1 × × (kIp− Ar)−1Br, (4.18b) Nup (x, t, k) = −iΩ−1Br†he−4itA†r 2 e2xA†r − Q r,Σe−4itA 2 re−2xArN r,Ω i−1 × × (kIp+ iA†r) −1 Cr†Σ−1, (4.18c) Ndn (x, t, k) = I2+ iCr h e4itA2re2xAr − N r,Ωe4itA † r 2 e−2xA†rQ r,Σ i−1 × × Nr,Ωe4itA † r 2 e−2xA†r(kI p+ iA†r) −1 Cr†Σ−1, (4.18d) as well as Mup (x, t, k) = I2+ iCl h e−4itA2le−2xAl− N l,Ωe−4itA † l 2 e2xA†lQ l,Σ i−1 × × Nl,Ωe−4itA † l 2 e2xA†l(kI p+ iA † l) −1 Cl†Σ−1, (4.19a) Mdn (x, t, k) = −iΩ−1Bl†he4itA†l 2 e−2xA†l − Q l,Σe4itA 2 le2xAlN l,Ω i−1 × × (kIp+ iA † l) −1C† lΣ −1, (4.19b)
Mup(x, t, k) = iCl h e−4itA2le−2xAl− N l,Ωe−4itA † l 2 e2xA†lQ l,Σ i−1 × × (kIp− iAl)−1Bl, (4.19c) Mdn(x, t, k) = I2− iΩ−1Bl† h e4itA†l 2 e−2xA†l − Ql,Σe4itA 2 le2xAlN l,Ω i−1 × × Ql,Σe4itA 2 le2xAl(kI p− iAl)−1Bl. (4.19d)
Taking suitable limits as x → ±∞, we obtain the scattering coefficients a(k) = I2+ iΩ−1Br†N −1 r,Ω(kIp− iAr)−1Br, (4.20a) a(k) = I2− iCrQ−1r,Σ(kIp+ iA†r) −1 Cr†Σ−1, (4.20b) c(k) = I2− iClQ−1l,Σ(kIp + iA†l)−1Cl†Σ−1, (4.20c) c(k) = I2+ iΩ−1B † lN −1 l,Ω(kIp− iAl)−1Bl, (4.20d)
provided the matrices Qr,Σ, Nr,Ω, Ql,Σ, and Nl,Ω are invertible. It is easily
verified that their invertibility implies that the matrix triplets (Ar, Br, Cr)
and (Al, Bl, Cl) are minimal. Using the Lyapunov equations we obtain the
inverted scattering coefficients
a(k)−1 = I2− iΩ−1Br†(kIp+ iA†r) −1 Nr,Ω−1Br, (4.21a) a(k)−1 = I2+ iCr(kIp− iAr)−1Q−1r,ΣC † rΣ −1 , (4.21b) c(k)−1 = I2+ iCl(kIp− iAl)−1Q−1l,ΣC † lΣ −1 , (4.21c) c(k)−1 = I2− iΩ−1B † l(kIp+ iA † l) −1 Nl,Ω−1Bl. (4.21d)
So far we have not used the complex symmetry of the potential Q(x, t) to derive symmetry relations for the matrix triplets (Ar, Br, Cr) and (Al, Bl, Cl).
This we intend to remedy presently. Starting from the Marchenko integral kernel representations (3.8) with zero reflection coefficient terms, under the condition of minimality of the triplets there exists, for (3.8a), a unique in-vertible p × p matrix S such that
ArS = SATr, SBr = CrT, B T
r = CrS−1. (4.22)
Taking transposes, we obtain the similarity relations ArST = STATr, S
TB
r = CrT, B T
r = CrST −1. (4.23)
Using the uniqueness of the similarity, we get ST = S. In other words, S
matrix, then Ar is a complex symmetric matrix and Br is the transpose of
Cr, though it might not be immediate how to relate the matrix triplet to the
norming constants as defined in [22]. The same conclusions can be drawn for the other three matrix triplet representations whenever minimal.
4.3
Four cases of the matrix triplet method
In case 1 (defocusing case, Σ = Ω = I2) there do not exist any multisoliton
solutions.
In case 2 (focusing case, Σ = −Ω = I2) the matrices Qr,Σ, −Nr,Ω, Ql,Σ,
and −Nl,Ω are positive selfadjoint and hence nonsingular [24, 6] iff the
cor-responding matrix triplet is minimal. This implies that the four matrices in (4.8) have a positive determinant for each x ∈ R. Thus in case 2 any matrix triplet, minimal or not, leads to a multisoliton solution without finite singularities which is exponentially decaying as x → ±∞ (see [6, 24]).
In cases 3 and 4 it is possible for the matrix triplet (Ar, Br, Cr) to be
minimal, whereas one (or both) of the matrices Qr,Σ and Nr,Ω is singular. In
this situation we may find reflectionless NLS solutions that decay exponen-tially as x → +∞ but do not decay as x → −∞. We refer to [22, Eq. (84)] for an example of this situation. In either case it is also possible that, for some instant t, one (or both) of the matrices Γr(x, t) and Γr(x, t) is singular
at certain finite x-values. In this situation the reflectionless solution has a pole at some finite x for this particular instant t and does not represent a soliton solution.
Let us illustrate these phenomena using well-known results for one-soliton solutions with norming constant C1 = (cc10 cc−10 ). As shown in [22, Sec. 4], a
one-soliton solution (without singularities) occurs under the necessary and sufficient condition that
( |c1|2+ |c−1|2− 2|c0|2 < 2|c1c−1− c20|, det C1 6= 0, |c1| = |c−1| and c20 = c1c−1, det C1 = 0, in Case 3 and ( |c1|2+ |c−1|2− 2|c0|2 > −2|c1c−1− c20|, det C1 6= 0,
any singular matrix C1, det C1 = 0,
in Case 4. Thus there do not exist norming constants C1 that do not lead to
5
Substitution in the matrix NLS equation
In this section we prove, by direct substitution, that the potentials Q(x, t) in (4.17b) and R(x, t) in (4.17d) satisfy the NLS equation (1.1). A similar substitution proved the potentials Q(x, t) and R(x, t) in (4.19a) and (4.19c) to satisfy the NLS equation [4].
Let us write Q(x, t) = −2ClexAlΓl(x, t)−1e4itA 2 lexAlB l, (5.1) where Γl(x, t) = Ip− e4itA 2 lexAlN l,Ωe−4itA † l 2 e2xA†lQ l,ΣexAl, (5.2)
p being the matrix order of Al. Then we can write Γl(x, t) as follows
Γl(x, t) = Ip− N (x, t)Q(x), (5.3) where N (x, t) = e4itA2 lexAlN l,Ωe−4itA † l 2 exA†l, Q(x) = exA † lQ l,ΣexAl.
It is then easily verified that
Nx = AlN + N A†l, (5.4a) Qx = A † lQ + QAl, (5.4b) Nt= 4i h A2lN − N A†l2i, (5.4c) (Γ−1l )x = −Γ−1l (Γl)xΓ−1l = 2Γ−1l (Al+ N A † lQ)Γ −1 l − Γ −1 l Al− AlΓ−1l , (5.4d) (Γ−1l )t= −Γ−1l (Γl)tΓ−1l = 4iΓ −1 l (A 2 lΓl− A2l + N (A † l) 2Q)Γ−1 l , (5.4e)
Then we readily differentiate (5.1) with respect to x to get Qx= −2ClexAlΓl−1Al+ AlΓ−1l − Γ −1 l (Γl)xΓ−1l e 4itA2 lexAlB l = −4ClexAlΓ−1l h Al+ N A † lQ i Γ−1l e4itA2lexAlB l, (5.5)
Let us now substitute Q and R in (4.17b) and (4.17d) into (1.1). Dif-ferentiating (5.1) and taking into account that Γl+ N Q = Ip and equation
(5.4c), we obtain iQt= 8ClexAlΓ−1l h A2lΓl+ A2lN Q − N A † l 2 QiΓ−1l e4itA2lexAlB l = 8ClexAlΓ−1l h A2l − N A†l2QiΓ−1l e4itA2lexAlB l. (5.6) Using (Al+ N A † lQ)x = AlN A † lQ + N A † lQAl+ 2N A † l 2 Q, we obtain Qxx = −4ClexAl n AlΓ−1l (Al+ N A † lQ)Γ −1 l + Γ −1 l (Al+ N A † lQ)Γ −1 l Al o e4itA2lexAlB l − 4ClexAlΓ−1l n AlN A†lQ + N A†lQAl+ 2N A†l 2 QoΓ−1l e4itA2lexAlB l − 4ClexAl h 2Γ−1l (Al+ N A † lQ)Γ −1 l − Γ −1 l Al− AlΓ−1l i (Al+ N A † lQ)Γ −1 l e 4itA2lexAlB l − 4ClexAlΓ−1l (Al+ N A † lQ) h 2Γ−1l (Al+ N A † lQ)Γ −1 l − Γ −1 l Al− AlΓ−1l i e4itA2lexAlB l = 8ClexAlΓ−1l (A2l − N A † l 2 Q)Γ−1l e4itA2lexAlB l − 16ClexAlΓ−1l (Al+ N A † lQ)Γ −1 l (Al+ N A † lQ)Γ −1 l e 4itA2 lexAlB l, (5.7)
where we have used (5.3). Finally, using that AlN + N A † l = e 4itA2 lexAlB lΩ−1B † le −4itA†l2 exA†l, A†lQ + QAl = exA † lC† lΣ −1 CexAl, we get −2QRQ = 16ClexAlΓ−1l (AlN + N A†l)Γ†l −1 (A†lQ + QAl)Γ−1l e4itA 2 lexAlBl = −16ClexAlΓ−1l h A2l − N A†l2QiΓ−1l e4itA2lexAlB l + 16ClexAlΓ−1l (Al+ N A † lQ)Γ −1 l (Al+ N A † lQ)Γ −1 l e 4itA2 lexAlB l. (5.8)
Since the final expressions in (5.6), (5.7), and (5.8) add up to zero, the potential Q(x, t) satisfies the matrix NLS equation (1.1).
6
Examples of reflectionless NLS solutions
In this section we compute the reflectionless NLS solutions explicitly for certain elementary matrix triplets based on the Jordan normal form. We also give necessary and sufficient condition for the minimality of these triplets.
It is well known that it is possible to choose the triplet (Al, Bl, Cl) in a
“canonical way.” Following the same arguments used in [4], for some appro-priate positive integer r, we have
Al= A1 0 . . . 0 0 A2 . . . 0 .. . ... . .. ... 0 0 · · · Ar , (6.1a) Bl = B1 .. . Br , Cl= C1 . . . Cr , (6.1b) where in the case of an eigenvalue λj of iAj with positive imaginary part the
corresponding blocks are given by
Aj := −iλjI2 −I2 0 . . . 0 0 0 −iλjI2 −I2 . . . 0 0 0 0 −iλjI2 . . . 0 0 .. . ... ... . .. ... ... 0 0 0 . . . −iλjI2 −I2 0 0 0 . . . 0 −iλjI2 . (6.1c)
Here Aj has size 2nj × 2nj, Bl has size 2nj × 2, and Cl has size 2 × 2nj.
Observe that 2(n1+ . . . + nr) = p.
To arrive at a reflectionless solution formula for a complex symmetric potential Q(x, t), we let En be the n × n matrix with I2 as its entries on the
trailing diagonal and 02×2as its other entries. In the matrix triplet (6.1) the
similarity transformation S as in (6.2) below making ATl similar to Al is the
direct sum
We then define Bl in terms of Cl by Bl = SClT or Bl = En1C T 1 En2C T 2 .. . EnrC T r ,
where we note that premultiplication of a block matrix with 2 × 2 entries having n rows by En leads to a reversal of the order of the rows in this
matrix. If Alis a diagonal matrix with 2 × 2 diagonal entries, we have S = Ip
and Bl = ClT. Unfortunately, in this way it is not easy to relate the matrix
triplet to the norming constants as defined in [22].
To define a minimal triplet for a complex symmetric potential whose entries are easily related to the norming constants, we now consider the matrix triplet (Al, Bl, Cl), where
Al= diag(−iλ1I2, . . . , −iλrI2)
for distinct algebraically simple eigenvalues λ1, . . . , λr ∈ C+ and
Bl = I2 .. . I2 , Cl = −i C1 . . . Cr .
Then the identities (4.22) (with the subscripts r replaced by l) imply that S = −i diag(CT
1, . . . , CrT) for suitable invertible 2 × 2 matrices C1, . . . , Cr.
Thus if we take the complex symmetric matrices C1, . . . , Cr as the norming
constants, then the triplet (Al, Bl, Cl) is minimal iff none of the norming
constants is a singular matrix.
Example 6.1 (One-soliton solution) Consider the matrix triplet Al = (η − iξ)I2, Bl = Ω, Cl= −iC1Ω−1,
where ξ ∈ R, η > 0, and C1 is a complex symmetric 2 × 2 matrix. Then (cf.
[22, Eq. (71)])
Q(x, t) = 2ie−2η(x+4ξt)e−4it(ξ2−η2)e−2iξx× × I2− e−4η(x+4ξt) 4η2 Ω −1 C1†Σ−1C1 −1 Ω−1C1†Σ−1 (6.2)
is a one-soliton solution. The triplet is minimal iff det C1 is nonzero. It is
easily verified that
Nup(x, t, k) = I2− i e−4η(x+4ξt) 2η[k − ξ − iη] I2− e−4η(x+4ξt) 4η2 Ω −1 C1†Σ−1C1 −1 Ω−1C1†Σ−1, so that Nup(x, t, ξ − iη) = I2− e−4η(x+4ξt) 4η2 Ω −1 C1†Σ−1C1 −1 .
Example 6.2 (Two-soliton solutions) We plot (figures 1, 2 and 3)) the three distinct elements of the complex symmetric matrix Q(x, t) for the same triplet (Ar, Br, Cr) in the cases 2, 3 and 4, where
Ar = 3 0 0 0 0 3 0 0 0 0 2 0 0 0 0 2 , Cr = −i 1 2 1 0 2 3 0 1 , Br= 1 0 0 1 1 0 0 1 .
We observe that the plots of case 4 is similar to the plots of the focusing case (case 2). Instead, in case 3, the plots show that a singularity arises in the following matrix (which appears in the expression of the soliton solution (4.17a)) h e2xA†re−4itA†r 2 − Qr,Σe−2xAre−4itA 2 rN r,Ω i .
This fact is not surprising because the case 3 is closed to the defocusing case where, as underlined in the previous section, no soliton solutions exists.
Note that Cr consists of two nonsingular block elements. We have not
found necessary and sufficient conditions on the two discrete eigenvalues and the associated norming constants in order that, in Case 3 or in Case 4, there are no finite singularities.
Let us now consider the matrix triplet (Ar, Br, Cr) as in (6.1) for distinct
Figure 1: An example of a two soliton solution in the focusing case.
Figure 2: An example of a two soliton solution in case 3. Note the presence of a finite singularity.
first of the identities (4.22) imply that S = diag(S1, . . . , Sr), where Sj has
size 2nj× 2nj and has the block Hankel form
Sj = S1;j S2;j S3;j . . . Snj;j S2;j S3;j S4;j . . . Snj;j 02×2 S3;j S4;j S5;j . . . 02×2 02×2 .. . 02×2 02×2 .. . Snj;j 02×2 . . . 02×2 02×2 Snj;j 02×2 . . . 02×2 02×2 .
Then S is invertible iff all matrices Snj;j are invertible. If we then choose
Cj = −i inj−1C1;j inj−2C2;j . . . iCnj−1;j Cnj;j
and BT
j = I2 02×2 . . . 02×2, then Sj;l = inl−j−1Cj;lT (l = 1, 2, . . . , nj). In
other words, the triplet (Al, Bl, Cl) is minimal iff the matrices Cnj;j among
the matrices generalizing the norming constants to multipole situations (i.e., to nj ≥ 2) are invertible.
Example 6.3 (Double pole soliton solutions) In the Cases 2, 3 and 4 we plot the three distinct elements of the complex symmetric matrix Q(x, t) for the same matrix triplet (Ar, Br, Cr), where
Ar = 3 0 −1 0 0 3 0 −1 0 0 3 0 0 0 0 3 , Cr = −i i 2i 1 0 2i 3i 0 1 , Br= 1 0 0 1 1 0 0 1 .
The plots show that in Case 3 (see figure 5) the elements of Q(x, t) have the same finite singularity and that in Case 4 (see figure 6) these elements do not have any finite singularities. Moreover, the plots of Case 2 and 4 are very similar because of the “structures” of these two cases are very similar. Also for the double poles solutionswe have not found necessary and sufficient conditions on the discrete eigenvalue and the two associated norming con-stants in order that, in Case 3 or in Case 4, there are no finite singularities. However, we observe that in the Case 4
Figure 4: An example of a double pole solution in the focsing case.
Figure 5: An example of a double pole solution in case 3. Note the presence of a finite singularity.
Acknowledgments
The research leading to this article has been partially supported by the Re-gione Autonoma Sardegna in the framework of the research programs Integro-Differential equations and non-local problems and Algorithms and Models for Imaging Science [AMIS], and by INdAM-GNFM (Istituto Nazionale di Alta Matematica, National Institute of Advanced Mathematics – Gruppo Nazionale per la Fisica Matematica, National Group for Mathematical Physics).
A
Integral equations for kernel functions
In this appendix we list the integral equations for the auxiliary functions K(x, y, t), K(x, y, t), L(x, y, t), and L(x, y, t). Their proof can be found in the literature [2, 6, 24]. For the sake of brevity, we omit the time variable in all equations.
We have the following Volterra integral equations for the kernel functions:
Kup(x, y) = − Z ∞ x dz Q(z)Kdn(z, z + y − x), (A.1a) Kdn(x, y) = −1 2R( 1 2[x + y]) − Z 12[x+y] x dz R(z)Kup(z, x + y − z), (A.1b) Kup (x, y) = −1 2Q( 1 2[x + y]) − Z 12[x+y] x dz Q(z)Kdn (z, x + y − z), (A.1c) Kdn (x, y) = − Z ∞ x dz R(z)Kup (z, z + y − x), (A.1d)
as well as Lup (x, y) = Z x −∞ dz Q(z)Ldn (z, z + y − x), (A.2a) Ldn (x, y) = 12R(12[x + y]) + Z 1 2[x+y] x dz R(z)Lup (z, x + y − z), (A.2b) Lup(x, y) = 12Q(12[x + y]) + Z 12[x+y] x dz Q(z)Ldn(z, x + y − z), (A.2c) Ldn(x, y) = Z x −∞ dz R(z)Lup(z, z + y − x). (A.2d) Equations (A.1) and (A.2) obviously imply (2.8).
Suppose that F (y) is a matrix function of y ≥ x ≥ x0or a matrix function
of y ≤ x ≤ x0. Then we define µ(F, x) = (R∞ x dy kF (y)k, x ≥ x0, Rx −∞dy kF (y)k, x ≤ x0.
Then (A.1) and (A.2) imply µ(Kup, x) ≤ Z ∞ x dz kQ(z)kµ(Kdn, z), (A.3a) µ(Kdn, x) ≤ Z ∞ x dz kR(z)k + Z ∞ x dz kR(z)kµ(Kup, z), (A.3b) µ(Kup, x) ≤ Z ∞ x dz kQ(z)k + Z ∞ x dz kQ(z)kµ(Kdn , z), (A.3c) µ(Kdn, x) ≤ Z ∞ x dz kR(z)kµ(Kup , z), (A.3d) as well as µ(Lup , x) ≤ Z x −∞ dz kQ(z)kµ(Ldn , x), (A.4a) µ(Ldn , x) ≤ Z x −∞ dz kR(z)k + Z x −∞ dz kR(z)kµ(Lup , x), (A.4b) µ(Lup, x) ≤ Z x −∞ dz kQ(z)k + Z x −∞ dz kQ(z)kµ(Ldn, z), (A.4c) µ(Ldn, x) ≤ Z x −∞ dz kR(z)kµ(Lup, z). (A.4d)
The unique solvability of (A.1) and (A.2) now follows immediately by apply-ing Gronwall’s inequality.
Using Gronwall’s inequality we get µ(Kup, x) + µ(Kdn, x) ≤ Z ∞ x dz kQ(z)k exp Z ∞ x dz kQ(z)k , (A.5a) µ(Kup , x) + µ(Kdn , x) ≤ Z ∞ x dz kQ(z)k exp Z ∞ x dz kQ(z)k , (A.5b) µ(Lup , x) + µ(Ldn , x) ≤ Z x −∞ dz kQ(z)k exp Z x −∞ dz kQ(z)k , (A.5c) µ(Lup, x) + µ(Ldn, x) ≤ Z x −∞ dz kQ(z)k exp Z x −∞ dz kQ(z)k , (A.5d) where we have used that kQ(z)k ≡ kR(z)k.
Equations (2.8a) and (2.8b) are to be interpreted as follows: The integral terms in (A.1) and (A.2) are continuous in y ∈ [x, +∞) and in y ∈ (−∞, x], respectively. As a result, lim z→x+ 2K dn (x, 2z − x) + R(z) = 0, (A.6a) lim z→x+k2K up (x, 2z − x) + Q(z)k = 0, (A.6b) lim z→x−k2L dn (x, 2z − x) − R(z)k = 0, (A.6c) lim z→x− 2Lup(x, 2z − x) − Q(z) = 0, (A.6d)
where we have substituted z = 12[x + y]. If the entries of Q(x) and Qx both
belong to L1(R, dx), then (2.8a) and (2.8b) are valid pointwise.
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