- Email: [email protected]
Complex short-pulse solutions by gauge transformation
Journal Pre-proof Complex short-pulse solutions by gauge transformation Cornelis van der Mee
PII: DOI: Reference:
S0393-0440(19)30220-7 https://doi.org/10.1016/j.geomphys.2019.103539 GEOPHY 103539
To appear in:
Journal of Geometry and Physics
Received date : 30 August 2019 Revised date : 8 October 2019 Accepted date : 5 November 2019 Please cite this article as: C.v.d. Mee, Complex short-pulse solutions by gauge transformation, Journal of Geometry and Physics (2019), doi: https://doi.org/10.1016/j.geomphys.2019.103539. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Complex short-pulse solutions by gauge transformation Cornelis van der Mee∗
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Abstract
In this article we apply the gauge transformation method for solving the Heisenberg ferromagnet equation by using the NLS Jost solutions to derive solutions of the complex short-pulse equation from the Jost functions of the complex sine-Gordon equation. The matrix triplet method is used to compute the most general reflectionless complex short-pulse solutions by solving the Marchenko integral equation for the complex sine-Gordon equation.
Introduction
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1
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The short-pulse equation
uxt = u + 61 (u3 )xx
(1.1)
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appears as the integrability condition of one of the pseudospherical surfaces studied in [27] (also [7]). This equation was also derived to describe the propagation of ultra-short optical pulses in silica optical fibers [30]. Equation (1.1) is known to be integrable [28] and its bi-Hamiltonian structure and conservation laws have been studied in detail [9, 10, 11]. Equation (1.1) can be related to the sine-Gordon equation and is known to have (anti)loop soliton solutions [29, 24], where, for certain t ∈ R, u(x, t) is a multi-valued function of x on certain bounded x-intervals. Loop soliton solutions have ∗ Dip. Matematica e Informatica, Universit`a di Cagliari, Viale Merello 92, 09123 Cagliari, Italy. Email: [email protected]
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(1.2)
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uxt = u + 12 (|u|2 ux )x
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also been obtained by alternative methods [34, 22, 25, 18, 8]. Well-posedness issues were studied in [26, 12, 23]. The complex short-pulse equation
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was derived and studied by Feng [16]. Like its real counterpart (1.1), it is integrable and can in fact be derived as the compatibility condition (Vx )t = (Vt )x of the Lax pair equations 1 iux V, (1.3a) Vx = −ik −iu∗x −1 i 1 1 1 2 2 − ik|u| − iu + k|u| u x 4k 2 2 2 Vt = V. (1.3b) − 12 iu∗ − 12 k|u|2u∗x − 4ki + 12 ik|u|2
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The one-soliton and two-soliton solutions of the complex short-pulse equation, the latter also as an interaction of two one-soliton solutions, have been derived by Feng [16] using Hirota’s method [21]. Our method for solving the initial-value problem for the complex shortpulse equation with solutions u such that u and ux vanish as x → ±∞, is analogous to the solution of the initial-value problem of Heisenberg’s ferromagnet equation (1.6) below. It departs from the change of variable Z ∞ q ξ(x, t) = x − 1 + |uy |2 − 1 (1.4) dy
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x
which allows us to replace (1.3a) by Vξ = −ik (m(ξ, t) · σ) V,
(1.5)
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where σ is the vector of Pauli matrices σ1 = ( 01 10 ), σ2 = ( 0i −i 0 ), and σ3 = 1 0 ( 0 −1 ), and m(ξ, t) = −(Im uξ )σ1 − (Re uξ )σ2 + p
1 σ3 1 + |ux |2
is a real vector of unit length depending on (ξ, t) ∈ R2 . Together with the equation Vt = −2ik 2 (m · σ) + ik(m × mξ · σ) V, 2
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mt = m × mξξ .
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(1.5) forms a pair of evolution equations whose compatibility condition is equivalent to Heisenberg’s ferromagnet equation [31, 36, 14] (1.6)
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Using the 2 × 2 matrix of Jost solutions Ψ(ξ, t, k) to the focusing nonlinear Schr¨odinger (NLS) equation iut + uξξ + 2|u|2u = 0, Zakharov and Takhtajan [36] have shown that the gauge transformation Ψ(ξ, t, 0)−1σ3 Ψ(ξ, t, 0)
(1.7)
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converts the NLS Jost solutions into solutions of Heisenberg’s ferromagnet equation (1.6). In a similar way we shall derive complex short-pulse solutions expressed in the variables ξ, t ∈ R by applying the gauge transformation (1.7) to the Jost solutions of the complex sine-Gordon equation. To do so, we shall 2 replace the time factor e4ik t governing the time evolution of the focusing NLS scattering data into the time factor e−it/2k governing the time evolution of the complex sine-Gordon equation. It then remains to write complex short-pulse solutions in terms of x, t ∈ R by inverting the change of variable (1.4). Loop and antiloop soliton and breather solutions arise if the variable transformation x 7→ ξ is not a 1-1 correspondence between real numbers and there exist bounded ξ-intervals in which the derivative dx/dξ is negative (see e.g. [8] for a discussion of these matters). Transformations of the type x 7→ ξ have been used before to facilitate describing large k asymptotics in the scattering theory of the Schr¨odinger equation −uxx + Qu = H 2 u on the line [6] and that of the Camassa-Holm equation [13]. Let us discuss the contents of this article. In Section 2 we introduce and study the Jost solutions, scattering coefficients, and reflection coefficients of (1.2) and derive the gauge transformation that allows us to obtain the complex short-pulse solutions from the Jost functions of the complex sineGordon equation. In Section 3 we apply the matrix triplet method to derive the reflectionless Jost solutions of the complex sine-Gordon equations and, by gauge transformation, the reflectionless complex short-pulse solutions. We analyze the loop structure of the complex short-pulse solution. In Section 4 we define the matrix triplets leading to certain loop soliton solutions and work out the one-soliton case analytically in detail.
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2
Gauge transformations
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Assuming that ux ∈ L1 (R) and hence that uξ ∈ L1 (R, dξ), we define the 2 ×2 Jost matrices as the unique solutions of the Volterra integral equation Z ξ −ikξσ3 Ψ± (ξ, t, k) = e − ik dη eik(η−ξ)σ3 (m(η, t) · σ − σ3 ) Ψ± (η, t, k), ±∞
(2.1) where (ξ, t) ∈ R . Equations (1.5) and (1.3b) imply that, for (ξ, t, k) ∈ R3 , the Jost matrix Ψ(ξ, t, k) is unitary and has unit determinant. Using (1) (2) superscripted Ψ± and Ψ± to denote the columns of the Jost matrices Ψ± , standard iteration [3, 33] implies the following: 2
(2)
(1)
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a) For each (ξ, t) ∈ R2 , e−ikξ Ψ+ (ξ, t, k) and eikξ Ψ− (ξ, t, k) are continuous in k ∈ C+ ∪ R and are analytic in k ∈ C+ . (1)
(2)
b) For each (ξ, t) ∈ R2 , eikξ Ψ+ (ξ, t, k) and e−ikξ Ψ− (ξ, t, k) are continuous in k ∈ C− ∪ R and are analytic in k ∈ C− .
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Moreover, Ψ± (ξ, t, 0) = I2 . Unfortunately, the presence of the k-factors to multiply the integrals in (2.1) makes it impossible to study the large kasymptotics of the Jost solutions directly. Now assume that uξ and uξξ both belong to L1 (R, dξ). Applying integration by parts to (2.1), we get
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Ψ± (ξ, t, k) = e−ikξσ3 − σ3 (m(ξ, t) · σ − σ3 ) Ψ± (ξ, t, k) Z ξ + σ3 dη eik(η−ξ)σ3 [mη · σ−ik (m(η, t) · σ−σ3 ) (m(η, t) · σ)] Ψ± (η, t, k). ±∞
Adding (2.1) and reshuffling terms we arrive at the integral equation Z ξ −ikξσ3 D(ξ, t)Ψ±(ξ, t, k) = e + dη eik(η−ξ)σ3 σ3 (mη · σ)Ψ± (η, t, k), (2.2)
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where
D(ξ, t) =
±∞
1 2
(I2 +σ3 (m(ξ, t) · σ))
has determinant 21 (1 + m3 (ξ, t)). In fact, Gronwall’s inequality implies # " Z ξ ∓1 1 dη kmη · σk . exp p kΨ± (ξ, t, k)k ≤ p 1 + m3 (ξ, t) 2 1 + m3 (ξ, t) ±∞
Since m3 (ξ, t) is nonnegative, we have in addition to a) and b) 4
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c) For each (ξ, t) ∈ R2 , e−ikξ Ψ+ (ξ, t, k) and eikξ Ψ− (ξ, t, k) tend to ( 01 ) and ( 10 ) as k → ∞ from within C+ ∪ R. (1)
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d) For each (ξ, t) ∈ R2 , eikξ Ψ− (ξ, t, k) and e−ikξ Ψ+ (ξ, t, k) tend to ( 10 ) and ( 01 ) as k → ∞ from within C− ∪ R. Theorem 2.1 Suppose uξ and uξξ belong to L1 (R, dξ). Then Z ξ −ikξσ3 −ikησ3 , Ψ± (ξ, t, k) = H± (ξ, t) e ∓ dη K± (ξ, η, t)e Rξ
±∞
dη kK± (ξ, η, t)k converges and H± (ξ, t) is a unitary matrix.
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where
(2.3)
±∞
Proof.
Substituting (2.3) into (2.2), using the identity eikcσ3 M = 12 M eikcσ3 + e−ikcσ3 + 21 σ3 Mσ3 eikcσ3 − e−ikcσ3
for each 2 × 2 matrix M and each c ∈ R, and postmultiplying the resulting equation by eikξσ3 , we get after reshuffling some terms Z ξ p 1 dη B± (η, t) D(ξ, t)H± (ξ, t) − I2 − 2
1 2
where
dζ ±∞
Z
ζ ±∞
±∞
−ik(ˆ η −ξ)σ3
±∞
dˆ η K± (ξ, ηˆ, t)e
dˆ η C±p (ζ, ηˆ, t)e−ik(ˆη−ζ)σ3 ∓
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∓
ξ
ξ
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= ±D(ξ, t)H± (ξ, t) Z
Z
1 2
Z
+
1 2
ξ
±∞
dζ
Z
Z
ξ
±∞ ζ
±∞
m dˆ η B± (ˆ η , t)e−2ik(ˆη−ξ)σ3
dˆ η C±m (ζ, ηˆ, t)e−ik(ˆη−2ξ+ζ)σ3 ,
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p B± (η, t) = σ3 (mη · σ)H± (η, t) + (mη · σ)H± (η, t)σ3 , m B± (η, t) = σ3 (mη · σ)H± (η, t) − (mη · σ)H± (η, t)σ3 , C±p (ζ, η, t) = σ3 (mζ · σ)H± (ζ, t)K±(ζ, η, t) + (mζ · σ)H± (ζ, t)K± (ζ, η, t)σ3, C±m (ζ, η, t) = σ3 (mζ · σ)H± (ζ, t)K±(ζ, η, t) − (mζ · σ)H± (ζ, t)K± (ζ, η, t)σ3.
Making the change of variable from ηˆ − ξ, 2(ˆ η − ξ), ηˆ − ζ, and ηˆ − 2ξ + ζ to η − ξ in the four Fourier integrals, we get the algebraic identity Z ξ p 1 dη B± (η, t) D(ξ, t)H± (ξ, t) = I2 + 2 ±∞
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plus the Volterra integral equations
±∞
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±∞
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m 1 D(ξ, t)H± (ξ, t)K± (ξ, η, t) ± 14 B± ( 2 [ξ + η], t) Z ξ Z ξ p 1 1 −2 dζ C± (ζ, ζ + η − ξ) − 2 dζ C±m (ζ, ξ + η − ζ, t) = 02×2 .
Using that H± (ξ, t) are unitary matrices and D(ξ, t) is a contraction, we Rξ define F (ξ)• = ∓ ±∞ dη kF (ξ, η)k and derive the estimates
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√ Z ξ ∓ 2 dζ kmζ · σk (1 + 2kK± (ζ, t)k•) , kK± (ξ, t)k• ≤ 1 + m3 (ξ, t) ±∞ √ where we have used that kD(ξ, t)−1k = 2/(1 + m3 (ξ, t)). By Gronwall’s inequality, the integral equation for K± (ξ, η, t) has a unique solution that is integrable with respect to the η-variable.
Applying integration by parts to remove k-factors in front of the Fourier integrals, we obtain 02×2 = [Ψ± ]ξ + ik(m(ξ, t) · σ)Ψ± (ξ, t, k)
= ik ((m(ξ, t) · σ)H± (ξ, t) − H± (ξ, t)σ3 ) e−ikξσ3
±∞
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Z
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+ ([H± ]ξ ± (m(ξ, t) · σ)H± (ξ, t)K± (ξ, ξ, t)σ3 ∓ H± (ξ, t)K± (ξ, ξ, t)) e−ikξσ3 Z ξ ∓ (m(ξ, t) · σ)H± (ξ, t) dη [K± ]η σ3 e−ikησ3 ∓ [H± ]ξ
ξ
±∞
−ikησ3
dη K± (ξ, η, t)e
∓ H± (ξ, t)
Z
ξ
±∞
dη [K± ]ξ e−ikησ3 .
By the Riemann-Lebesgue lemma, we get for the leading terms as k → ±∞
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(m(ξ, t) · σ)H± (ξ, t) − H± (ξ, t)σ3 = 02×2 , [H± ]ξ ∓ H± (ξ, t)K± (ξ, ξ, t) ± (m(ξ, t) · σ)H± (ξ, t)K± (ξ, ξ, t)σ3 = 02×2 .
(2.4a) (2.4b)
Equation (2.4a) implies the key identity m(ξ, t) · σ = H± (ξ, t)σ3 H± (ξ, t)−1 . 6
(2.5)
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Putting Ψ± (ξ, t, k) = e
∓
Z
ξ
±∞
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−ikξσ3
SG
K± (ξ, η, t)e−ikησ3 = H± (ξ, t)−1Ψ± (ξ, η, t), (2.6)
ΨSG ±
ξ
= −H±−1 [H± ]ξ H±−1 Ψ± − ikH±−1 (m · σ)Ψ± = −H±−1 [H± ]ξ − ikH±−1 (m · σ)H±−1 ΨSG ± SG −1 = −ikσ3 − H± [H± ]ξ Ψ± = (−ikσ3 ∓ [K± (ξ, ξ, t) − σ3 K± (ξ, ξ, t)σ3]) ΨSG ± ,
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we obtain
which is a Zakharov-Shabat system with focusing potential Q(ξ, t) = −Q(ξ, t)† = ∓ [K± (ξ, ξ, t) − σ3 K± (ξ, ξ, t)σ3] .
(2.7)
Equation (2.4a) implies that H− (ξ, t)−1 H+ (ξ, t) commutes with σ3 and hence is a diagonal matrix. Since it is also a unitary matrix of unit determinant, there exists a real scalar function α(ξ, t) such that (2.8)
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H− (ξ, t)−1H+ (ξ, t) = eiα(ξ,t)σ3 . Next, using (2.4b) we get
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H± (ξ, t)−1[H± ]ξ = −Q(ξ, t),
where Q(ξ, t) is given by (2.7). Differentiating (2.8) with respect to ξ we get iαξ σ3 eiασ3 = −H−−1 [H− ]ξ eiασ3 + H−−1 [H+ ]ξ = Qeiασ3 − eiασ3 Q,
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where the left-hand side is a diagonal matrix and the right-hand side is an off-diagonal matrix. Equating either side to 02×2 , we see that α is a real constant and that eiασ3 commutes with Q. Hence, α ≡ 0 and the matrix functions H+ (ξ, t) and H− (ξ, t) coincide. Henceforth we shall often write H(ξ, t) for H± (ξ, t). Let us define the scattering matrix a(k, t) b(k, t) S(k, t) = b(k, t) a(k, t) 7
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as follows:
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Ψ− (ξ, t, k) = Ψ+ (ξ, t, k)S(k, t).
(2.9)
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Using that Ψ± (ξ, t, k) = V (x(ξ, t), t)C± (k, t)−1 where V (x, t, k) satisfies the Lax pair equations (1.3), it follows from the asymptotic form of the 4 × 4 matrix in (1.3b) that i St = (σ3 S − Sσ3 ) , 4k so that S(k, t) = e(it/4k)σ3 S(k, 0)e−(it/4k)σ3 . (2.10)
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As a result, the diagonal elements a and a of the scattering matrix are time independent, whereas its off-diagonal elements satisfy b(k, t) = e−it/2k b(k, 0),
b(k, t) = eit/2k b(k, 0).
Since H± (ξ, t) coincide, it is easily verified that −1 −1 ΨSG − (ξ, t, k) = H− (ξ, t) Ψ− (ξ, t, k) = H− (ξ, t) Ψ+ (ξ, t, k)S(k, t) = H+ (ξ, t)−1 Ψ+ (ξ, t, k)S(k, t) = ΨSG + (ξ, t, k)S(k, t).
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Consequently, S(k, t) is the scattering matrix of the focusing ZakharovShabat system SG [ΨSG (2.11) ± ]ξ = (−ikσ3 + Q) Ψ± and at the same time satisfies the time evolution relation (2.10).
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This leads to the following algorithm for solving the short-pulse equation: 1. Compute one of the Jost matrices ΨSG ± (ξ, t, k) of the focusing ZakharovShabat system (2.11), where the scattering matrix has the time evolution property (2.10). This may be achieved alternatively by applying the Riemann-Hilbert method [3, 2] or the Marchenko method [3, 33].
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2. Put
Z H(ξ, t) = I2 ∓
ξ
±∞
where
SG
−ikξσ3
Ψ± (ξ, t, k) = e
∓
8
Z
dη K± (ξ, η, t) ξ
±∞
−1
,
dη K± (ξ, η, t)e−ikησ3 .
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3. Compute
= H(ξ, t)σ3 H(ξ, t)−1.
dx σ3 dξ
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m(ξ, t) · σ = −(Im uξ )σ1 − (Re uξ )σ2 +
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This step is exactly the gauge transformation introduced [36] to solve the Heisenberg ferromagnet equation by using the NLS Jost solutions, apart from the different time factor.
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4. Compute u(ξ, t) from its 1-2 element by integration with respect to ξ, assuming that u vanishes as ξ → +∞. Only the solutions that also vanish as ξ → −∞ are acceptable. 5. Write x as a function of ξ by integrating its 1-1 element with respect to ξ, assuming that x ∼ ξ as ξ → +∞. 6. Write ξ as a function of (x, t) ∈ R2 and express the short-pulse solution in the variables x and t.
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It remains to identify the nonlinear evolution equation that can be solved by using the Zakharov-Shabat scattering data with time factor eit/2k . Let C and S be two scalar functions (C real-valued, but in general S complexvalued) such that C(0) = 1 and S(0) = 0 and the system of differential equations Cξ = − 12 (uξ S ∗ + u∗ξ S),
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Sξ = uξ C,
Sξ∗ = u∗ξ C,
(2.12)
called the complex sine-Gordon equation [20] is valid. Then it is easily verified that the compatibility condition of the Lax pair i C S −ik − 12 uξ X= 1 ∗ , T = , (2.13) u ik 4k S ∗ −C 2 ξ
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is equivalent to the nonlinear evolution equation uξt = S. If u is real-valued, then we readily verify that C = cos(u) and S = sin(u), thus turning (2.13) into the well-known Lax pair for the (real) sine-Gordon equation uξt = sin(u) [1, 2]. If u and u∗ are proportional, then it is easily verified that the Lax pair (2.13) is compatible with the nonlinear evolution equation uξt =
sin(|u|) u. |u| 9
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Marchenko method and matrix triplets
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The asymptotic form of the matrix T above as ξ → ±∞ implies that the corresponding scattering matrix satisfies (2.10). Thus the gauge transformation introduced in [36] converts the Jost matrix solving the complex sine-Gordon equation into the solution to the complex short-pulse equation. If the Cauchy problem for the complex short-pulse equation (1.2) has a global in time (classical) solution, then the transformation x 7→ ξ is a 11 correspondence and the derivative dx/dξ is always positive [26]. If the derivative dx/dξ vanishes at certain isolated values of ξ ∈ R, then the graph of u(x, t) contains a loop if x(ξ, t) changes sign and displays a cusp with vertical tangent if x(ξ, t) does not change sign.
Throughout this article we make the assumption of absence of spectral singularities, i.e., of nonexistence of real zeros to the scattering coefficients a(k) and a(k) (which equals a(k)∗ for k ∈ R). In terms of the entries of the Jost matrices Ψ± (ξ, t, k), (2.9) and det Ψ± (ξ, t, k) = 1 imply the following: a(k) = Ψ+,22 (ξ, t, k)Ψ−,11(ξ, t, k) − Ψ+,12 (ξ, t, k)Ψ−,21(ξ, t, k), a(k) = Ψ+,11 (ξ, t, k)Ψ−,22(ξ, t, k) − Ψ+,21 (ξ, t, k)Ψ−,12(ξ, t, k),
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e−it/2k b(k) = Ψ+,11 (ξ, t, k)Ψ−,21(ξ, t, k) − Ψ+,21 (ξ, t, k)Ψ−,11(ξ, t, k),
eit/2k b(k) = Ψ+,22 (ξ, t, k)Ψ−,12(ξ, t, k) − Ψ+,12 (ξ, t, k)Ψ−,22(ξ, t, k).
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Thus a(k) is continuous in k ∈ C+ ∪ R, is analytic in k ∈ C+ , and tends to 1 as k → ∞ from within C+ ∪R. Analogously, a(k) is continuous in k ∈ C− ∪R, is analytic in k ∈ C− , and tends to 1 as k → ∞ from within C− ∪ R. Finally, b(k) and b(k) are continuous in k ∈ R and vanish as k → ±∞. Introducing the reflection coefficients ρ = (b/a), ρ = (b/a), r = −b/a, and r = −b/a, we obtain their time evolution ρ(k, t) = eit/2k ρ(k, 0),
r(k, t) = eit/2k r(k, 0),
r(k, t) = e−it/2k r(k, 0).
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ρ(k, t) = e−it/2k ρ(k, 0),
Let us denote by W the Wiener algebra of constants plus Fourier transˆ W = |c| + khk1 . forms of functions in L1 (R), endowed with the norm kc + hk Then W is a commutative Banach algebra with unit element with respect to the natural summation and the product operation ˆ 1 )(c2 + h ˆ 2 ) = c1 c2 + c2 h ˆ 1 + c1 h ˆ 2 + (h\ (c1 + h 1 ∗ h2 ), 10
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−∞
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where the asterisk denotes the convolution product. According to a clasˆ sical result [35, 19], the invertible elements of W are exactly those c + h R∞ ikη for which the constants c and the Fourier transform c + −∞ dη e h(η) are nonzero for every k ∈ R. Equation (2.6) plus the convergence of the inteRξ grals ±∞ dη kK± (ξ, η, t)k for each (ξ, t) ∈ R2 imply that the entries of the ikξσ3 matrices ΨSG belong to W. As a result, the scattering coefficients ± (ξ, t, k)e −it/2k a(k), a(k), e b(k), and eit/2k b(k) also belong to W. Hence, e−it/2k ρ(k), it/2k it/2k ρ(k), e r(k), and e−it/2k r(k) belong to W. Thus, there exist ρˆ(·, t), e ˆ t), rˆ(·, t), and ˆr(·, t) in L1 (R) such that ρ(·, Z ∞ Z ∞ −ikη ˆ t), ρ(k, t) = dη e ρˆ(η, t), ρ(k, t) = dη eikη ρ(η, −∞ −∞ Z ∞ Z ∞ ikη r(k, t) = dη e−ikη ˆr(η, t). r(k, t) = dη e rˆ(η, t), −∞
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The unitarity and unimodularity of S(k, t) for (k, t) ∈ R2 imply that ρˆ and −ρˆ as well as rˆ and −ˆr are each other’s complex conjugates. Assuming the absence of spectral singularities, the scattering coefficient a(k) has only finitely many zeros k ∈ C+ and no real zeros, whereas a(k) has only finitely many zeros k ∈ C− (the complex conjugates of those of a(k) in C+ ) and no real zeros. We can then write down the usual coupled system of Marchenko integral equations as the single integral equation [4, 33] Z ∞ K+ (ξ, η, t) + F + (ξ + η, t) + dζ K+ (ξ, ζ, t)F + (ζ + η, t) = 02×2 , (3.1) where
urn
ξ
0 −F+ (ξ + η, t)∗ F + (ξ + η, t) = F+ (ξ + η, t) 0 1 −1 0 −ρ(ξ + η, t)∗ + C r e−ξAr e− 2 tAr B r = ρ(ξ + η, t) 0
A†r 0p×p , 0p×p Ar
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and
Ar =
Br =
0p×1 −Cr† , Br 0p×1
Cr =
Br† 01×p . 01×p Cr
By the same token, we can study the integral equation Z ξ K− (ξ, η, t) + F − (ξ + η, t) + dζ K− (ξ, ζ, t)F − (ζ + η, t) = 02×2 , −∞
11
(3.2)
(3.3)
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where 0 F− (ξ + η, t) F − (ξ + η, t) = −F− (ξ + η, t)∗ 0 1 −1 0 r(ξ + η, t) + C l eξAl e− 2 tAl B l = ∗ −r(ξ + η, t) 0 and Al =
Al 0p×p , 0p×p A†l
Bl =
(3.4)
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of
0p×1 Bl , −Cl† 0p×1
Cl =
Cl 01×p . 01×p Bl†
2
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In these expressions the NLS time factor e4itAr,l (as in [4]) has been replaced 1
−1
by the sine-Gordon time factor e− 2 tAr,l (as in [5, 15]). In this article we solve the reflectionless Marchenko equation (3.1) [with ρˆ ≡ 0 in (3.2)] by separation of variables. We drop the subscript r and rely on [4, 5] for the proofs of the invertibility of some of the matrices while solving (3.1). Substituting the expression (3.2) into (3.1) we obtain 1
−1
K+ (ξ, η, t) = −W + (ξ, t)e− 2 tA B,
al
where
−ξA
W + (ξ, t) = Ce
+
Z
∞
dζ K+ (ξ, ζ, t)Ce−ζA
ξ 1
−1
Here
urn
= Ce−ξA − W + (ξ, t)e−ξAe− 2 tA P e−ζA .
P =
Z
∞
−ζA
dζ e
−ζA
BCe
0
=
Jo
is the unique solution of the Sylvester equation
0p×p −Q N 0p×p
AP + P A = BC
(3.5)
and Q and N are the unique solutions of the Lyapunov equations A† Q + QA = C † C,
AN + NA† = BB † .
12
(3.6)
Journal Pre-proof
Hence, K+ (ξ, η, t) = −Ce
−1 1 1 −1 −ξA − 2 tA−1 −ξA I2p + e e Pe e−ηA e− 2 tA B
of
−ξA
p ro
= −CΠ+ (ξ, t)−1 e−(η−ξ)A B, where
1
Π± (ξ, t) = e2ξA e 2 tA As a result, H+ (ξ, t)
−1
= I2 +
Z
ξ
∞
−1
± P.
dη K+ (ξ, η, t) = I2 − CΠ+ (ξ, t)−1 A−1 B,
Pr e-
H+ (ξ, t) = I2 + CA−1 Π− (ξ, t)−1 B,
(3.7a) (3.7b)
where the Sylvester equation (3.5) is used to derive (3.7b) from (3.7a). Consequently, dx/dξ iuξ = H+ (ξ, t)σ3 H+ (ξ, t)−1. (3.8) −iu∗ξ −dx/dξ
al
Let us now substitute (3.7) into (3.8) and simplify the resulting expression, something we failed to do for Heisenberg’s ferromagnet equation [14]. Put Σ3 = σ3 ⊗ Ip = Ip ⊕ (−Ip ). Then Σ3 commutes with A, anticommutes with P , and satisfies Σ3 B = −Bσ3 , CΣ3 = σ3 C, while Σ3 Π± (ξ, t) = Π∓ (ξ, t)Σ3 . Then, using the Sylvester equation (3.5) and the identity [Π− (ξ, t)−1]ξ = −Π− (ξ, t)−1 [Π− (ξ, t)]ξ Π− (ξ, t)−1 ,
urn
we compute
H+ (ξ, t)σ3 H+ (ξ, t)−1 = σ3 + CA−1 Π− (ξ, t)−1Bσ3 − σ3 CΠ+ (ξ, t)−1A−1 B + CA−1 Π− (ξ, t)−1 BCΠ− (ξ, t)−1A−1 Bσ3
Jo
= σ3 + CA−1 Π− (ξ, t)−1Bσ3 + CΠ− (ξ, t)−1A−1 Bσ3 − CA−1 Π− (ξ, t)−1× 1 2ξA 2 tA−1 Π− (ξ, t)−1A−1 Bσ3 × AΠ− (ξ, t) + Π− (ξ, t)A − 2Ae e 1
−1
= σ3 + 2CA−1 Π− (ξ, t)−1 Ae2ξA e 2 tA Π− (ξ, t)−1 A−1 Bσ3
= σ3 + CA−1 Π− (ξ, t)−1 [Π− (ξ, t)]ξ Π− (ξ, t)−1A−1 Bσ3 ∂ = σ3 − ∂ξ CA−1 Π− (ξ, t)−1 A−1 Bσ3 ∂ σ3 CA−1 Π+ (ξ, t)−1 A−1 B . = σ3 + ∂ξ 13
Journal Pre-proof
Thus (3.8) implies that
p ro
of
x(ξ, t) = ξ ± CA−1 Π± (ξ, t)−1A−1 B 11 , x(ξ, t) = ξ ± CA−1 Π± (ξ, t)−1A−1 B 22 , u(ξ, t) = −i CA−1 Π± (ξ, t)−1A−1 B 12 , u(ξ, t)∗ = −i CA−1 Π± (ξ, t)−1A−1 B 21 .
(3.9a) (3.9b) (3.9c) (3.9d)
Pr e-
Now observe that the matrices P , Q, and N are nonsingular iff the order p of the matrix A equals the sum of the orders of the zeros of a(k) in C+ [4, 33]. Consequently, if this is the case, then ( −1 CA−1 P −1 A−1 B 11 = B † A† N −1 A−1 B > 0, lim (x − ξ) = (3.10) −1 ξ→−∞ CA−1 P −1 A−1 B 22 = CA−1 Q−1 A† C † > 0,
where Q and N satisfy the Lyapunov equations (3.6). Finally, for fixed t ∈ R we have lim u(ξ, t) = ∓i CA−1 P −1 A−1 B 12 = 0. (3.11) ξ→−∞
4
urn
al
Thus reflectionless complex short-pulse solutions vanish as ξ → ±∞. If the matrix triplet (A, B, C) consists of real matrices, then the shortpulse solutions u are purely imaginary and the real functions −iu satisfy the short-pulse equation (1.1). The gauge transformation (3.8) relates the Jost matrices for the (real) sine-Gordon equation to the solutions of (1.1). This connection between the sine-Gordon equation and (1.1) can also be understood from the transformations in [29].
Using matrix triplets
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Hitherto we have derived reflectionless complex short-pulse solutions in a rather abstract way. However, in the situation studied most, where the scattering coefficient a(k) has only the distinct simple zeros k1 , . . . , kp in C+ and N1 , . . . , Np denote the corresponding norming constants, we choose
14
Journal Pre-proof
the matrix triplet (A, B, C) as follows: A = diag(ik1∗ , . . . , ikp∗ , −ik1 , . . . , −ikp ), 0 iN1∗ .. .. . . 1 ... 1 0 ... 0 0 iNp∗ B= . , C = 0 . . . 0 iN1 . . . iNp 1 0 . .. .. . 1 0
p ro
of
(4.1a)
(4.1b)
Pr e-
Using the Jordan normal form of A = A† ⊕ A, the matrix triplet (A, B, C) can be generalized to the case where at least one of the zeros k1 , . . . , kp ∈ C+ of a(k) is multiple [4, 5, 33]. In the one-soliton case, with eigenvalue k1 = µ1 + iν1 with ν1 > 0 and norming constant N1 , we adopt as a special case of (4.1) the matrix triplet ∗ 1 0 ik1 0 0 iN1∗ , C= . A= , B= 1 0 0 iN1 0 −ik1
al
Then the solution P of the Sylvester equation (3.5) is as follows: ! |N1 |2 0 − 2ν1 . P = 1 0 2ν1
1 −1 e2ξA e 2 tA
urn
Moreover, Π± (ξ, t) =
±P =
2
e2ik1 ξ−it/2k1 ∓ |N2ν11| ± 2ν11 e−2ik1 ξ+it/2k1 ∗
∗
!
.
Furthermore,
1 |N1 |2 |N1 |2 2 2ξA 2 tA−1 = W (ξ, t) + , ± P = e4ν1 ξ eν1 t/|k1 | + ∆(ξ, t) = det e e 1 4ν12 4ν12
Jo
where
4ν1 ξ ν1 t/|k1 |2
W1 (ξ, t) = e
e
4ν1 ξ+
=e
t 4|k1 |2
.
Moreover,
Π± (ξ, t)
−1
1 = ∆(ξ, t)
2
e−2ik1 ξ+it/2k1 ± |N2ν11| ∗ ∗ e2ik1 ξ−it/2k1 ∓ 2ν11 15
!
.
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Consequently, we have the unitary matrices
Pr e-
p ro
of
! ! −2ik ξ+it/2k N1∗ |N1 |2 1 1 1 0 1 0 e k1∗ 2ν1 H+ (ξ, t)−1 = I2 − ∗ ∗ i ∆(ξ, t) 0 iN1 − 2ν11 e2ik1 ξ−it/2k1 0 k1 ! N1∗ −2ik1 ξ+it/2k1 i|N1 |2 e 1 2k1 ν1 k1∗ = I2 − ∗ ∗ −i|N1 |2 ∆(ξ, t) − Nk 1 e2ik1 ξ−it/2k1 2k1∗ ν1 1 ! ∗ 2 N1 1| −2ik1 ξ+it/2k1 − e 1 − 2k1i|N ∗ ν1 ∆(ξ,t) k1 ∆(ξ,t) ; = 2 N1 1| 2ik1∗ ξ−it/2k1∗ e 1 + 2k∗i|N k1 ∆(ξ,t) ν ∆(ξ,t) 1 1 ! ! −i −|N1 |2 ∗ −2ik ξ+it/2k 1 1 0 1 ∗ 0 iN e k1 1 2ν1 H+ (ξ, t) = I2 + ∗ ∗ 1 1 1 0 ∆(ξ, t) 0 −N e2ik1 ξ−it/2k1 k1 2ν1 ! N1∗ −2ik1 ξ+it/2k1 i|N1 |2 e 1 2k1∗ ν1 k1∗ = I2 + ∗ ∗ −i|N1 |2 ∆(ξ, t) − N1 e2ik1 ξ−it/2k1 k1 2k1 ν1 ! N1∗ i|N1 |2 −2ik1 ξ+it/2k1 e 1 + 2k∗ ν1 ∆(ξ,t) k1∗ ∆(ξ,t) 1 . = 2 N1 1| 2ik1∗ ξ−it/2k1∗ 1 − 2k1i|N − k1 ∆(ξ,t) e ν1 ∆(ξ,t)
al
A∗ B 2 2 Put H+ (ξ, t)−1 = BA∗ −B −B ∗ A , and |A| + |B| = 1. A∗ , H+ (ξ, t) = Then (dx/dξ) = |A|2 − |B|2 , iuξ = −2BA∗ , and −iu∗ξ = −2AB ∗ . Consequently,
urn
2 2 dx N1∗ −2ik1 ξ+it/2k1 2 1| = 1 − 2k1i|N − e k∗ ∆(ξ,t) , ν1 ∆(ξ,t) 1 dξ i|N1 |2 −2i N1∗ −2ik1 ξ+it/2k1 1+ ∗ . e uξ = ∗ k1 ∆(ξ, t) 2k1 ν1 ∆(ξ, t)
Recalling that
4ν1 ξ ν1 t/|k1 |2
Jo
W1 (ξ, t) = e
e
4ν1 ξ+
=e
t 4|k1 |2
,
|N1 |2 ∆(ξ, t) = W1 (ξ, t) + , 4ν12
we obtain
dx = dξ
W1 (ξ, t) +
|N1 |2 4ν12
−
i|N1 |2 k1∗ 2|k1 |2 ν1
∆(ξ, t)2
16
2
−
|N1 |2 W1 (ξ, t) |k1 |2
Journal Pre-proof
−
|N1 |2 2ν12
|N1 |2 2|k1 |2
2
+
|N1 |2 µ1 2|k1 |2 ν1
2
∆(ξ, t)2 2|N1 |2 − |k1 |2 W1 (ξ, t) +
−
|N1 |2 W1 (ξ, t) |k1 |2
|N1 |4 16ν14
∆(ξ, t)2
p ro
=
W1 (ξ, t)2 +
|N1 |2 4ν12
of
=
W1 (ξ, t) +
Pr e-
whose numerator is a quadratic polynomial in W1 (ξ, t). The discriminant of this polynomial is as follows: 2 ! 4 ν1 2|N1 | −1 + 2 , discr. = 2 2 |k1 | ν1 |k1 | which is negative and hence leads to an always positive (dx/dξ) iff arg(k1 ) ∈ (0, π4 ) ∪ ( 3π , π). 4 If arg(k1 ) equals
π 4
or
3π , 4
then
2
1| W1 (ξ, t)2 + |N dx 16ν14 = dξ ∆(ξ, t)2
2
2
urn
al
1| , then the coefficient |N2ν12| − 2|N of is always positive. If π4 < arg(k1 ) < 3π 4 |k1 |2 1 W1 (ξ, t) of W1 (ξ, t) in the quadratic polynomial is negative. This polynomial of W1 (ξ, t) then has the two positive zeros s 2 2 2 |N1 | |N1 | |N1 |4 |N1 |2 |N1 |2 Z± (N1 , k1 ) = ± − − − |k1 |2 4ν12 |k1 |2 4ν12 16ν14 s 2 ν12 ν1 |N1 |2 1 1 |N1 |2 ± − − . = 2 2 2 ν1 |k1 | 4 |k1 |ν1 |k1 | 2
Jo
We thus find a positive value of dx/dξ, unless 1 t 1 ln Z− (N1 , k1 ) ≤ ξ + ≤ ln Z+ (N1 , k1 ). 2 4ν1 4|k1 | 4ν1
Thus loop solitons only occur if
π 4
< arg(k1 ) <
17
3π . 4
(4.2)
Journal Pre-proof
1
2
2|N1 | h |k1|2
W1 (ξ, t)
p ro
=1−
W1 (ξ, t) +
|N1 |2 4ν12
Since ξ = x + o(1) as x → +∞, we obtain
=ξ+ where ξ0 =
1 2ν1
|N1 |2 1 2 2|k1 | ν1 W1 (ξ, t) + 2ν1 |k1 |2
1
4ν1 (ξ−ξ0 +
1+e
|N1 |2 4ν12
=ξ+
i2 .
|N1 |2 2|k1 |2 ν1
Pr e-
x(ξ, t) = ξ +
|N1 |4 16ν14
of
Let us now integrate the function 2 2|N1 |2 |N1 | 2 W1 (ξ, t) + 2ν 2 − |k1 |2 W1 (ξ, t) + dx 1 = h i 2 2 dξ W1 (ξ, t) + |N4ν12|
1
t 4ν1 (ξ+ ) 4|k1 |2 e
+
|N1 |2 4ν12
, t ) 4|k1 |2
ln(|N1 |/2ν1 ). Moreover, x − ξ → (2ν1 /|k1|2 ) as ξ → −∞.
al
Acknowledgments
urn
The author is greatly indebted to Barbara Prinari for her hospitality during a visit to the University of Buffalo. 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 INdAM-GNFM.
References
Jo
[1] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett. 30, 1262–1264 (1973).
[2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, The inverse scattering transform. Fourier analysis for nonlinear problems, Stud. Appl. Math. 53, 249–315 (1974). 18
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[3] M.J. Ablowitz, B. Prinari, and A.D. Trubatch, Discrete and Continuous Nonlinear Schr¨odinger Systems, Cambridge Univ. Press, Cambridge, 2004.
p ro
[4] T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the focusing nonlinear Schr¨odinger equation, Inverse Problems 23, 2171– 2195 (2007). [5] T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the sine-Gordon equation, J. Math. Phys. 51, 123521 (2010), 27 pp.
Pr e-
[6] T. Aktosun, M. Klaus, and C. van der Mee, Scattering and inverse scattering in one-dimensional nonhomogeneous media, J. Math. Phys. 33, 1717–1744 (1992). [7] R. Beals, M. Rabelo, and K. Tenenblat, B¨acklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math. 81, 121–151 (1989). [8] A. Boutet de Monvel, D. Shepelsky, and L. Zielinski, The short pulse equation by a Riemann-Hilbert approach, Lett. Math. Phys. 107, 1345–1373 (2016).
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[9] J.C. Brunelli, The short pulse hierarchy, J. Math. Phys. 46, 123507 (2005).
urn
[10] J.C. Brunelli, The bi-Hamiltonian structure of the short pulse equation, Phys. Lett. A 353, 475–478 (2006). [11] J.C. Brunelli and S. Sakovich, Hamiltonian integrability of twocomponent short pulse equations, J. Math. Phys. 54, 012701 (2013). [12] G.M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation, Z. angew. Math. Phys. 66, 1529–1557 (2015).
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[13] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A 457, 953–970 (2001). [14] F. Demontis, S. Lombardo, M. Sommacal, C. van der Mee, and F. Vargiu, Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation, Commun. Nonlin. Sci. Numer. Simul. 64, 35–65 (2018). 19
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[15] F. Demontis and C. van der Mee, Closed form solutions to the matrix Sine-Gordon equation, IMA J. Appl. Math. 77, 308–315 (2012). [16] Bao-Feng Feng, Complex short pulse and coupled complex short pulse equations, Physica D 297, 62–75 (2015).
p ro
[17] H.C. Fogedby, Solitons and magnons in the classical Heisenberg chain, J. Phys. A 13, 1467–1499 (1980).
Pr e-
[18] G. Gambino, U. Tanriver, P. Guha, A.G. Choudhury, and S.R. Choudhury, Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation: phase-plane, multiinfinite series and variational approaches, Commun. Nonlinear Sci. Numer. Simul. 20, 375–388 (2015). [19] I. Gelfand, To the theory of normed rings. II. On absolutely convergent trigonometric series and integrals, C.R. (Doklady) Acad. Sci. URSS (N.S.) 25, 570–572 (1939). [20] Hang Nian-Ning, On the method for solving the complex sine-Gordon equation, Lettere al Nuovo Cimento 38, 60–64 (1983).
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[21] R. Hirota, The Direct Method in Soliton Theory, Cambridge Univ. Press, Cambridge, 2004.
urn
[22] V.K. Kuetche, T.B. Bouetou, and T.C. Kofane, On two-loop soliton solution of the Sch¨afer-Wayne short-pulse equation using Hirota‘s method and Hodnett-Moloney approach, J. Phys. Soc. Japan 76, 024004 (2007). [23] Y. Liu, D. Pelinovsky, and A. Sakovich, Wave breaking in the shortpulse equation, Dynamics of PDE 6, 291–310 (2009).
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[24] Y. Matsuno, Multiloop soliton and multibreather solutions of the short pulse model equation, J. Phys. Soc. Japan 76, 084003 (2007).
[25] E.J. Parkes, Some periodic and solitary travelling-wave solutions of the short-pulse equation, Chaos, Solitons & Fractals 38, 154–159 (2008).
20
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[26] D. Pelinovsky and A. Sakovich, Global well-posedness of the shortpulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations 35, 613–629 (2010).
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[27] M.L. Rabelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math. 81, 221–248 (1989). [28] A. Sakovich and S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Japan 74, 239–241 (2005). [29] A. Sakovich and S. Sakovich, Solitary wave solutions of the short pulse equation, J. Phys. A: Math. Gen. 39, L361–L367 (2006).
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[30] T. Sch¨afer and C.E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D 196, 90–105 (2004). [31] L.A. Takhtajan, Integration of the continuous Heisenberg spin chain through the inverse scattering method, Phys. Lett. 64A, 235–237 (1977).
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[32] H.T. Tchoukouansi, V.K. Kuetche, and T.C. Kofane, Exact soliton solutions to a new coupled integrable short light-pulse system, Chaos, Solitons & Fractals 68, 10–19 (2014). [33] C. van der Mee, Nonlinear Evolution Models of Integrable Type, SIMAI e-Lecture Notes 11, SIMAI, Torino, 2013.
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[34] K.K. Victor, B.B. Thomas, and T.C. Kofane, On exact solutions of the Sch¨afer-Wayne short pulse equation: WKI eigenvalue problems, J. Phys. A: Math. Theor. 40, 5585–5596 (2007). [35] N. Wiener, The Fourier integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933.
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[36] V.F. Zakharov and L.A. Takhtajan, Equivalence of the nonlinear Schr¨odinger equation and the equation of a Heisenberg ferromagnet, Theor. Math. Phys. 38(1), 17–23 (1979).
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PII: DOI: Reference:
S0393-0440(19)30220-7 https://doi.org/10.1016/j.geomphys.2019.103539 GEOPHY 103539
To appear in:
Journal of Geometry and Physics
Received date : 30 August 2019 Revised date : 8 October 2019 Accepted date : 5 November 2019 Please cite this article as: C.v.d. Mee, Complex short-pulse solutions by gauge transformation, Journal of Geometry and Physics (2019), doi: https://doi.org/10.1016/j.geomphys.2019.103539. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Manuscript
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Complex short-pulse solutions by gauge transformation Cornelis van der Mee∗
Pr e-
Abstract
In this article we apply the gauge transformation method for solving the Heisenberg ferromagnet equation by using the NLS Jost solutions to derive solutions of the complex short-pulse equation from the Jost functions of the complex sine-Gordon equation. The matrix triplet method is used to compute the most general reflectionless complex short-pulse solutions by solving the Marchenko integral equation for the complex sine-Gordon equation.
Introduction
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1
urn
The short-pulse equation
uxt = u + 61 (u3 )xx
(1.1)
Jo
appears as the integrability condition of one of the pseudospherical surfaces studied in [27] (also [7]). This equation was also derived to describe the propagation of ultra-short optical pulses in silica optical fibers [30]. Equation (1.1) is known to be integrable [28] and its bi-Hamiltonian structure and conservation laws have been studied in detail [9, 10, 11]. Equation (1.1) can be related to the sine-Gordon equation and is known to have (anti)loop soliton solutions [29, 24], where, for certain t ∈ R, u(x, t) is a multi-valued function of x on certain bounded x-intervals. Loop soliton solutions have ∗ Dip. Matematica e Informatica, Universit`a di Cagliari, Viale Merello 92, 09123 Cagliari, Italy. Email: [email protected]
1
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(1.2)
p ro
uxt = u + 12 (|u|2 ux )x
of
also been obtained by alternative methods [34, 22, 25, 18, 8]. Well-posedness issues were studied in [26, 12, 23]. The complex short-pulse equation
Pr e-
was derived and studied by Feng [16]. Like its real counterpart (1.1), it is integrable and can in fact be derived as the compatibility condition (Vx )t = (Vt )x of the Lax pair equations 1 iux V, (1.3a) Vx = −ik −iu∗x −1 i 1 1 1 2 2 − ik|u| − iu + k|u| u x 4k 2 2 2 Vt = V. (1.3b) − 12 iu∗ − 12 k|u|2u∗x − 4ki + 12 ik|u|2
al
The one-soliton and two-soliton solutions of the complex short-pulse equation, the latter also as an interaction of two one-soliton solutions, have been derived by Feng [16] using Hirota’s method [21]. Our method for solving the initial-value problem for the complex shortpulse equation with solutions u such that u and ux vanish as x → ±∞, is analogous to the solution of the initial-value problem of Heisenberg’s ferromagnet equation (1.6) below. It departs from the change of variable Z ∞ q ξ(x, t) = x − 1 + |uy |2 − 1 (1.4) dy
urn
x
which allows us to replace (1.3a) by Vξ = −ik (m(ξ, t) · σ) V,
(1.5)
Jo
where σ is the vector of Pauli matrices σ1 = ( 01 10 ), σ2 = ( 0i −i 0 ), and σ3 = 1 0 ( 0 −1 ), and m(ξ, t) = −(Im uξ )σ1 − (Re uξ )σ2 + p
1 σ3 1 + |ux |2
is a real vector of unit length depending on (ξ, t) ∈ R2 . Together with the equation Vt = −2ik 2 (m · σ) + ik(m × mξ · σ) V, 2
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mt = m × mξξ .
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(1.5) forms a pair of evolution equations whose compatibility condition is equivalent to Heisenberg’s ferromagnet equation [31, 36, 14] (1.6)
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Using the 2 × 2 matrix of Jost solutions Ψ(ξ, t, k) to the focusing nonlinear Schr¨odinger (NLS) equation iut + uξξ + 2|u|2u = 0, Zakharov and Takhtajan [36] have shown that the gauge transformation Ψ(ξ, t, 0)−1σ3 Ψ(ξ, t, 0)
(1.7)
Jo
urn
al
Pr e-
converts the NLS Jost solutions into solutions of Heisenberg’s ferromagnet equation (1.6). In a similar way we shall derive complex short-pulse solutions expressed in the variables ξ, t ∈ R by applying the gauge transformation (1.7) to the Jost solutions of the complex sine-Gordon equation. To do so, we shall 2 replace the time factor e4ik t governing the time evolution of the focusing NLS scattering data into the time factor e−it/2k governing the time evolution of the complex sine-Gordon equation. It then remains to write complex short-pulse solutions in terms of x, t ∈ R by inverting the change of variable (1.4). Loop and antiloop soliton and breather solutions arise if the variable transformation x 7→ ξ is not a 1-1 correspondence between real numbers and there exist bounded ξ-intervals in which the derivative dx/dξ is negative (see e.g. [8] for a discussion of these matters). Transformations of the type x 7→ ξ have been used before to facilitate describing large k asymptotics in the scattering theory of the Schr¨odinger equation −uxx + Qu = H 2 u on the line [6] and that of the Camassa-Holm equation [13]. Let us discuss the contents of this article. In Section 2 we introduce and study the Jost solutions, scattering coefficients, and reflection coefficients of (1.2) and derive the gauge transformation that allows us to obtain the complex short-pulse solutions from the Jost functions of the complex sineGordon equation. In Section 3 we apply the matrix triplet method to derive the reflectionless Jost solutions of the complex sine-Gordon equations and, by gauge transformation, the reflectionless complex short-pulse solutions. We analyze the loop structure of the complex short-pulse solution. In Section 4 we define the matrix triplets leading to certain loop soliton solutions and work out the one-soliton case analytically in detail.
3
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2
Gauge transformations
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Assuming that ux ∈ L1 (R) and hence that uξ ∈ L1 (R, dξ), we define the 2 ×2 Jost matrices as the unique solutions of the Volterra integral equation Z ξ −ikξσ3 Ψ± (ξ, t, k) = e − ik dη eik(η−ξ)σ3 (m(η, t) · σ − σ3 ) Ψ± (η, t, k), ±∞
(2.1) where (ξ, t) ∈ R . Equations (1.5) and (1.3b) imply that, for (ξ, t, k) ∈ R3 , the Jost matrix Ψ(ξ, t, k) is unitary and has unit determinant. Using (1) (2) superscripted Ψ± and Ψ± to denote the columns of the Jost matrices Ψ± , standard iteration [3, 33] implies the following: 2
(2)
(1)
Pr e-
a) For each (ξ, t) ∈ R2 , e−ikξ Ψ+ (ξ, t, k) and eikξ Ψ− (ξ, t, k) are continuous in k ∈ C+ ∪ R and are analytic in k ∈ C+ . (1)
(2)
b) For each (ξ, t) ∈ R2 , eikξ Ψ+ (ξ, t, k) and e−ikξ Ψ− (ξ, t, k) are continuous in k ∈ C− ∪ R and are analytic in k ∈ C− .
al
Moreover, Ψ± (ξ, t, 0) = I2 . Unfortunately, the presence of the k-factors to multiply the integrals in (2.1) makes it impossible to study the large kasymptotics of the Jost solutions directly. Now assume that uξ and uξξ both belong to L1 (R, dξ). Applying integration by parts to (2.1), we get
urn
Ψ± (ξ, t, k) = e−ikξσ3 − σ3 (m(ξ, t) · σ − σ3 ) Ψ± (ξ, t, k) Z ξ + σ3 dη eik(η−ξ)σ3 [mη · σ−ik (m(η, t) · σ−σ3 ) (m(η, t) · σ)] Ψ± (η, t, k). ±∞
Adding (2.1) and reshuffling terms we arrive at the integral equation Z ξ −ikξσ3 D(ξ, t)Ψ±(ξ, t, k) = e + dη eik(η−ξ)σ3 σ3 (mη · σ)Ψ± (η, t, k), (2.2)
Jo
where
D(ξ, t) =
±∞
1 2
(I2 +σ3 (m(ξ, t) · σ))
has determinant 21 (1 + m3 (ξ, t)). In fact, Gronwall’s inequality implies # " Z ξ ∓1 1 dη kmη · σk . exp p kΨ± (ξ, t, k)k ≤ p 1 + m3 (ξ, t) 2 1 + m3 (ξ, t) ±∞
Since m3 (ξ, t) is nonnegative, we have in addition to a) and b) 4
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(2)
(1)
of
c) For each (ξ, t) ∈ R2 , e−ikξ Ψ+ (ξ, t, k) and eikξ Ψ− (ξ, t, k) tend to ( 01 ) and ( 10 ) as k → ∞ from within C+ ∪ R. (1)
(2)
p ro
d) For each (ξ, t) ∈ R2 , eikξ Ψ− (ξ, t, k) and e−ikξ Ψ+ (ξ, t, k) tend to ( 10 ) and ( 01 ) as k → ∞ from within C− ∪ R. Theorem 2.1 Suppose uξ and uξξ belong to L1 (R, dξ). Then Z ξ −ikξσ3 −ikησ3 , Ψ± (ξ, t, k) = H± (ξ, t) e ∓ dη K± (ξ, η, t)e Rξ
±∞
dη kK± (ξ, η, t)k converges and H± (ξ, t) is a unitary matrix.
Pr e-
where
(2.3)
±∞
Proof.
Substituting (2.3) into (2.2), using the identity eikcσ3 M = 12 M eikcσ3 + e−ikcσ3 + 21 σ3 Mσ3 eikcσ3 − e−ikcσ3
for each 2 × 2 matrix M and each c ∈ R, and postmultiplying the resulting equation by eikξσ3 , we get after reshuffling some terms Z ξ p 1 dη B± (η, t) D(ξ, t)H± (ξ, t) − I2 − 2
1 2
where
dζ ±∞
Z
ζ ±∞
±∞
−ik(ˆ η −ξ)σ3
±∞
dˆ η K± (ξ, ηˆ, t)e
dˆ η C±p (ζ, ηˆ, t)e−ik(ˆη−ζ)σ3 ∓
urn
∓
ξ
ξ
al
= ±D(ξ, t)H± (ξ, t) Z
Z
1 2
Z
+
1 2
ξ
±∞
dζ
Z
Z
ξ
±∞ ζ
±∞
m dˆ η B± (ˆ η , t)e−2ik(ˆη−ξ)σ3
dˆ η C±m (ζ, ηˆ, t)e−ik(ˆη−2ξ+ζ)σ3 ,
Jo
p B± (η, t) = σ3 (mη · σ)H± (η, t) + (mη · σ)H± (η, t)σ3 , m B± (η, t) = σ3 (mη · σ)H± (η, t) − (mη · σ)H± (η, t)σ3 , C±p (ζ, η, t) = σ3 (mζ · σ)H± (ζ, t)K±(ζ, η, t) + (mζ · σ)H± (ζ, t)K± (ζ, η, t)σ3, C±m (ζ, η, t) = σ3 (mζ · σ)H± (ζ, t)K±(ζ, η, t) − (mζ · σ)H± (ζ, t)K± (ζ, η, t)σ3.
Making the change of variable from ηˆ − ξ, 2(ˆ η − ξ), ηˆ − ζ, and ηˆ − 2ξ + ζ to η − ξ in the four Fourier integrals, we get the algebraic identity Z ξ p 1 dη B± (η, t) D(ξ, t)H± (ξ, t) = I2 + 2 ±∞
5
Journal Pre-proof
plus the Volterra integral equations
±∞
p ro
±∞
of
m 1 D(ξ, t)H± (ξ, t)K± (ξ, η, t) ± 14 B± ( 2 [ξ + η], t) Z ξ Z ξ p 1 1 −2 dζ C± (ζ, ζ + η − ξ) − 2 dζ C±m (ζ, ξ + η − ζ, t) = 02×2 .
Using that H± (ξ, t) are unitary matrices and D(ξ, t) is a contraction, we Rξ define F (ξ)• = ∓ ±∞ dη kF (ξ, η)k and derive the estimates
Pr e-
√ Z ξ ∓ 2 dζ kmζ · σk (1 + 2kK± (ζ, t)k•) , kK± (ξ, t)k• ≤ 1 + m3 (ξ, t) ±∞ √ where we have used that kD(ξ, t)−1k = 2/(1 + m3 (ξ, t)). By Gronwall’s inequality, the integral equation for K± (ξ, η, t) has a unique solution that is integrable with respect to the η-variable.
Applying integration by parts to remove k-factors in front of the Fourier integrals, we obtain 02×2 = [Ψ± ]ξ + ik(m(ξ, t) · σ)Ψ± (ξ, t, k)
= ik ((m(ξ, t) · σ)H± (ξ, t) − H± (ξ, t)σ3 ) e−ikξσ3
±∞
urn
Z
al
+ ([H± ]ξ ± (m(ξ, t) · σ)H± (ξ, t)K± (ξ, ξ, t)σ3 ∓ H± (ξ, t)K± (ξ, ξ, t)) e−ikξσ3 Z ξ ∓ (m(ξ, t) · σ)H± (ξ, t) dη [K± ]η σ3 e−ikησ3 ∓ [H± ]ξ
ξ
±∞
−ikησ3
dη K± (ξ, η, t)e
∓ H± (ξ, t)
Z
ξ
±∞
dη [K± ]ξ e−ikησ3 .
By the Riemann-Lebesgue lemma, we get for the leading terms as k → ±∞
Jo
(m(ξ, t) · σ)H± (ξ, t) − H± (ξ, t)σ3 = 02×2 , [H± ]ξ ∓ H± (ξ, t)K± (ξ, ξ, t) ± (m(ξ, t) · σ)H± (ξ, t)K± (ξ, ξ, t)σ3 = 02×2 .
(2.4a) (2.4b)
Equation (2.4a) implies the key identity m(ξ, t) · σ = H± (ξ, t)σ3 H± (ξ, t)−1 . 6
(2.5)
Journal Pre-proof
Putting Ψ± (ξ, t, k) = e
∓
Z
ξ
±∞
of
−ikξσ3
SG
K± (ξ, η, t)e−ikησ3 = H± (ξ, t)−1Ψ± (ξ, η, t), (2.6)
ΨSG ±
ξ
= −H±−1 [H± ]ξ H±−1 Ψ± − ikH±−1 (m · σ)Ψ± = −H±−1 [H± ]ξ − ikH±−1 (m · σ)H±−1 ΨSG ± SG −1 = −ikσ3 − H± [H± ]ξ Ψ± = (−ikσ3 ∓ [K± (ξ, ξ, t) − σ3 K± (ξ, ξ, t)σ3]) ΨSG ± ,
Pr e-
p ro
we obtain
which is a Zakharov-Shabat system with focusing potential Q(ξ, t) = −Q(ξ, t)† = ∓ [K± (ξ, ξ, t) − σ3 K± (ξ, ξ, t)σ3] .
(2.7)
Equation (2.4a) implies that H− (ξ, t)−1 H+ (ξ, t) commutes with σ3 and hence is a diagonal matrix. Since it is also a unitary matrix of unit determinant, there exists a real scalar function α(ξ, t) such that (2.8)
al
H− (ξ, t)−1H+ (ξ, t) = eiα(ξ,t)σ3 . Next, using (2.4b) we get
urn
H± (ξ, t)−1[H± ]ξ = −Q(ξ, t),
where Q(ξ, t) is given by (2.7). Differentiating (2.8) with respect to ξ we get iαξ σ3 eiασ3 = −H−−1 [H− ]ξ eiασ3 + H−−1 [H+ ]ξ = Qeiασ3 − eiασ3 Q,
Jo
where the left-hand side is a diagonal matrix and the right-hand side is an off-diagonal matrix. Equating either side to 02×2 , we see that α is a real constant and that eiασ3 commutes with Q. Hence, α ≡ 0 and the matrix functions H+ (ξ, t) and H− (ξ, t) coincide. Henceforth we shall often write H(ξ, t) for H± (ξ, t). Let us define the scattering matrix a(k, t) b(k, t) S(k, t) = b(k, t) a(k, t) 7
Journal Pre-proof
as follows:
of
Ψ− (ξ, t, k) = Ψ+ (ξ, t, k)S(k, t).
(2.9)
p ro
Using that Ψ± (ξ, t, k) = V (x(ξ, t), t)C± (k, t)−1 where V (x, t, k) satisfies the Lax pair equations (1.3), it follows from the asymptotic form of the 4 × 4 matrix in (1.3b) that i St = (σ3 S − Sσ3 ) , 4k so that S(k, t) = e(it/4k)σ3 S(k, 0)e−(it/4k)σ3 . (2.10)
Pr e-
As a result, the diagonal elements a and a of the scattering matrix are time independent, whereas its off-diagonal elements satisfy b(k, t) = e−it/2k b(k, 0),
b(k, t) = eit/2k b(k, 0).
Since H± (ξ, t) coincide, it is easily verified that −1 −1 ΨSG − (ξ, t, k) = H− (ξ, t) Ψ− (ξ, t, k) = H− (ξ, t) Ψ+ (ξ, t, k)S(k, t) = H+ (ξ, t)−1 Ψ+ (ξ, t, k)S(k, t) = ΨSG + (ξ, t, k)S(k, t).
al
Consequently, S(k, t) is the scattering matrix of the focusing ZakharovShabat system SG [ΨSG (2.11) ± ]ξ = (−ikσ3 + Q) Ψ± and at the same time satisfies the time evolution relation (2.10).
urn
This leads to the following algorithm for solving the short-pulse equation: 1. Compute one of the Jost matrices ΨSG ± (ξ, t, k) of the focusing ZakharovShabat system (2.11), where the scattering matrix has the time evolution property (2.10). This may be achieved alternatively by applying the Riemann-Hilbert method [3, 2] or the Marchenko method [3, 33].
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2. Put
Z H(ξ, t) = I2 ∓
ξ
±∞
where
SG
−ikξσ3
Ψ± (ξ, t, k) = e
∓
8
Z
dη K± (ξ, η, t) ξ
±∞
−1
,
dη K± (ξ, η, t)e−ikησ3 .
Journal Pre-proof
3. Compute
= H(ξ, t)σ3 H(ξ, t)−1.
dx σ3 dξ
of
m(ξ, t) · σ = −(Im uξ )σ1 − (Re uξ )σ2 +
p ro
This step is exactly the gauge transformation introduced [36] to solve the Heisenberg ferromagnet equation by using the NLS Jost solutions, apart from the different time factor.
Pr e-
4. Compute u(ξ, t) from its 1-2 element by integration with respect to ξ, assuming that u vanishes as ξ → +∞. Only the solutions that also vanish as ξ → −∞ are acceptable. 5. Write x as a function of ξ by integrating its 1-1 element with respect to ξ, assuming that x ∼ ξ as ξ → +∞. 6. Write ξ as a function of (x, t) ∈ R2 and express the short-pulse solution in the variables x and t.
al
It remains to identify the nonlinear evolution equation that can be solved by using the Zakharov-Shabat scattering data with time factor eit/2k . Let C and S be two scalar functions (C real-valued, but in general S complexvalued) such that C(0) = 1 and S(0) = 0 and the system of differential equations Cξ = − 12 (uξ S ∗ + u∗ξ S),
urn
Sξ = uξ C,
Sξ∗ = u∗ξ C,
(2.12)
called the complex sine-Gordon equation [20] is valid. Then it is easily verified that the compatibility condition of the Lax pair i C S −ik − 12 uξ X= 1 ∗ , T = , (2.13) u ik 4k S ∗ −C 2 ξ
Jo
is equivalent to the nonlinear evolution equation uξt = S. If u is real-valued, then we readily verify that C = cos(u) and S = sin(u), thus turning (2.13) into the well-known Lax pair for the (real) sine-Gordon equation uξt = sin(u) [1, 2]. If u and u∗ are proportional, then it is easily verified that the Lax pair (2.13) is compatible with the nonlinear evolution equation uξt =
sin(|u|) u. |u| 9
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Marchenko method and matrix triplets
Pr e-
3
p ro
of
The asymptotic form of the matrix T above as ξ → ±∞ implies that the corresponding scattering matrix satisfies (2.10). Thus the gauge transformation introduced in [36] converts the Jost matrix solving the complex sine-Gordon equation into the solution to the complex short-pulse equation. If the Cauchy problem for the complex short-pulse equation (1.2) has a global in time (classical) solution, then the transformation x 7→ ξ is a 11 correspondence and the derivative dx/dξ is always positive [26]. If the derivative dx/dξ vanishes at certain isolated values of ξ ∈ R, then the graph of u(x, t) contains a loop if x(ξ, t) changes sign and displays a cusp with vertical tangent if x(ξ, t) does not change sign.
Throughout this article we make the assumption of absence of spectral singularities, i.e., of nonexistence of real zeros to the scattering coefficients a(k) and a(k) (which equals a(k)∗ for k ∈ R). In terms of the entries of the Jost matrices Ψ± (ξ, t, k), (2.9) and det Ψ± (ξ, t, k) = 1 imply the following: a(k) = Ψ+,22 (ξ, t, k)Ψ−,11(ξ, t, k) − Ψ+,12 (ξ, t, k)Ψ−,21(ξ, t, k), a(k) = Ψ+,11 (ξ, t, k)Ψ−,22(ξ, t, k) − Ψ+,21 (ξ, t, k)Ψ−,12(ξ, t, k),
al
e−it/2k b(k) = Ψ+,11 (ξ, t, k)Ψ−,21(ξ, t, k) − Ψ+,21 (ξ, t, k)Ψ−,11(ξ, t, k),
eit/2k b(k) = Ψ+,22 (ξ, t, k)Ψ−,12(ξ, t, k) − Ψ+,12 (ξ, t, k)Ψ−,22(ξ, t, k).
urn
Thus a(k) is continuous in k ∈ C+ ∪ R, is analytic in k ∈ C+ , and tends to 1 as k → ∞ from within C+ ∪R. Analogously, a(k) is continuous in k ∈ C− ∪R, is analytic in k ∈ C− , and tends to 1 as k → ∞ from within C− ∪ R. Finally, b(k) and b(k) are continuous in k ∈ R and vanish as k → ±∞. Introducing the reflection coefficients ρ = (b/a), ρ = (b/a), r = −b/a, and r = −b/a, we obtain their time evolution ρ(k, t) = eit/2k ρ(k, 0),
r(k, t) = eit/2k r(k, 0),
r(k, t) = e−it/2k r(k, 0).
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ρ(k, t) = e−it/2k ρ(k, 0),
Let us denote by W the Wiener algebra of constants plus Fourier transˆ W = |c| + khk1 . forms of functions in L1 (R), endowed with the norm kc + hk Then W is a commutative Banach algebra with unit element with respect to the natural summation and the product operation ˆ 1 )(c2 + h ˆ 2 ) = c1 c2 + c2 h ˆ 1 + c1 h ˆ 2 + (h\ (c1 + h 1 ∗ h2 ), 10
Journal Pre-proof
−∞
Pr e-
p ro
of
where the asterisk denotes the convolution product. According to a clasˆ sical result [35, 19], the invertible elements of W are exactly those c + h R∞ ikη for which the constants c and the Fourier transform c + −∞ dη e h(η) are nonzero for every k ∈ R. Equation (2.6) plus the convergence of the inteRξ grals ±∞ dη kK± (ξ, η, t)k for each (ξ, t) ∈ R2 imply that the entries of the ikξσ3 matrices ΨSG belong to W. As a result, the scattering coefficients ± (ξ, t, k)e −it/2k a(k), a(k), e b(k), and eit/2k b(k) also belong to W. Hence, e−it/2k ρ(k), it/2k it/2k ρ(k), e r(k), and e−it/2k r(k) belong to W. Thus, there exist ρˆ(·, t), e ˆ t), rˆ(·, t), and ˆr(·, t) in L1 (R) such that ρ(·, Z ∞ Z ∞ −ikη ˆ t), ρ(k, t) = dη e ρˆ(η, t), ρ(k, t) = dη eikη ρ(η, −∞ −∞ Z ∞ Z ∞ ikη r(k, t) = dη e−ikη ˆr(η, t). r(k, t) = dη e rˆ(η, t), −∞
al
The unitarity and unimodularity of S(k, t) for (k, t) ∈ R2 imply that ρˆ and −ρˆ as well as rˆ and −ˆr are each other’s complex conjugates. Assuming the absence of spectral singularities, the scattering coefficient a(k) has only finitely many zeros k ∈ C+ and no real zeros, whereas a(k) has only finitely many zeros k ∈ C− (the complex conjugates of those of a(k) in C+ ) and no real zeros. We can then write down the usual coupled system of Marchenko integral equations as the single integral equation [4, 33] Z ∞ K+ (ξ, η, t) + F + (ξ + η, t) + dζ K+ (ξ, ζ, t)F + (ζ + η, t) = 02×2 , (3.1) where
urn
ξ
0 −F+ (ξ + η, t)∗ F + (ξ + η, t) = F+ (ξ + η, t) 0 1 −1 0 −ρ(ξ + η, t)∗ + C r e−ξAr e− 2 tAr B r = ρ(ξ + η, t) 0
A†r 0p×p , 0p×p Ar
Jo
and
Ar =
Br =
0p×1 −Cr† , Br 0p×1
Cr =
Br† 01×p . 01×p Cr
By the same token, we can study the integral equation Z ξ K− (ξ, η, t) + F − (ξ + η, t) + dζ K− (ξ, ζ, t)F − (ζ + η, t) = 02×2 , −∞
11
(3.2)
(3.3)
Journal Pre-proof
where 0 F− (ξ + η, t) F − (ξ + η, t) = −F− (ξ + η, t)∗ 0 1 −1 0 r(ξ + η, t) + C l eξAl e− 2 tAl B l = ∗ −r(ξ + η, t) 0 and Al =
Al 0p×p , 0p×p A†l
Bl =
(3.4)
p ro
of
0p×1 Bl , −Cl† 0p×1
Cl =
Cl 01×p . 01×p Bl†
2
Pr e-
In these expressions the NLS time factor e4itAr,l (as in [4]) has been replaced 1
−1
by the sine-Gordon time factor e− 2 tAr,l (as in [5, 15]). In this article we solve the reflectionless Marchenko equation (3.1) [with ρˆ ≡ 0 in (3.2)] by separation of variables. We drop the subscript r and rely on [4, 5] for the proofs of the invertibility of some of the matrices while solving (3.1). Substituting the expression (3.2) into (3.1) we obtain 1
−1
K+ (ξ, η, t) = −W + (ξ, t)e− 2 tA B,
al
where
−ξA
W + (ξ, t) = Ce
+
Z
∞
dζ K+ (ξ, ζ, t)Ce−ζA
ξ 1
−1
Here
urn
= Ce−ξA − W + (ξ, t)e−ξAe− 2 tA P e−ζA .
P =
Z
∞
−ζA
dζ e
−ζA
BCe
0
=
Jo
is the unique solution of the Sylvester equation
0p×p −Q N 0p×p
AP + P A = BC
(3.5)
and Q and N are the unique solutions of the Lyapunov equations A† Q + QA = C † C,
AN + NA† = BB † .
12
(3.6)
Journal Pre-proof
Hence, K+ (ξ, η, t) = −Ce
−1 1 1 −1 −ξA − 2 tA−1 −ξA I2p + e e Pe e−ηA e− 2 tA B
of
−ξA
p ro
= −CΠ+ (ξ, t)−1 e−(η−ξ)A B, where
1
Π± (ξ, t) = e2ξA e 2 tA As a result, H+ (ξ, t)
−1
= I2 +
Z
ξ
∞
−1
± P.
dη K+ (ξ, η, t) = I2 − CΠ+ (ξ, t)−1 A−1 B,
Pr e-
H+ (ξ, t) = I2 + CA−1 Π− (ξ, t)−1 B,
(3.7a) (3.7b)
where the Sylvester equation (3.5) is used to derive (3.7b) from (3.7a). Consequently, dx/dξ iuξ = H+ (ξ, t)σ3 H+ (ξ, t)−1. (3.8) −iu∗ξ −dx/dξ
al
Let us now substitute (3.7) into (3.8) and simplify the resulting expression, something we failed to do for Heisenberg’s ferromagnet equation [14]. Put Σ3 = σ3 ⊗ Ip = Ip ⊕ (−Ip ). Then Σ3 commutes with A, anticommutes with P , and satisfies Σ3 B = −Bσ3 , CΣ3 = σ3 C, while Σ3 Π± (ξ, t) = Π∓ (ξ, t)Σ3 . Then, using the Sylvester equation (3.5) and the identity [Π− (ξ, t)−1]ξ = −Π− (ξ, t)−1 [Π− (ξ, t)]ξ Π− (ξ, t)−1 ,
urn
we compute
H+ (ξ, t)σ3 H+ (ξ, t)−1 = σ3 + CA−1 Π− (ξ, t)−1Bσ3 − σ3 CΠ+ (ξ, t)−1A−1 B + CA−1 Π− (ξ, t)−1 BCΠ− (ξ, t)−1A−1 Bσ3
Jo
= σ3 + CA−1 Π− (ξ, t)−1Bσ3 + CΠ− (ξ, t)−1A−1 Bσ3 − CA−1 Π− (ξ, t)−1× 1 2ξA 2 tA−1 Π− (ξ, t)−1A−1 Bσ3 × AΠ− (ξ, t) + Π− (ξ, t)A − 2Ae e 1
−1
= σ3 + 2CA−1 Π− (ξ, t)−1 Ae2ξA e 2 tA Π− (ξ, t)−1 A−1 Bσ3
= σ3 + CA−1 Π− (ξ, t)−1 [Π− (ξ, t)]ξ Π− (ξ, t)−1A−1 Bσ3 ∂ = σ3 − ∂ξ CA−1 Π− (ξ, t)−1 A−1 Bσ3 ∂ σ3 CA−1 Π+ (ξ, t)−1 A−1 B . = σ3 + ∂ξ 13
Journal Pre-proof
Thus (3.8) implies that
p ro
of
x(ξ, t) = ξ ± CA−1 Π± (ξ, t)−1A−1 B 11 , x(ξ, t) = ξ ± CA−1 Π± (ξ, t)−1A−1 B 22 , u(ξ, t) = −i CA−1 Π± (ξ, t)−1A−1 B 12 , u(ξ, t)∗ = −i CA−1 Π± (ξ, t)−1A−1 B 21 .
(3.9a) (3.9b) (3.9c) (3.9d)
Pr e-
Now observe that the matrices P , Q, and N are nonsingular iff the order p of the matrix A equals the sum of the orders of the zeros of a(k) in C+ [4, 33]. Consequently, if this is the case, then ( −1 CA−1 P −1 A−1 B 11 = B † A† N −1 A−1 B > 0, lim (x − ξ) = (3.10) −1 ξ→−∞ CA−1 P −1 A−1 B 22 = CA−1 Q−1 A† C † > 0,
where Q and N satisfy the Lyapunov equations (3.6). Finally, for fixed t ∈ R we have lim u(ξ, t) = ∓i CA−1 P −1 A−1 B 12 = 0. (3.11) ξ→−∞
4
urn
al
Thus reflectionless complex short-pulse solutions vanish as ξ → ±∞. If the matrix triplet (A, B, C) consists of real matrices, then the shortpulse solutions u are purely imaginary and the real functions −iu satisfy the short-pulse equation (1.1). The gauge transformation (3.8) relates the Jost matrices for the (real) sine-Gordon equation to the solutions of (1.1). This connection between the sine-Gordon equation and (1.1) can also be understood from the transformations in [29].
Using matrix triplets
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Hitherto we have derived reflectionless complex short-pulse solutions in a rather abstract way. However, in the situation studied most, where the scattering coefficient a(k) has only the distinct simple zeros k1 , . . . , kp in C+ and N1 , . . . , Np denote the corresponding norming constants, we choose
14
Journal Pre-proof
the matrix triplet (A, B, C) as follows: A = diag(ik1∗ , . . . , ikp∗ , −ik1 , . . . , −ikp ), 0 iN1∗ .. .. . . 1 ... 1 0 ... 0 0 iNp∗ B= . , C = 0 . . . 0 iN1 . . . iNp 1 0 . .. .. . 1 0
p ro
of
(4.1a)
(4.1b)
Pr e-
Using the Jordan normal form of A = A† ⊕ A, the matrix triplet (A, B, C) can be generalized to the case where at least one of the zeros k1 , . . . , kp ∈ C+ of a(k) is multiple [4, 5, 33]. In the one-soliton case, with eigenvalue k1 = µ1 + iν1 with ν1 > 0 and norming constant N1 , we adopt as a special case of (4.1) the matrix triplet ∗ 1 0 ik1 0 0 iN1∗ , C= . A= , B= 1 0 0 iN1 0 −ik1
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Then the solution P of the Sylvester equation (3.5) is as follows: ! |N1 |2 0 − 2ν1 . P = 1 0 2ν1
1 −1 e2ξA e 2 tA
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Moreover, Π± (ξ, t) =
±P =
2
e2ik1 ξ−it/2k1 ∓ |N2ν11| ± 2ν11 e−2ik1 ξ+it/2k1 ∗
∗
!
.
Furthermore,
1 |N1 |2 |N1 |2 2 2ξA 2 tA−1 = W (ξ, t) + , ± P = e4ν1 ξ eν1 t/|k1 | + ∆(ξ, t) = det e e 1 4ν12 4ν12
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where
4ν1 ξ ν1 t/|k1 |2
W1 (ξ, t) = e
e
4ν1 ξ+
=e
t 4|k1 |2
.
Moreover,
Π± (ξ, t)
−1
1 = ∆(ξ, t)
2
e−2ik1 ξ+it/2k1 ± |N2ν11| ∗ ∗ e2ik1 ξ−it/2k1 ∓ 2ν11 15
!
.
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Consequently, we have the unitary matrices
Pr e-
p ro
of
! ! −2ik ξ+it/2k N1∗ |N1 |2 1 1 1 0 1 0 e k1∗ 2ν1 H+ (ξ, t)−1 = I2 − ∗ ∗ i ∆(ξ, t) 0 iN1 − 2ν11 e2ik1 ξ−it/2k1 0 k1 ! N1∗ −2ik1 ξ+it/2k1 i|N1 |2 e 1 2k1 ν1 k1∗ = I2 − ∗ ∗ −i|N1 |2 ∆(ξ, t) − Nk 1 e2ik1 ξ−it/2k1 2k1∗ ν1 1 ! ∗ 2 N1 1| −2ik1 ξ+it/2k1 − e 1 − 2k1i|N ∗ ν1 ∆(ξ,t) k1 ∆(ξ,t) ; = 2 N1 1| 2ik1∗ ξ−it/2k1∗ e 1 + 2k∗i|N k1 ∆(ξ,t) ν ∆(ξ,t) 1 1 ! ! −i −|N1 |2 ∗ −2ik ξ+it/2k 1 1 0 1 ∗ 0 iN e k1 1 2ν1 H+ (ξ, t) = I2 + ∗ ∗ 1 1 1 0 ∆(ξ, t) 0 −N e2ik1 ξ−it/2k1 k1 2ν1 ! N1∗ −2ik1 ξ+it/2k1 i|N1 |2 e 1 2k1∗ ν1 k1∗ = I2 + ∗ ∗ −i|N1 |2 ∆(ξ, t) − N1 e2ik1 ξ−it/2k1 k1 2k1 ν1 ! N1∗ i|N1 |2 −2ik1 ξ+it/2k1 e 1 + 2k∗ ν1 ∆(ξ,t) k1∗ ∆(ξ,t) 1 . = 2 N1 1| 2ik1∗ ξ−it/2k1∗ 1 − 2k1i|N − k1 ∆(ξ,t) e ν1 ∆(ξ,t)
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A∗ B 2 2 Put H+ (ξ, t)−1 = BA∗ −B −B ∗ A , and |A| + |B| = 1. A∗ , H+ (ξ, t) = Then (dx/dξ) = |A|2 − |B|2 , iuξ = −2BA∗ , and −iu∗ξ = −2AB ∗ . Consequently,
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2 2 dx N1∗ −2ik1 ξ+it/2k1 2 1| = 1 − 2k1i|N − e k∗ ∆(ξ,t) , ν1 ∆(ξ,t) 1 dξ i|N1 |2 −2i N1∗ −2ik1 ξ+it/2k1 1+ ∗ . e uξ = ∗ k1 ∆(ξ, t) 2k1 ν1 ∆(ξ, t)
Recalling that
4ν1 ξ ν1 t/|k1 |2
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W1 (ξ, t) = e
e
4ν1 ξ+
=e
t 4|k1 |2
,
|N1 |2 ∆(ξ, t) = W1 (ξ, t) + , 4ν12
we obtain
dx = dξ
W1 (ξ, t) +
|N1 |2 4ν12
−
i|N1 |2 k1∗ 2|k1 |2 ν1
∆(ξ, t)2
16
2
−
|N1 |2 W1 (ξ, t) |k1 |2
Journal Pre-proof
−
|N1 |2 2ν12
|N1 |2 2|k1 |2
2
+
|N1 |2 µ1 2|k1 |2 ν1
2
∆(ξ, t)2 2|N1 |2 − |k1 |2 W1 (ξ, t) +
−
|N1 |2 W1 (ξ, t) |k1 |2
|N1 |4 16ν14
∆(ξ, t)2
p ro
=
W1 (ξ, t)2 +
|N1 |2 4ν12
of
=
W1 (ξ, t) +
Pr e-
whose numerator is a quadratic polynomial in W1 (ξ, t). The discriminant of this polynomial is as follows: 2 ! 4 ν1 2|N1 | −1 + 2 , discr. = 2 2 |k1 | ν1 |k1 | which is negative and hence leads to an always positive (dx/dξ) iff arg(k1 ) ∈ (0, π4 ) ∪ ( 3π , π). 4 If arg(k1 ) equals
π 4
or
3π , 4
then
2
1| W1 (ξ, t)2 + |N dx 16ν14 = dξ ∆(ξ, t)2
2
2
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1| , then the coefficient |N2ν12| − 2|N of is always positive. If π4 < arg(k1 ) < 3π 4 |k1 |2 1 W1 (ξ, t) of W1 (ξ, t) in the quadratic polynomial is negative. This polynomial of W1 (ξ, t) then has the two positive zeros s 2 2 2 |N1 | |N1 | |N1 |4 |N1 |2 |N1 |2 Z± (N1 , k1 ) = ± − − − |k1 |2 4ν12 |k1 |2 4ν12 16ν14 s 2 ν12 ν1 |N1 |2 1 1 |N1 |2 ± − − . = 2 2 2 ν1 |k1 | 4 |k1 |ν1 |k1 | 2
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We thus find a positive value of dx/dξ, unless 1 t 1 ln Z− (N1 , k1 ) ≤ ξ + ≤ ln Z+ (N1 , k1 ). 2 4ν1 4|k1 | 4ν1
Thus loop solitons only occur if
π 4
< arg(k1 ) <
17
3π . 4
(4.2)
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1
2
2|N1 | h |k1|2
W1 (ξ, t)
p ro
=1−
W1 (ξ, t) +
|N1 |2 4ν12
Since ξ = x + o(1) as x → +∞, we obtain
=ξ+ where ξ0 =
1 2ν1
|N1 |2 1 2 2|k1 | ν1 W1 (ξ, t) + 2ν1 |k1 |2
1
4ν1 (ξ−ξ0 +
1+e
|N1 |2 4ν12
=ξ+
i2 .
|N1 |2 2|k1 |2 ν1
Pr e-
x(ξ, t) = ξ +
|N1 |4 16ν14
of
Let us now integrate the function 2 2|N1 |2 |N1 | 2 W1 (ξ, t) + 2ν 2 − |k1 |2 W1 (ξ, t) + dx 1 = h i 2 2 dξ W1 (ξ, t) + |N4ν12|
1
t 4ν1 (ξ+ ) 4|k1 |2 e
+
|N1 |2 4ν12
, t ) 4|k1 |2
ln(|N1 |/2ν1 ). Moreover, x − ξ → (2ν1 /|k1|2 ) as ξ → −∞.
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Acknowledgments
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The author is greatly indebted to Barbara Prinari for her hospitality during a visit to the University of Buffalo. 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 INdAM-GNFM.
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