Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation Vincent Caudrelier1 Aikaterini Gkogkou2 and Barbara Prinari 2

2025-05-03 0 0 3.73MB 62 页 10玖币

侵权投诉

Soliton interactions and Yang-Baxter maps for the complex

coupled short-pulse equation

Vincent Caudrelier1, Aikaterini Gkogkou2, and Barbara Prinari*2

1School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

2Department of Mathematics, University at Buffalo, Buffalo, NY 14260, USA

Abstract

The complex coupled short pulse equation (ccSPE) describes the propagation of

ultra-short optical pulses in nonlinear birefringent ﬁbers. The system admits a variety

of vector soliton solutions: fundamental solitons, fundamental breathers, composite

breathers (generic or non-generic), as well as so-called self-symmetric composite soli-

tons. In this work, we use the dressing method and the Darboux matrices correspond-

ing to the various types of solitons to investigate soliton interactions in the focusing

ccSPE. The study combines refactorization problems on generators of certain ratio-

nal loop groups, and long-time asymptotics of these generators, as well as the main

refactorization theorem for the dressing factors which leads to the Yang-Baxter prop-

erty for the refactorization map and the vector soliton interactions. Among the results

obtained in this paper, we derive explicit formulas for the polarization shift of funda-

mental solitons which are the analog of the well-known formulas for the interaction

of vector solitons in the Manakov system. Our study also reveals that upon interacting

with a fundamental breather, a fundamental soliton becomes a fundamental breather

and, conversely, that the interaction of two fundamental breathers generically yields

two fundamental breathers with a polarization shifts, but may also result into a funda-

mental soliton and a fundamental breather. Explicit formulas for the coefﬁcients that

characterize the fundamental breathers, as well as for their polarization vectors are

obtained. The interactions of other types of solitons are also derived and discussed in

detail and illustrated with plots. New Yang-Baxter maps are obtained in the process.

1 Introduction

Mathematical models of nonlinear wave propagation can often be reduced to a class of

nonlinear partial differential equations known as integrable systems. One of the most

widely studied integrable systems is the nonlinear Schrödinger (NLS) equation, which in

the last 50 years has been shown to be a universal model for weakly dispersive nonlinear

wave trains, with physical applications ranging from deep water waves, plasma physics

and nonlinear optics, to magneto-static spin waves, low temperature physics and Bose-

Einstein condensation. On the other hand, the propagation of ultra-short optical pulses

*Corresponding author: bprinari@buffalo.edu

arXiv:2210.02265v2 [nlin.SI] 15 May 2023

(width∼10−15sand much smaller than the carrier frequency) in nonlinear media is better

described by the so-called “complex short-pulse” equation (cSPE):

uxt =u+σ

2(|u|2ux)x,σ=±1(1)

where u=u(x,t)is a complex function representing the electric ﬁeld associated to the

propagating optical pulse. The cSPE was introduced relatively recently in [1], and like

NLS, the sign of σdistinguishes the two dispersion regimes (σ=1corresponding to the

anomalous dispersion regime, or focusing cSPE, and σ=−1to normal dispersion, or

defocusing cSPE). If one restricts u(x,t)to be a real function (representing, in this case,

the magnitude of the electric ﬁeld), the above equation reduces to the (real) short-pulse

equation (SPE), which was originally introduced in the context of differential geometry

[2], and was later derived as a model for the propagation of ultra-short pulses in nonlinear

silica optics [3]. Equations of short-pulse type:

Qxτ=4iQ −2i(RQQx)x,Rxτ=4iR −2i(QRRx)x

were obtained in the earlier works [5, 6] through the negative Wadati-Konno-Ichikawa

ﬂow [7–10]. These equations reduce to (1) for R=−σQ∗but with a complex time

t=4iτ.

A key feature of the SPE and the cSPE is that, in addition to standard smooth solitons,

both admit loop soliton solutions, which are not single-valued, and “cuspons”, and also

solutions that oscillate between single- and multi-valued states. For applications to bire-

fringent ﬁbers, two orthogonally polarized modes have to be considered, and in analogy

to the Manakov system [4], which is the extension of the NLS equation to 2-components,

several generalizations of the SPE were proposed in the literature for the propagation of

polarized ultra-short pulse in anisotropic media. While there is a sizeable amount of liter-

ature on the SPE, on its two- and multi-component generalizations and discretization (see

[5–9, 12–32]), the study of the cSPE and of its vector version, the complex coupled SPE

(ccSPE) also introduced in [1], namely:

uxt =u+σ

2(||u||2ux)x,u= (u1,u2)T,σ=±1 , (2)

where u(x,t)is a two-component complex vector function and σagain distinguishes

between the focusing and defocusing equations, is obviously much more recent and less

extensive. Like NLS and the Manakov system, the defocusing cSPE and ccSPE only

admit dark solitons, i.e., solitons on a non-zero background. Soliton solutions for the

focusing cSPE equation have been constructed in [1, 33–37], and dark soliton solutions

of the defocusing cSPE have been obtained in [38, 39]. The inverse scattering transform

(IST) to solve the initial-value problem for the focusing cSPE equation was developed

in [40], and the long-time asymptotic behavior was analyzed in [41]. As to the focusing

ccSPE, several types of solutions were presented in [42–45], and the IST was developed

in [46].

The main goal of this work is to study interactions of vector solitons of the focusing

ccSPE. It is known [47, 48] that the interactions between solitons in the Manakov model,

or more generally vector NLS, give rise to maps on their polarization vectors which pro-

vide solutions of the set-theoretical Yang-Baxter equation [49]. In this context and also

in the context of discrete integrable systems, such maps are known as Yang-Baxter maps

[50]. They arise in a much larger variety of contexts, and we refer the interested reader to

[51] for an overview of various key areas where the set-theoretical Yang-Baxter equation

(and its companion, the set-theoretical reﬂection equation [47, 52]) can arise. From the

point of view of soliton dynamics, such maps ensure that multicomponent soliton inter-

actions are elastic and that the scattering of a multisoliton solution factorizes consistently

into a succession of two-soliton interactions. This is a well-known key feature of scalar

solitons, but it is more intricate to derive in the multicomponent case. Nevertheless, the

interplay between multicomponent integrable equations and Yang-Baxter maps is well

documented and ﬁnds its roots in the refactorization properties appearing in the under-

lying dressing method [53]. This was used extensively e.g. in [47]. In this paper, we

investigate Yang-Baxter maps for the focusing ccSPE, and use them to unravel the nature

of the corresponding soliton interactions. An essential new feature compared to the vector

NLS case is the variety of possible one-soliton solution that the model admits: fundamen-

tal solitons, fundamental breathers, composite breathers (generic or non-generic), as well

as so-called self-symmetric solitons. In a ﬁrst instance, by considering the interaction

of two fundamental solitons, we derive a formula analogous to Manakov’s result for the

polarization shift of interacting vector NLS solitons. This gives a ﬁrst example of the

Yang-Baxter maps involved in ccSPE. To get the full picture, we take advantage of the

ideas illustrated above, and classify the possible dressing factors creating those various

types of solitons. We then derive the “master” Yang-Baxter map arising from the refactor-

ization of the most general elementary dressing factors. Combining this with a long-time

asymptotic analysis of all the possible two-soliton solutions yields the various maps on the

polarizations of the solitons. All of them enjoy the Yang-Baxter property, being derived

from the “master” Yang-Baxter map, but take on different explicit forms.

For the rest of this work, we restrict our attention to the focusing ccSPE (so we as-

sume σ=1, and simply refer to Eq. (2) with σ=1as the ccSPE), and consider so-

lutions that are rapidly decaying as |x| → ∞. The structure of the paper is as follows.

In Sec. 2 we give a brief overview of the IST for the ccSPE as developed in [46], and

of its one-soliton solutions, which include fundamental solitons, fundamental breathers,

and composite breathers, depending on the rank and structure of the norming constant

associated to the soliton. We also discuss in detail the case of self-symmetric discrete

eigenvalues, and derive the explicit expression of a self-symmetric soliton. In Sec. 3 we

discuss the reductions of the ccSPE to the case of real solutions. In Sec. 4 we provide

the explicit expressions of the (matrix) transmission coefﬁcients corresponding to a 1-

fundamental soliton solution, a 1-fundamental breather solution, and a 1-self-symmetric

soliton solution. In Sec. 5 we use Manakov’s method [4] to investigate the pairwise inter-

actions of two fundamental solitons, and also the interaction of self-symmetric solitons.

Sec. 6 reviews the main idea of the dressing method and the notion of dressing factors

(or Darboux-Bäcklund matrices), as well as the main refactorization theorem for such

dressing factors which leads to the Yang-Baxter property for the refactorization map. It

also contains the classiﬁcation of the elementary dressing factors necessary to build the

three types of solitons in the ccSPE, as well as their various degenerations. Finally, the

long-time analysis of various two-soliton solutions leads to the derivation of the various

Yang-Baxter maps on the polarization vectors of the solitons. Our study reveals that upon

interacting with a fundamental breather, a fundamental soliton becomes a fundamental

breather and, conversely, that the interaction of two fundamental breathers generically

yields two fundamental breathers with a polarization shifts, but may also result into a

fundamental soliton and a fundamental breather. Explicit formulas for the coefﬁcients

that characterize the fundamental breathers, as well as for their polarization vectors are

obtained. The interactions of other types of solitons are also discussed in detail and illus-

trated with plots. Finally, Appendices A-E provide more technical details regarding the

derivation of the explicit expression of the analytic scattering coefﬁcients in various cases,

hodograph transformation and exact two-soliton solutions.

2 Overview of the IST and one-soliton solutions

Below, we give a succinct overview of the IST for the ccSPE as developed in [46]. The

ccSPE (2) with σ=1possess the following Lax pair:

Φx=XΦ=−ikI2kUx

−kVxikI2Φ,(3a)

Φt=TΦ=i

4kI2−i

2kUV −i

2U+1

2kUVUx

−i

2V−1

2kVUVx−i

4kI2+i

2kVU Φ,(3b)

where the matrices Uand Vare given by

U=−iu1−iu2

iu∗

2−iu∗

1,V=U†,(4)

and Indenotes the n×nidentity matrix. In [46], the gauge transformation

Φ(x,t,k) = P†(x,t)Φ(x,t,k),(5)

with Pchosen so that it diagonalizes the matrix iX/k, namely

P=pI2−α

α†I2,p2=1+q

2q,q=q1+||ux||2,α=iUx

1+q,(6a)

was used to control the behavior of the eigenfunctions at k=0and k=∞. Indeed, with

such a choice for P, the gauge transformation (5) reduces the Lax pair (3) to

Φx+Qxˆ

Φ=ˆ

Xˆ

Φ,ˆ

Φt+Qtˆ

Φ=ˆ

Tˆ

Φ,(7)

where

Qx=ikqΣ3,Qt=1

4ik +i

2kq||u||2Σ3,Σ3=diag(I2,−I2),(8a)

X=



2(1+q)qxI2−1

2q(1+q)UxVxx i

2qUxx −iqx

2q(1+q)Ux

2qVxx −iqx

2q(1+q)Vx−1

2(1+q)qxI2+1

2q(1+q)VxxUx

,(8b)

T=i

4kq (1−q)I2−iUx

iVx−(1−q)I2+(8c)

+p2

2(αtα†−αα†

t)−i(Uα†+αV)2αt−iU +iαVα

−2α†

t−iV +iα†Uα†(α†

tα−α†αt) + i(α†U+Vα).

Eqs. (8a) can be integrated explicitly, giving Q(x,t) = iθ(x,t,k)Σ3where

θ(x,t,k) = kξ(x,t)−t/4k,ξ(x,t) = x−

∞

q1+||uy||2−1dy .(9)

Then, under the assumption U(x,t)→0sufﬁciently rapidly as x→ ±∞, one can show

that ˆ

X,ˆ

T→0in this limits and hence deﬁne the Jost eigenfunctions

Φ±(x,t,k) = ˆ

Φ±,1(x,t,k),ˆ

Φ±,2(x,t,k)∼I4e−iθ(x,t,k)Σ3,x→ ±∞,(10)

as simultaneous solutions of the Lax pair (7). It is convenient to consider modiﬁed eigen-

functions with constant asymptotic behavior

M±(x,t,k) = (M±,1(x,t,k),M±,2(x,t,k))=ˆ

Φ±(x,t,k)eiθ(x,t,k)Σ3∼I4,x→ ±∞,

(11)

and one can prove that the 4×2columns M−,1,M+,2 are analytic for k∈C+and con-

tinuous for k∈R, and the columns M+,1,M−,2 are analytic for k∈C−and continuous

for k∈R. Since ˆ

Φ+and ˆ

Φ−are two fundamental solutions of the Lax pair for any

k∈R, one can deﬁne a 4×4matrix S(k)(independent of x,t) such that

Φ−(x,t,k) = ˆ

Φ+(x,t,k)S(k),S(k) = a(k)¯

b(k)

b(k)¯

a(k),k∈R,(12)

whose 2×2blocks are such that a(k)(respectively, ¯

a(k)) is analytic in C+(respectively,

in C−) and continuous for k∈R, while b(k),¯

b(k)are in general only deﬁned for k∈R.

Equation (12) for k∈Rcan be written as

M−,1(x,t,k)a−1(k) = M+,1(x,t,k) + M+,2(x,t,k)e2iθ(x,t,k)ρ(k),(13a)

M−,2(x,t,k)¯

a−1(k) = M+,2(x,t,k) + M+,1(x,t,k)e−2iθ(x,t,k)¯

ρ(k),(13b)

where the functions M−,1(x,t,k)a−1(k)and M−,2(x,t,k)¯

a−1(k)are meromorphic in

the upper/lower half k-plane respectively, and

ρ(k) = b(k)a−1(k),¯

ρ(k) = ¯

b(k)¯

a−1(k)k∈R,(14)

are the (matrix) reﬂection coefﬁcients.

For future reference, we note that one can also express the columns of ˆ

Φ+in terms of

the columns of ˆ

Φ−as

Φ+(x,t,k) = ˆ

Φ−(x,t,k)S−1(k),S−1(k) = ¯

c(k)d(k)

d(k)c(k),k∈R,(15)

where c,d,¯

c,¯

dare 2×2matrix functions of k, and (15) can be written in terms of the

analytic groups of columns of the modiﬁed eigenfunctions as

M+,1(x,t,k)¯

c−1(k) = M−,1(x,t,k) + M−,2(x,t,k)e2iθ(x,t,k)¯

r(k),(16a)

M+,2(x,t,k)c−1(k) = M−,2(x,t,k) + M−,1(x,t,k)e−2iθ(x,t,k)r(k),(16b)

where r(k) = d(k)c−1(k)and ¯

r(k) = ¯

d(k)¯

c−1(k)are the (matrix) reﬂection coefﬁcients

from the right deﬁned for system (15). For future convenience, we refer to a(k),¯

a(k)as

the (inverses) of the “left” transmission coefﬁcients, and to c(k),¯

c(k)as the (inverses) of

the “right” transmission coefﬁcients.

The Lax pair (3) admits two symmetries, k→k∗and k→ −k∗, which induce

corresponding symmetries in the scattering data. Speciﬁcally, the ﬁrst symmetry implies

ρ(k) = −ρ†(k),k∈R, det ¯

a(k) = det a†(k∗),k∈C−,(17a)

and the second symmetry gives

a∗(−k∗) = σ2a(k)σ2,k∈C+,¯

a∗(−k∗) = σ2¯

a(k)σ2,k∈C−,(17b)

ρ∗(−k) = σ2ρ(k)σ2,¯

ρ∗(−k) = σ2¯

ρ(k)σ2,k∈R,(17c)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

SolitoninteractionsandYang-Baxtermapsforthecomplexcoupledshort-pulseequationVincentCaudrelier1,AikateriniGkogkou2,andBarbaraPrinari*21SchoolofMathematics,UniversityofLeeds,Leeds,LS29JT,UK2DepartmentofMathematics,UniversityatBuffalo,Buffalo,NY14260,USAAbstractThecomplexcoupledshortpulseequation(ccSPE...

展开>> 收起<<

Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation Vincent Caudrelier1 Aikaterini Gkogkou2 and Barbara Prinari 2.pdf

共62页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation Vincent Caudrelier1 Aikaterini Gkogkou2 and Barbara Prinari 2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: