Multi-soliton solutions and the Cauchy problem for a two-component short pulse system

Zhaqilao, Zhaqilao; Hu, Qiaoyi; Zhang, Qiao

doi:10.1088/1361-6544/aa7e9c

Cited by 20 publications

(9 citation statements)

References 60 publications

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“…Now, if the norming constant 𝐶 𝑛 has proportional columns, that is, 𝐶 𝑛 = (𝛾 𝛾 𝛾, 𝜅𝛾 𝛾 𝛾), where 𝛾 𝛾 𝛾 = (𝛼 𝑛 , 𝛽 𝑛 ) 𝑇 with 𝛼 𝑛 , 𝛽 𝑛 , 𝜅 ∈ ℂ, then one can prove that Equations ( 47) and ( 48) combined give a constraint for the multiplicative constant 𝜅, that is, 𝐶 𝑛 is nontrivial iff 𝜅 = ±𝑖. The fundamental breather solution is then given by (30) with 𝜇 = 1 and 𝜅 = ±𝑖.…”

Section: Real Solutions Of the Ccspementioning

confidence: 99%

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

2023

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The complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property 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 fundamental solitons that 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 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 derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.

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Section: Real Solutions Of the Ccspementioning

confidence: 99%

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

2023

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“…For applications to birefringent fibers, two orthogonally polarized modes have to be considered, and in analogy to the Manakov system, 18 which is the extension of the NLS equation to two components or to multicomponent, several extensions of the SPE were proposed in the literature for the propagation of polarized ultra-short pulse in anisotropic media. [19][20][21][22] Multisoliton solutions to the two-or multi-component SPE have been obtained, 23 and Darboux transformations 24 and the dressing method have been investigated. 25 Moreover, the Cauchy problem for the two-component SPE was considered as well in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…The cSPE equation can be classified into focusing or defocusing type, depending on the sign of the nonlinear term, similar to the NLS equation. The focusing cSPE equation admits bright soliton solutions, which have been constructed in pfaffian form 26 or determinant form, 24,27 as well as double pole solutions, 28 multibreather and higher-order rogue wave solutions. 29 The gauge transformation of the focusing cSPE equation was recently studied in Ref.…”

Section: Introductionmentioning

confidence: 99%

Inverse scattering transform for the complex coupled short‐pulse equation

et al. 2021

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In this paper, we develop the Riemann-Hilbert approach to the inverse scattering transform (IST) for the complex coupled short-pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real shortpulse equation (SPE) and complex short-pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one-soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather.

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“…In 2017 Z. Zhaqilao et al derived an N-fold Darboux transformation from the Lax pair of the two-component short pulse system with loop. [22] A generalization that allows to include description and interaction of opposite directed waves is connected with idea of joint account of the correspondent spaces of "hybrid" electric-magnetic amplitudes [7,8]. The projecting operators (PO) method [8] works at arbitrary dispersion and nonlinearity.…”

Section: Introduction On Nonlinear Pulse Propagation Theorymentioning

confidence: 99%

Interaction of orthogonal-polarized waves in 1D metamaterial with Kerr nonlinearity

Ampilogov,

Leble

2018

Preprint

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A theoretical study of wave propagation in 1D metamaterial is presented. A system of nonlinear evolution equation for electromagnetic waves with both polarizations account is derived by means of projection operators method for general nonlinearity and dispersion. The system describes interaction of opposite directed waves with a given polarization. The particular case of Kerr nonlinearity and Drude dispersion is considered. In such approximation it results in the correspondent systems of nonlinear equations that generalizes the Schäfer-Wayne one. Particular solutions in case of slow-varying envelopes are found, plotted and analyzed in gigahertz range. Travelling wave solution for the system of equation of interaction of orthogonal-polarized waves is also obtained and the correspondent nonlinear dispersion relations are written in explicit form.

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Multi-soliton solutions and the Cauchy problem for a two-component short pulse system

Cited by 20 publications

References 60 publications

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

Inverse scattering transform for the complex coupled short‐pulse equation

Interaction of orthogonal-polarized waves in 1D metamaterial with Kerr nonlinearity

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