A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

Feng, Bao‐Feng; Ling, Liming; Zhu, Zuo-Nong

doi:10.1098/rspa.2020.0853

Cited by 15 publications

(7 citation statements)

References 67 publications

(85 reference statements)

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“…This induces an additional symmetry on the Lax operators 𝑋 * (𝑥, 𝑡, −𝑘 * ) = 𝑋(𝑥, 𝑡, 𝑘), 𝑇 * (𝑥, 𝑡, −𝑘 * ) = 𝑇(𝑥, 𝑡, 𝑘), (38) and assuming uniqueness of solution of the equations of the Lax pair with prescribed boundary conditions as 𝑥 → ±∞, then the last symmetry implies…”

Section: Real Solutions Of the Ccspementioning

confidence: 99%

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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%

“…[1,[33][34][35][36][37], and dark soliton solutions of the defocusing cSPE have been obtained in Refs. [38,39]. The inverse scattering transform (IST) to solve the initial-value problem for the focusing cSPE equation was developed in Ref.…”

Section: Introductionmentioning

confidence: 99%

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

2023

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“…Remarkably, when we take the large-period limits, both of them degenerate to the Peregrine soliton [26], which is localized both in time and space and turns into a prototype of RWs. It turns out that this idea has been widely adopted in constructing RW solutions of many other integrable equations and their multi-component generalizations [15,27].…”

Section: The Nonlinear Schrödinger Equation (Nlse)mentioning

confidence: 99%

“…It is well-known that MI is one of the most ubiquitous phenomena in nature and commonly appears in many physical contexts such as water waves, plasma waves and electromagnetic transmission lines [13]. Whereas recent theoretical developments indicated that the presence of baseband MI supports the generation of rogue waves (RW) [14], breathers also appear to be a significant strategy in deriving RW solutions of many integrable equations [16,15].…”

Section: Introductionmentioning

confidence: 99%

Multi-breather solutions to the Sasa-Satsuma equation

Wu,

Wei,

Shi

et al. 2021

Preprint

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General breather solution to the Sasa-Satsuma (SS) equation is systematically investigated in this paper. We firstly transform the SS equation into a set of three Hirota bilinear equations under proper plane wave background. Starting from a specially arranged tau-function of the Kadomtsev-Petviashvili hierarchy and a set of eleven bilinear equations satisfied, we implement a series steps of reduction procedure, i.e., C-type reduction, dimension reduction and complex conjugate reduction, and reduce these eleven equations to three bilinear equations for the SS equation.Meanwhile, general breather solution to the SS equation is found in determinant of even order. The one-and two-breather solutions are calculated and analyzed in details.

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“…Following the integrable discretization work of the two pioneers Ablowitz and Hirota [6,7], there are an abundant discrete soliton equations that are proposed and studied by some well-established methods [8][9][10][11]. Therefore, a series of exact solutions including the discrete soliton solution, breather, positon solution and the rogue wave solutions are investigated [12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

Yang

Tian²

2022

Nonlinear Dyn

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In this paper, we investigate the smooth positon and breather-positon solutions of the a generalized integrable discrete nonlinear Schrödinger (NLS) equation by the generalized Darboux transformation (DT). Starting from the zero seed solution, the Nth-order smooth positon solutions are obtained by degenerate DT. The breather solutions including Akhmediev breather, Kuznetsov-Ma breather and spacetime periodic breather are derived from the nonzero seed solution. Then the breather-positon solutions are constructed by gradual Taylor series expansion of the eigenfunctions in breather solutions. We study the effect of higher-order nonlinear terms on these discrete smooth positon solutions and breatherpositon solutions, which demonstrates that the interacting region of soliton-positon and breather-positon are highly compressed by higher-order nonlinear effects, but the distance between the two positons have an opposite effect in two waveforms.

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A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

Cited by 15 publications

References 67 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

Multi-breather solutions to the Sasa-Satsuma equation

Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

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