Determinant representation of Darboux transformation for the AKNS system

He, Jingsong; Zhang, Ling; Cheng, Yi; Li, Yishen

doi:10.1007/s11425-006-2025-1

Cited by 100 publications

(105 citation statements)

References 21 publications

Supporting

Mentioning

102

Contrasting

Unclassified

Order By: Relevance

“…In general, this indeterminate form is unavoidable to construct RWs for many equations by the DT. Thus it is worthwhile to provide a direct proof of the above conjecture based on the determinant representation of the DT for the NLS [13,33,56].…”

Section: Introductionmentioning

confidence: 98%

The height of an n th-order fundamental rogue wave for the nonlinear Schrödinger equation

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 98%

The height of an n th-order fundamental rogue wave for the nonlinear Schrödinger equation

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…We consider the simplest onefold T. Comparing to the DT of the NLS equation [43,60], we set that T has the following form: …”

Section: (A) the Onefold Darboux Transformationmentioning

confidence: 99%

The Darboux transformation of the Kundu–Eckhaus equation

Qiu

Zhang³

et al. 2015

Proc. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

We construct an analytical and explicit representation of the Darboux transformation (DT) for the KunduEckhaus (KE) equation. Such solution and n-fold DT T n are given in terms of determinants whose entries are expressed by the initial eigenfunctions and 'seed' solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues. ., all these parameters will be defined latter. These solutions have a parameter β, which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the β on the RW solution. We observe two interesting results on localization characters of β, such that if β is increasing from a/2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if β is increasing to a/2. We define an area of the RW solution and find that this area associated with c = 1 is invariant when a and β are changing.

show abstract

“…Considering the application of the representation for the n-fold DT by means of the determinant [60] of eigenfunctions with different eigenvalues in the following context, we need to introduce 2n eigenfunctions by…”

Section: One-fold Darboux Transformation Of Nls-mb Equationsmentioning

confidence: 99%

“…In this subsection, our prime aim is to establish the determinant representation of the n-fold DT for NLS-MB equations as we have done for the case of NLS equation [60]. According to the form of T 1 in eq.…”

Section: N-fold Darboux Transformation For Nls-mb Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Publisher's Note:N-order bright and dark rogue waves in a resonant erbium-doped fiber system [Phys. Rev. E86, 066603 (2012)]

He¹,

Xu²,

Porsezian³

2013

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

The rogue waves in a resonant erbium-doped fibre system governed by a coupled system of the nonlinear Schrödinger equation and the Maxwell-Bloch equation (NLS-MB equations) are given explicitly by a Taylor series expansion about the breather solutions of the normalized slowly varying amplitude of the complex field envelope E, polarization p and population inversion η. The norder breather solutions of the three fields are constructed using Darboux transformation (DT) by assuming periodic seed solutions. What is more, the n-order rogue waves are given by determinant forms with n + 3 free parameters. Furthermore, the possible connection between our rouge waves and the generation of supercontinuum generation is discussed.

show abstract

Determinant representation of Darboux transformation for the AKNS system

Cited by 100 publications

References 21 publications

The height of an n th-order fundamental rogue wave for the nonlinear Schrödinger equation

The height of an n th-order fundamental rogue wave for the nonlinear Schrödinger equation

The Darboux transformation of the Kundu–Eckhaus equation

Publisher's Note:N-order bright and dark rogue waves in a resonant erbium-doped fiber system [Phys. Rev. E86, 066603 (2012)]

Contact Info

Product

Resources

About