On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

Zhaqilao, Zhaqilao

doi:10.1016/j.physleta.2013.01.044

Cited by 56 publications

(30 citation statements)

References 26 publications

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“…In particular, the DT of the KE equation is not completely established [20,25] because there exists an overall factor A A (or α N ) involved with complicated integrations, which produces the difficulty in the construction of multi-fold DT. Therefore, the DT [20,25] of the KE equation is not fully understood and then needed to be improved by eliminating integration in order to get higher order RWs. In recent years, the concept of a RW, which was first introduced to describe a suddenly appeared high-wall of water in deep ocean [26][27][28][29], has been gradually extended to different fields [30][31][32].…”

Section: Indiamentioning

confidence: 99%

The Darboux transformation of the Kundu–Eckhaus equation

Qiu

Zhang³

et al. 2015

Proc. R. Soc. A.

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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.

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Section: Indiamentioning

confidence: 99%

The Darboux transformation of the Kundu–Eckhaus equation

Qiu

Zhang³

et al. 2015

Proc. R. Soc. A.

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“…Recently, the generalized DT [17] was proposed, which is a new and simpler method to construct the higher-order rational solution of the rogue waves of (1). By making use of the generalized DT, Zhaqilao [18] investigated the first-order and the second-order solutions of the rogue waves for the generalized nonlinear Schrödinger equation, Song et al [19] obtained the higher-order solutions of the rogue waves for the fourth-order dispersive nonlinear Schrödinger equation, and Lv and Lin [20] solved the three coupled higher-order nonlinear Schrödinger equations with the achievement of -soliton formula and derived the lump solutions [21]. Nonlinear phenomena can be described mathematically on the basis of the corresponding nonlinear evolution equations, through which we can study the potential nonlinear dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…For the second-order to fourth-order nonlinear Schrö-dinger equations, a lot of research has been done, including the rogue waves [18,19], soliton solutions [29][30][31], modulation instability, integrability [32][33][34], and rational solutions [35]. However, to the best of our knowledge, the rogue wave solution for the fifth-order nonlinear Schrödinger equation has never been reported in the literature.…”

Section: Introductionmentioning

confidence: 99%

Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

Song

Zhang

Wang

et al. 2017

Journal of Function Spaces

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The rogue wave solutions are discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Using the Darboux matrix, the generalized Darboux transformation is constructed and a recursive formula is derived. Based on the transformation, the first-order to the third-order rogue wave solutions are obtained. Then, the nonlinear dynamics of the first-order to the third-order rogue waves are studied on the basis of some free parameters. Several new structures of the rogue waves are found using numerical simulation. The conclusions will be a supportive tool to study the rogue waves better.

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“…[29,30,31,32,33,34,35,36]. Motivated by this contemporary development, in this paper, we intend to construct the N-th order RW solution of this model.…”

Section: Introductionmentioning

confidence: 99%

Generalized Darboux transformation and Nth order rogue wave solution of a general coupled nonlinear Schrödinger equations

Priya

Senthilvelan

2015

Communications in Nonlinear Science and Numerical Simulation

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We construct a generalized Darboux transformation (GDT) of a general coupled nonlinear Schrödinger (GCNLS) system. Using GDT method we derive a recursive formula and present determinant representations for N-th order rogue wave solution of this system. Using these representations we derive first, second and third order rogue wave solutions with certain free parameters. By varying these free parameters we demonstrate the formation of triplet, triangle and hexagonal patterns of rogue waves.

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On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

Cited by 56 publications

References 26 publications

The Darboux transformation of the Kundu–Eckhaus equation

The Darboux transformation of the Kundu–Eckhaus equation

Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

Generalized Darboux transformation and Nth order rogue wave solution of a general coupled nonlinear Schrödinger equations

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