New Types of Solitary Wave Solutions for the Higher Order Nonlinear Schrödinger Equation

Li, Zhonghao; Li, Lü; Tian, Huiping; Zhou, Guanghui

doi:10.1103/physrevlett.84.4096

Cited by 230 publications

(77 citation statements)

References 17 publications

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“…Its dynamics is similar with the W-shaped soliton reported in [8]. But the solution form is distinctive from the ones in [8] which has a nonrational form, and their distribution shapes are distinguished too. Furthermore, we discuss the stability of the rational Wshaped soliton through numerical stimulation method.…”

supporting

confidence: 49%

“…This is quite different from the rational solution presented in [12,13]. The soliton's shape is similar to the "W"-shaped soliton presented in [8]. But the middle highest hump is much higher than the background, and its |E| 2 value is four times the background's density value, which is distinctive from the ones presented in [8].…”

Section: Two Explicit Cases For the Rational W-shaped Soliton Somentioning

confidence: 48%

“…The soliton's shape is similar to the "W"-shaped soliton presented in [8]. But the middle highest hump is much higher than the background, and its |E| 2 value is four times the background's density value, which is distinctive from the ones presented in [8]. To demonstrate these differences, we calculate the hump's value |E| 2 h = 4 and its valleys' values are both |E| 2 v = 5 8 , with c = 1.…”

Section: Two Explicit Cases For the Rational W-shaped Soliton Somentioning

confidence: 56%

“…It does not describe the dynamics of RW, in contrast to the rational ones of the simplified NLS, which have been used to describe RW phenomena photographically [14][15][16][17]. Its dynamics is similar with the W-shaped soliton reported in [8]. But the solution form is distinctive from the ones in [8] which has a nonrational form, and their distribution shapes are distinguished too.…”

mentioning

confidence: 92%

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Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

Zhao

Ling

2014

Phys. Rev. E

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We study localized wave on continuous wave background analytically in a nonlinear fiber with higher order effects such as higher order dispersion, Kerr dispersion, and stimulated inelastic scattering. We present an exact rational W-shaped soliton solutions, whose structural properties depend on the frequency of the background field. The hump value increase with the decrease of the background frequency in the certain regime. The highest value of the W-shaped soliton can be nine times the background's, and the distribution shape is identical with the one of well-known eyes-shaped rogue wave with its maximum peak. The numerical stimulations indicate that the W-shaped soliton is stable with small perturbations.

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supporting

confidence: 49%

Section: Two Explicit Cases For the Rational W-shaped Soliton Somentioning

confidence: 48%

Section: Two Explicit Cases For the Rational W-shaped Soliton Somentioning

confidence: 56%

mentioning

confidence: 92%

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Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

Zhao

Ling

2014

Phys. Rev. E

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“…This wave features a soliton-like structure with a stable peak |E 1s | p = 3a, and two stable valleys |E 1s | v = 0, which can be called as w-shaped traveling wave [26]. It is worth noting that such unique solution (5) is specific to the HE since it cannot exist in the standard NLSE.…”

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confidence: 99%

State transition induced by higher-order effects and background frequency

et al. 2015

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State transition between the Peregrine rogue wave and w-shaped traveling wave induced by higherorder effects and background frequency is studied. We find that this intriguing transition, described by an exact explicit rational solution, is consistent with the modulation instability (MI) analysis that involves MI region and stability region in low perturbation frequency region. In particular, the link between the MI growth rate and transition characteristic analytically demonstrates that, the size characteristic of transition is positively associated with the reciprocal of zero-frequency growth rate. Further, we investigate the case for nonlinear interplay of multi-localized waves. It is interesting that the interaction of second-order waves in stability region features a line structure, rather than an elastic interaction between two w-shaped traveling waves. [7,8] of weakly modulated plane wave. More specifically, rogue waves exist only in the MI subregion where the instability frequency band is close to the vanishing frequency [9]. In this regard, rogue wave has a consistent structure in the standard nonlinear Schrödinger equation (NLSE) where the MI characteristic remains invariant in nature. Nevertheless, the MI often exhibits some interesting features when the additional physical effects are taken into consideration, such as cross-phase modulation [10, 11], higher-order perturbation terms [12], etc. Thus, it is important to study the rogue wave property induced by the features of MI growth rate distribution.Recent studies demonstrate that rogue wave can exhibit structural diversity beyond the reach of the standard NLSE in presence of higher-order effects [13][14][15][16][17][18][19][20][21][22][23][24]. However, to our knowledge, less attention has been paid to analyzing rogue wave property in combination with the distribution characteristic of corresponding MI growth rate. Here we find that, with some higher-order perturbation terms (the third-order dispersion and delayed nonlinear response term), the MI growth rate shows a non-uniform distribution characteristic in low perturbation frequency region, in particular, it opens up a stability region as background frequency changes (see Fig. 1). Thus we anticipate that there will be some interesting physical properties as a rogue wave evolves and approaches the stability region.As a starting point, we address the problem via a completely integrable higher-order NLSE-the Hirota equa- * Electronic address: zyyang@nwu.edu.cn † Electronic address: zhaolichen3@163.com tion (HE) [25]-which involves the higher-order perturbation effects. In dimensionless form, the HE readswhere z is the propagation variable, t is the retarded time in a moving frame with the group velocity, and E(t, z) is the slowly varying envelope of the wave field. The real parameter β is introduced to be responsible for the thirdorder dispersion and delayed nonlinear response term, respectively. If β = 0, it reduces to the standard NLSE. The existence of rogue waves in the HE has been recently dem...

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Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities

Triki

Biswas

2011

Math. Meth. Appl. Sci.

106

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Communicated by S. GeorgievThe dynamics of dark solitons is studied within the framework of a generalized nonlinear Schrödinger equation. The specific cases of parabolic law and dual-power law nonlinearity are considered. The solitary wave ansatz method is used to carry out the integration. All the physical parameters in the soliton solutions are obtained as functions of the dependent model coefficients. Parametric conditions for the existence of envelope solitons are given.

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New Types of Solitary Wave Solutions for the Higher Order Nonlinear Schrödinger Equation

Cited by 230 publications

References 17 publications

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

State transition induced by higher-order effects and background frequency

Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities

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