Modulational instability and dynamics of implicit higher-order rogue wave solutions for the Kundu equation

2018 ◽  
Vol 32 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Xiao-Yong Wen ◽  
Guoqiang Zhang
Keyword(s):  
Rogue Wave ◽  
Modulational Instability ◽  
Higher Order ◽  
Rational Solutions ◽  
Optical Pulses ◽  
Propagation Process ◽  
First Order ◽  
Ultrashort Optical Pulses ◽  
Dynamical Behaviors ◽  
Perturbation Formula

Under investigation in this paper is the Kundu equation, which may be used to describe the propagation process of ultrashort optical pulses in nonlinear optics. The modulational instability of the plane-wave for the possible reason of the formation of the rogue wave (RW) is studied for the system. Based on our proposed generalized perturbation [Formula: see text]-fold Darboux transformation (DT), some new higher-order implicit RW solutions in terms of determinants are obtained by means of the generalized perturbation [Formula: see text]-fold DT, when choosing different special parameters, these results will reduce to the RW solutions of the Kaup–Newell (KN) equation, Chen–Lee–Liu (CLL) equation and Gerjikov–Ivanov (GI) equation, respectively. The relevant wave structures are shown graphically, which display abundant interesting wave structures. The dynamical behaviors and propagation stability of the first-order and second-order RW solutions are discussed by using numerical simulations, the higher-order nonlinear terms for the Kundu equation have an impact on the propagation instability of the RW. The method can also be extended to find the higher-order RW or rational solutions of other integrable nonlinear equations.

scholarly journals Multi-rogue Wave Solutions for a Generalized Integrable Discrete Nonlinear Schrodinger Equation With Higher-order Excitations

2021 ◽  
Author(s):  
Ma Li-Yuan ◽  
Yang Jun ◽  
Zhang Yan-Li
Keyword(s):  
Modulation Instability ◽  
Rogue Wave ◽  
Order Term ◽  
Higher Order ◽  
Discrete Version ◽  
Nls Equation ◽  
Third Order ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Dynamical Behaviors ◽  
Continuous Waves

Abstract In this paper, we construct the discrete rogue wave(RW) solutions for a higher-order or generalized integrable discrete nonlinear Schr¨odinger(NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation are constructed. Second, the dynamical behaviors of first-, second- and third-order RWsolutions are investigated in corresponding to the unique spectral parameter λ, higher-order term coefficient γ, and free constants dk, fk (k = 1, 2, · · · ,N), which exhibit affluent wave structures. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz-Ladik(AL) equation are illustrated in figures. Moreover, numerical experiments are explored, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.

scholarly journals Applying Non-Steady-State Photo-Emf Technique to Detection of Higher Order Auto Correlation Functions of Ultrashort Optical Pulses

2012 ◽  
Vol 10 (2) ◽  
Author(s):  
P. Moreno Zarate ◽  
Alexandre S. Shcherbakov ◽  
Svetlana Mansurova
Keyword(s):  
Steady State ◽  
Correlation Functions ◽  
Higher Order ◽  
Optical Pulses ◽  
Detection Scheme ◽  
Advantages And Disadvantages ◽  
Ultrashort Optical Pulses ◽  
Auto Correlation ◽  
Emf Technique ◽  
Electro Motive Force

Here we examine a new application of the adaptive detectors based on non-steady-state photo-electro-motive force effect for the detection of higher order  correlation functions, aiming the estimation of the parameters of ultra short optical pulses arranged in high-repetition trains. For this purpose three beam interferometer scheme with two signal beams modulated at different frequencies is proposed. Theoretical analysis of non-steady-state photo-EMF current generated by light distribution formed by superposition of three waves is performed and the possibility to detect simultaneously second and higher order correlation  function is demonstrated.  Potential advantages and disadvantages of such detection scheme for measuring the higher order auto-correlations functions are discussed.

scholarly journals Rogue Wave and Multiple Lump Solutions of the (2+1)-Dimensional Benjamin-Ono Equation in Fluid Mechanics

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Zhonglong Zhao ◽  
Lingchao He ◽  
Yubin Gao
Keyword(s):  
Rogue Wave ◽  
Rogue Waves ◽  
Wave Form ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
First Order ◽  
Wave Solutions ◽  
Rogue Wave Solutions ◽  
Lump Wave

In this paper, the bilinear method is employed to investigate the rogue wave solutions and the rogue type multiple lump wave solutions of the (2+1)-dimensional Benjamin-Ono equation. Two theorems for constructing rogue wave solutions are proposed with the aid of a variable transformation. Four kinds of rogue wave solutions are obtained by means of Theorem 1. In Theorem 2, three polynomial functions are used to derive multiple lump wave solutions. The 3-lump solutions, 6-lump solutions, and 8-lump solutions are presented, respectively. The 3-lump wave has a “triangular” structure. The centers of the 6-lump wave form a pentagram around a single lump wave. The 8-lump wave consists of a set of seven first order rogue waves and one second order rogue wave as the center. The multiple lump wave develops into low order rogue wave as parameters decline to zero. The method presented in this paper provides a uniform method for investigating high order rational solutions. All the results are useful in explaining high dimensional dynamical phenomena of the (2+1)-dimensional Benjamin-Ono equation.

Higher-Order Rogue Waves for a Fifth-Order Dispersive Nonlinear Schrödinger Equation in an Optical Fibre

2015 ◽  
Vol 70 (5) ◽  
pp. 365-374 ◽  
Author(s):  
Qi-Min Wang ◽  
Yi-Tian Gao ◽  
Chuan-Qi Su ◽  
Yu-Jia Shen ◽  
Yu-Jie Feng ◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Fifth Order

AbstractIn this article, a fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort optical pulses, up to the attosecond duration, in an optical fibre. Rogue wave solutions are derived by virtue of the generalised Darboux transformation. Rogue wave structures and interaction are discussed through (i) the analyses on the higher-order rogue waves, the cubic, quartic, quintic, group-velocity, and phase-parameter effects; (ii) a higher-order rogue wave consisting of the first-order rogue waves via the interaction; (iii) characteristics of the rogue waves which are summarised, including the maximum/minimum values of the rogue waves and the number of the first-order rogue waves for composing the higher-order rogue wave; and (iv) spatial-temporal patterns which are illustrated and compared with those of the ‘self-focusing’ nonlinear Schrödinger equation. We find that the quintic terms increase the time of appearance for the first-order rogue waves which form the higher-order rogue wave, and that the quintic terms affect the interaction among the first-order rogue waves, which elongates the distance of appearance for the higher-order rogue wave.

Multi-Soliton and Rogue-Wave Solutions of the Higher-Order Hirota System for an Erbium-Doped Nonlinear Fiber

2014 ◽  
Vol 69 (10-11) ◽  
pp. 521-531 ◽  
Author(s):  
Da-Wei Zuo ◽  
Yi-Tian Gao ◽  
Yu-Hao Sun ◽  
Yu-Jie Feng ◽  
Long Xue
Keyword(s):  
Wave Propagation ◽  
Darboux Transformation ◽  
Rogue Wave ◽  
Higher Order ◽  
Soliton Interaction ◽  
Nls Equation ◽  
First Order ◽  
Wave Solutions ◽  
Erbium Doped ◽  
Nonlinear Fiber

AbstractThe nonlinear Schrödinger (NLS) equation appears in fluid mechanics, plasma physics, etc., while the Hirota equation, a higher-order NLS equation, has been introduced. In this paper, a higher-order Hirota system is investigated, which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order dispersion. By virtue of the Darboux transformation and generalized Darboux transformation, multi-soliton solutions and higher-order rogue-wave solutions are derived, beyond the published first-order consideration.Wave propagation and interaction are analyzed: (i) Bell-shape solitons, bright- and dark-rogue waves are found; (ii) the two-soliton interaction is elastic, i. e., the amplitude and velocity of each soliton remain unchanged after the interaction; (iii) the coefficient in the system affects the direction of the soliton propagation, patterns of the soliton interaction, distance, and direction of the first-order rogue-wave propagation, as well as the range and direction of the second-order rogue-wave interaction.

Fission and fusion interaction phenomena of the discrete kink multi-soliton solutions for the Chen–Lee–Liu lattice equation

2018 ◽  
Vol 32 (19) ◽  
pp. 1850211
Author(s):  
Nan Liu ◽  
Xiao-Yong Wen ◽  
Yaqing Liu
Keyword(s):  
Phase Modulation ◽  
Soliton Solutions ◽  
Lattice Equation ◽  
Rational Solutions ◽  
Optical Pulses ◽  
Small Noise ◽  
Perturbation Formula ◽  
Integrable Discretization ◽  
Soliton Fission ◽  
Kink Soliton

Chen–Lee–Liu (CLL) lattice equation is an integrable discretization of the CLL equation which can be used to model the evolution of the self-steepening optical pulses without self-phase modulation. In this paper, the discrete N-fold Darboux transformation (DT) is used to derive the discrete kink multi-soliton solutions in terms of determinant for CLL lattice equation. Soliton fission and fusion interaction structures of such solutions are shown graphically. The details of their evolution are investigated by using numerical simulations, showing that a small noise with amplitude less than or equal to 0.01 produces a strong oscillation and instability of these kink soliton solutions. The discrete generalized perturbation [Formula: see text]-fold DT is constructed to express some rational solutions in terms of the determinants of CLL lattice equation by modifying the discrete N-fold DT. Infinitely many conservation laws for CLL lattice equation are constructed based on its Lax representation. Results in this paper might be helpful for understanding the propagation of optical pulses.

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