Rogue waves in higher-order systems: Lagrangian approach

AbstractGeneral line rogue waves in the Mel’nikov equation are derived via the Hirota bilinear method, which are given in terms of determinants whose matrix elements have plain algebraic expressions. It is shown that fundamental rogue waves are line rogue waves, which arise from the constant background with a line profile and then disappear into the constant background again. By means of the regulation of free parameters, two subclass of nonfundamental rogue waves are generated, which are called as multirogue waves and higher-order rogue waves. The multirogue waves consist of several fundamental line rogue waves, which arise from the constant background and then decay back to the constant background. The higher-order rogue waves start from a localised lump and retreat back to it. The dynamical behaviours of these line rogue waves are demonstrated by the density and the three-dimensional figures.

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Rogue waves in presence of higher order effects

Advanced Photonics & Renewable Energy ◽

10.1364/np.2010.nma7 ◽

2010 ◽

Author(s):

Nail Akhmediev ◽

Adrian Ankiewicz ◽

J. M. Soto-Crespo

Keyword(s):

Rogue Waves ◽

Higher Order ◽

Order Effects ◽

Higher Order Effects

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Complex nonlinearities of rogue waves in generalized inhomogeneous higher-order nonlinear Schrödinger equation

Nonlinear Dynamics ◽

10.1007/s11071-015-2170-6 ◽

2015 ◽

Vol 82 (1-2) ◽

pp. 489-500 ◽

Cited By ~ 12

Author(s):

N. Song ◽

W. Zhang ◽

M. H. Yao

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Rogue Waves ◽

Higher Order ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger

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Multi-rogue waves for a higher-order nonlinear Schrödinger equation in optical fibers

Applied Mathematics and Computation ◽

10.1016/j.amc.2013.05.031 ◽

2013 ◽

Vol 220 ◽

pp. 176-184 ◽

Cited By ~ 4

Author(s):

Fajun Yu

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Optical Fibers ◽

Schrodinger Equation ◽

Rogue Waves ◽

Higher Order ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger

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Strong and weak interactions of rational vector rogue waves and solitons to any n -component nonlinear Schrödinger system with higher-order effects

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2021.0670 ◽

2022 ◽

Vol 478 (2257) ◽

Author(s):

Weifang Weng ◽

Guoqiang Zhang ◽

Zhenya Yan

Keyword(s):

Fourth Order ◽

Deep Ocean ◽

Rogue Waves ◽

Higher Order ◽

Strong Interactions ◽

Order Effects ◽

Rational Solutions ◽

Nonlinear Schrödinger ◽

Rational Vector ◽

Higher Order Effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate any n -component fourth-order nonlinear Schrödinger ( n -FONLS) system with non-zero backgrounds containing the n -Hirota equation and the n -Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of the n -FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of the n -FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of the n -FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.

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