Numerical analysis of the Hirota equation: Modulational instability, breathers, rogue waves, and interactions

2020 ◽  
Vol 30 (1) ◽  
pp. 013114 ◽  
Author(s):  
Li Wang ◽  
Zhenya Yan ◽  
Boling Guo
Keyword(s):  
Numerical Analysis ◽  
Modulational Instability ◽  
Rogue Waves ◽  
Hirota Equation

scholarly journals Dust-Acoustic Rogue Waves in an Electron-Positron-Ion-Dust Plasma Medium

Galaxies ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 31
Author(s):  
Md. Habibur Rahman ◽  
Nure Alam Chowdhury ◽  
Abdul Mannan ◽  
A. A. Mamun
Keyword(s):  
Numerical Analysis ◽  
Modulational Instability ◽  
Rogue Waves ◽  
Wave Number ◽  
Plasma Medium ◽  
Critical Wave Number ◽  
Electron Positron ◽  
Dust Acoustic ◽  
Component Plasma ◽  
Dust Plasma

In this work, the modulational instability of dust-acoustic (DA) waves (DAWs) is theoretically studied in a four-component plasma medium with electrons, positrons, ions, and negative dust grains. The nonlinear and dispersive coefficients of the nonlinear Schrödinger equation (NLSE) are used to recognize the stable and unstable parametric regimes of the DAWs. It can be seen from the numerical analysis that the amplitude of the DA rogue waves decreases with increasing populations of positrons and ions. It is also observed that the direction of the variation of the critical wave number is independent (dependent) of the sign (magnitude) of q. The applications of the outcomes from the present investigation are briefly addressed.

scholarly journals On the influence of wind on extreme wave events

2007 ◽  
Vol 7 (1) ◽  
pp. 123-128 ◽  
Author(s):  
J. Touboul
Keyword(s):  
Numerical Analysis ◽  
Spectral Method ◽  
Modulational Instability ◽  
Rogue Waves ◽  
High Order ◽  
Instability Wave ◽  
Extreme Wave ◽  
Wave Fields ◽  
The Impact ◽  
High Order Spectral Method

Abstract. This work studies the impact of wind on extreme wave events, by means of numerical analysis. A High Order Spectral Method (HOSM) is used to generate freak, or rogue waves, on the basis of modulational instability. Wave fields considered here are chosen to be unstable to two kinds of perturbations. The evolution of components during the propagation of the wave fields is presented. Their evolution under the action of wind, modeled through Jeffreys' sheltering mechanism, is investigated and compared to the results without wind. It is found that wind sustains rogue waves. The perturbation most influenced by wind is not necessarily the most unstable.

Rational vector rogue waves for the n-component Hirota equation with non-zero backgrounds

2021 ◽  
Vol 427 ◽  
pp. 133005
Author(s):  
Weifang Weng ◽  
Guoqiang Zhang ◽  
Li Wang ◽  
Minghe Zhang ◽  
Zhenya Yan
Keyword(s):  
Rogue Waves ◽  
Hirota Equation ◽  
Rational Vector

Semiclassical relativistic fluid theory for electrostatic envelope modes in dense electron–positron–ion plasmas: Modulational instability and rogue waves

2013 ◽  
Vol 79 (6) ◽  
pp. 1089-1094 ◽  
Author(s):  
IOANNIS KOURAKIS ◽  
MICHAEL MC KERR ◽  
ATA UR-RAHMAN
Keyword(s):  
Fluid Model ◽  
Modulational Instability ◽  
Rogue Waves ◽  
Relativistic Electrons ◽  
Relativistic Fluid ◽  
Ion Plasma ◽  
Electron Positron ◽  
Electrons And Positrons ◽  
Perturbative Method ◽  
Fluid Theory

AbstractA fluid model is used to describe the propagation of envelope structures in an ion plasma under the influence of the action of weakly relativistic electrons and positrons. A multiscale perturbative method is used to derive a nonlinear Schrödinger equation for the envelope amplitude. Criteria for modulational instability, which occurs for small values of the carrier wavenumber (long carrier wavelengths), are derived. The occurrence of rogue waves is briefly discussed.

scholarly journals Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modified KdV Equation

2021 ◽  
Author(s):  
Lin Huang ◽  
Nannan Lv
Keyword(s):  
Periodic Solution ◽  
Numerical Analysis ◽  
Exact Solutions ◽  
Soliton Solution ◽  
Vries Equation ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Modified Kdv Equation ◽  
Extended Complex

Abstract We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton solution, soliton molecules, positon solution, rational positon solution, rational solution, periodic solution and rogue waves solution. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental patterns, triangular patterns and ring patterns. For the fundamental patterns, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves.

On Signatures and Features of Modulational Instability in Ocean Waves

2019 ◽  
Author(s):  
Alexander V. Babanin
Keyword(s):  
Water Waves ◽  
Modulational Instability ◽  
Ocean Waves ◽  
Rogue Waves ◽  
Spectral Energy ◽  
Nonlinear Superposition ◽  
Relative Significance ◽  
Wave Fields ◽  
Ice Breakup ◽  
Surface Water Waves

Abstract Modulational instability of nonlinear waves in dispersive environments is known across a broad range of physical media, from nonlinear optics to waves in plasmas. Since it was discovered for the surface water waves in the early 60s, it was found responsible for, or able to contribute to the topics of breaking and rogue waves, swell, ice breakup, wave-current interactions and perhaps even spray production. Since the early days, however, the argument continues on whether the modulational instability, which is essentially a one-dimensional phenomenon, is active in directional wave fields (that is whether the realistic directional spectra are narrow enough to maintain such nonlinear behaviours). Here we discuss the distinct features of the evolution of nonlinear surface gravity waves, which should be attributed as signatures to this instability in oceanic wind-generated wave fields. These include: wave-breaking threshold in terms of average steepness; upshifting of the spectral energy prior to breaking; oscillations of wave asymmetry and skewness; energy loss from the carrier waves in the course of the breaking. We will also refer to the linear/nonlinear superposition of waves which is often considered a counterpart (or competing) mechanism responsible for breaking or rogue waves in the ocean. We argue that both mechanisms are physically possible and the question of in situ abnormal waves is a problem of their relative significance in specific circumstances.

A weakly coupled Hirota equation and its rogue waves

2019 ◽  
Vol 34 (22) ◽  
pp. 1950179 ◽  
Author(s):  
Huijuan Zhou ◽  
Chuanzhong Li
Keyword(s):  
Optical Fibers ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Taylor Series Expansion ◽  
Rogue Waves ◽  
Nls Equation ◽  
Hirota Equation ◽  
Weakly Coupled ◽  
Parameter Values ◽  
Higher Order Dispersion

The Hirota equation, a modified nonlinear Schrödinger (NLS) equation, takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. Its wave propagation is like in the ocean and optical fibers can be viewed as an approximation which is more accurate than the NLS equation. By considering the potential application of two mode nonlinear waves in nonlinear fibers under a certain case, we use the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text] and [Formula: see text] to define a weakly coupled Hirota equation (called Frobenius Hirota equation) including its Lax pair, in this paper. Afterwards, Darboux transformation of the Frobenius Hirota equation is constructed. The Darboux transformation implies the new solutions of ([Formula: see text], [Formula: see text]) generated from the known solution ([Formula: see text], [Formula: see text]). The new solutions ([Formula: see text], [Formula: see text]) provide soliton solutions, breather solutions of the Frobenius Hirota equation. Further, rogue waves of the Frobenius Hirota equation are given explicitly by a Taylor series expansion of the breather solutions. In particular, by choosing different parameter values for the rogue waves, we can get different images.

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