The Riemann problem method for solving a perturbed nonlinear Schrodinger equation describing pulse propagation in optic fibres

We study the properties of the chaotic wave fields generated in the frame of the Sasa-Satsuma equation (SSE). Modulation instability results in a chaotic pattern of small-scale filaments with a free parameter-the propagation constant k. The average velocity of the filaments is approximately given by the group velocity calculated from the dispersion relation for the plane-wave solution. Remarkably, our results reveal the reason for the skewed profile of the exact SSE rogue-wave solutions, which was one of their distinctive unexplained features. We have also calculated the probability density functions for various values of the propagation constant k, showing that probability of appearance of rogue waves depends on k.

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“…There is a number of publications dealing with the solutions of the SSE [33][34][35][36][37][38][39]. The form of Eq.…”

Section: Introductionmentioning

confidence: 99%

“…The form of Eq. (1) has been used in a series of works by Mihalache et al [33][34][35]. The form of the SSE in the work by Wright III [36] is slightly different from the original version:…”

Section: Introductionmentioning

confidence: 99%

Rogue waves of the Sasa-Satsuma equation in a chaotic wave field

et al. 2014

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“…In this case, the generalized NLS equation (2) under (24) can also include Sasa-Satsuma (SS) equation [47][48][49][50][51][52] Fig. 3 The second-order rogue wave (23) under the free functions given by (22) with α = 1, ε = 3, ε 2 = 0.1, ρ 0 = 1, χ 0 = 2, χ 1 = 0.6, w 0 = 0.01, β 0 = 0.1, a = 0.15 and β = 100: a The motions of the hump and valleys; b the value evolutions of the hump and valleys; c the 2D contour plot; d the density evolution of the rogue wave Fig.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms

2015

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We investigate optical temporal rogue waves of the generalized nonlinear Schrödinger (NLS) equation with higher-order odd and even terms and space-and time-modulated coefficients, which includes the NLS equation, Lakshmanan-Porsezian-Daniel (LPD) equation, Hirota equation, Chen-Lee-Liu equation, and Kaup-Newell derivative NLS equation. Based on the similarity reduction method, the generalized NLS equation can be reduced to the integrable LPDHirota equation under a set of constraint conditions, from which the solutions of generalized NLS equation can be obtained in the basis of solutions of the LPDHirota equation and the similarity transformation. In particular, the first-and second-order self-similar rogue wave solutions of the generalized NLS equation are derived under different parameters, and the contour profiles and density evolutions of self-similar rogue wave solutions are given to study their wave structures and dynamic properties. At the same time, the motions of the hump and valleys related to the self-similar rogue waves are also given, from which we can control and manage the self-similar rogue waves by adjusting the third-order dispersion term. These results may be useful in nonlinear optics and related fields. KeywordsThe generalized nonlinear Schrödinger (NLS) equation · The LPD-Hirota equation · Similarity reduction method · Rogue wave solutions · Motions of the hump and valleys

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“…with 11) where c 1 , c 2 , ν 1 = 0, ν 2 = 0 obey the relations (1.8), corresponding to the properties…”

Section: Oscillatory Soliton Solutionsmentioning

confidence: 99%

Oscillatory solitons of U(1)-invariant modified Korteweg-de Vries equations. II. Asymptotic behavior and constants of motion

Anco

Mia

Willoughby

2015

Journal of Mathematical Physics

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Depending on the modulation frequency, the speeds of oscillatory waves (1-solitons) can be positive, negative, or zero, in contrast to the strictly positive speed of ordinary solitons. When the speed is zero, an oscillatory wave is a time-periodic standing wave. Oscillatory 2-solitons with non-zero wave speeds are shown to describe overtake collisions of a fast wave and a slow wave moving in the same direction, or head-on collisions of two waves moving in opposite directions. When one wave speed is zero, oscillatory 2-solitons are shown to describe collisions in which a moving wave overtakes a standing wave. An asymptotic analysis using moving coordinates is carried out to show that, in all collisions, the speeds and modulation frequencies of the individual waves are preserved, while the phases and positions undergo a shift such that the center of momentum of the two waves moves at a constant speed. The primary constants of motion as well as some other features of the nonlinear interaction of the colliding waves are discussed.

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The Riemann problem method for solving a perturbed nonlinear Schrodinger equation describing pulse propagation in optic fibres

Cited by 62 publications

References 48 publications

Rogue waves of the Sasa-Satsuma equation in a chaotic wave field

Rogue waves of the Sasa-Satsuma equation in a chaotic wave field

Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms

Oscillatory solitons of U(1)-invariant modified Korteweg-de Vries equations. II. Asymptotic behavior and constants of motion

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