Few-cycle optical rogue waves: Complex modified Korteweg–de Vries equation

Abstract

In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology.

On the extension of solutions of the real to complex KdV equation and a mechanism for the construction of rogue waves

“…Rogue waves generated in the optical systems are sharp, rare and extremely high power pulses that share the main features. [16][17][18][19][20][21][22][23][24][25][26] They are also known as walls of water by seafarers. Very recently, they are shown to be accompanied by breathing offering a possible explanation for their generation.…”

Abnormal single or composite dissipative solitons generation

“…Recently, people pay more attention to some unusual optical solitons such as the similaritons which can maintain their overall shapes but with their widths and amplitudes changing with the distance, the vortex optical solitons and azimuthons with phase singularities and angular momentum [9,10], and optical rogue wave exhibiting very high amplitude, highly spatiotemporal localized structure, and sudden appearance and disappearance without any signs [11][12][13][14][15]. Specifically, the dissipative optical soltons, which generally appear in the dissipative systems where energy supplies and loss are involved, have recently trigged growing research interests for the reason that they can be exploited to design high performance passively mode-locked fiber lasers with extremely high pulse energy and peak power [16].…”

The Darboux transformation of the Kundu–Eckhaus equation

“…Naturally, the higher order RW solutions have more peaks and exhibit several interesting patterns [39][40][41][42][43][44]. In addition to the NLS equation, there are many other equations admitting RW (or Peregrine-type) solutions such as the modified Korteweg-de Vries equation, the Fokas-Lenells equation, the derivative NLS equation, the long-wave-short-wave resonance equation, the vector NLS, the Davey-Stewartson equation and the KP-I equation [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], etc.…”

“…So we only study this problem for the first-order RW solution. Recently, the contour line method has been introduced as an efficient tool to analyse the localization characters of the RW solution in [53]. According to this method, on the background plane with height c 2 , a contour line of |u…”

“…So getting the higher order RW solutions and their evolution in terms of different patterns are actually challenging uphill problems. -We have defined two characters (the length and the width) [53] of the first-order RW solution for the NLS equation. Naturally, can we define an area of this doubly localized solution of the KE equation on the (t, x) plane?…”