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

TL;DR

This paper models few-cycle optical pulses using the complex mKdV equation, constructing higher-order rogue wave solutions and classifying their structures to aid in ultra-short pulse generation.

Contribution

It introduces a generalized Darboux transformation for the complex mKdV equation and systematically derives and classifies higher-order rogue wave solutions.

Findings

Constructed first-, second-, third-order rogue wave solutions

Classified rogue waves into fundamental, triangular, and ring patterns

Provided analytical formulas for rogue wave dimensions

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 non-standard 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 RW of the complex mKdV and the NLS 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 ultra-short pulse technology.