Rogue waves on an elliptic function background in complex modified Korteweg–de Vries equation

Abstract

With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg-de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfunctions and eigenvalues. To begin we construct the elliptic function background. Then, on top of this background, we create a rogue wave. We demonstrate the outcome for three distinct elliptic modulus values. We find that when we increase the modulus value the amplitude of rogue waves on the dn-periodic background decreases whereas it increases in the case of cn-periodic background.

Localized nonlinear waves on spatio-temporally controllable backgrounds for a (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq model in water waves

“…Thus the methodology and results presented in this manuscript serve the required purposes and they will be helpful for characterising the dynamical behaviour of nonlinear waves on different backgrounds. Particularly, the present route of extracting nonlinear wave solutions has an extra advantage over Darboux transformation and other methods for NLS, KdV, sG, Hirota and their coupled family of equations [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45], in terms of reducing the mathematical/computational complexity as well as a richer variety of solution profiles. To be precise, we have utilized simple exponential and polynomial type test functions as initial seed solutions to obtain the kink soliton (7) and rogue wave (10d), respectively, which manifested themself to produce various wave phenomena due to the available arbitrary backgrounds (8).…”

“…Particularly, the physical motivation to look for such nonlinear waves on non-uniform/varying backgrounds starts from the situation of randomly varying surface or deep water waves to inhomogeneous plasma, layered magnetic materials, inhomogeneous optical media, and atomic condensate system [22][23][24][25]. As a result of this search, some localized nonlinear waves on varying backgrounds are investigated in recent times, which include the rogue waves on cnoidal, periodic, and solitary wave backgrounds in one-dimensional models such as focusing NLS model [26][27][28][29], derivative NLS equation [30][31][32], higher-order nonlinear Schrödinger equation [33,34], higher-order modified KdV equation [35], modified KdV models [36,37], Hirota equation [38,39], Gerdjikov-Ivanov model [40], sine-Gordon equation [41,42], Fokas model [43], and coupled cubic-quintic NLS equation [44] as well as vector Chen-Lee-Liu NLS model [45]. Mostly, the method used in these studies is nothing but the Darboux transformation which requires Lax pair and involves complex mathematical calculations.…”

Modulational instability and rogue waves in one-dimensional nonlinear acoustic metamaterials: case of diatomic model

“…Indeed, solitons play a significant role in a great number of systems and appear in various forms: kinks, compactons, solitons, pulse, Rogue Waves [35][36][37]. A RW in particular is a nonlinear wave which is localised in space and time.…”