Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation

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.…”

Modulation instability, rogue waves of the seventh-order nonlinear Schrödinger equation on a periodic background

“…For example, Zhang et al [29] constructed the rogue wave solution of fourth-order NLS equation under the periodic background. Sinthuja et al [30] constructed the rogue wave solution of the fifth-order NLS equation under the periodic background.…”

Whitham modulation theory and periodic solutions for the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

“…In Ref. [21], combining the Darboux transformation with the nonlinearization of the spectral problem, the rogue wave solutions on the periodic wave background of Eq. (1.1) have been constructed by using two types of Jacobian elliptic functions.…”