Modulation instability and rogue waves for the sixth-order nonlinear Schrödinger equation with variable coefficients on a periodic background

In this research, we delve into a generalized highly dispersive (HD) nonlinear Schrödinger equation, enriched with cubic-quintic-septic-nonic (CQSN) nonlinearities. The core of our investigation revolves around the perturbation of plane waves, aiming to understand their stability characteristics in such a complex medium. We investigate the influence of various factors such as the amplitude of the plane wave, perturbed wave number, nonic nonlinear term, and fourth-order dispersion term. Our findings indicate that increasing the amplitude of the plane wave widens the modulation instability (MI) bands and amplifies the MI growth rate. In contrast, increasing the nonic nonlinear term has opposing effects, narrowing the MI bands and diminishing the amplitude of the MI growth rate. Increasing the fourth-order dispersion term does not affect the amplitude of the MI growth rate but narrows the MI bands. The observed pattern of increasing and then decreasing MI intensity with rising K can be attributed to the complex interplay among phase matching conditions, dispersion effects, and nonlinear saturation. Initially, higher K enhances phase matching and boosts MI growth. However, as K increases further, the combined influence of dispersion and nonlinear effects can diminish the effectiveness of phase matching, resulting in a reduction in MI intensity. A significant portion of our work is dedicated to identifying and analyzing modulated rational, polynomial Jacobi elliptic function solutions, and the emergence of optical solitons within this framework. These findings provide new insights into the nonlinear dynamics underpinning the generalized HDNLSE, enriched with CQSN nonlinearities, offering valuable contributions to the theoretical understanding of such phenomena.

Section: Y Ay By Y Y Y Y My H Y Y H Y Yy H Y Ymentioning

confidence: 99%

Modulational instability, modulated wave, and optical solitons for a generalized highly dispersive cubic-quintic-septic-nonic medium with self-frequency shift and self-steepening nonlinear terms

Yomba

2024

“…Modulation Instability (MI) of short pulses was identified in an optical fiber experiment via the recording of symmetrical side bands in the averaged optical spectrum and oscillations. On the other hand, modulation instability analysis for rogue waves have been carried out through pioneering works [46][47][48][49][50]. The observation of rogue waves in an optical system reported based on extremely broadband radiation is generated from a narrow broadband input [51].…”

Section: Introductionmentioning

confidence: 99%

Numerical investigation on nonautonomous optical rogue waves and Modulation Instability analysis for a nonautonomous system

Saravana Veni,

Mani Rajan,

Bertrand Tabi

et al. 2024

In this paper, we report existence of optical rogue waves in the focussing non - autonomousnonlinear Schr ̈odinger equation (NLSE) through numerical studies of modulation instability (MI).The dynamics of non-autonomous rogue waves discussed and its associated modulation instabilitythrough linear stability analysis taken place followed by pulse splitting behaviour due to non -autonomous coefficient. We prove that the excitation of rogue waves with certain conditions in thebase band modulation instability regime. The above analysis of complex dynamics in terms of MIprocesses has allowed to experiments to excite the nonlinear superposition of rogue wave solutionsusing a modulated plane wave optical field injected into optical fiber which offer the evidence forexcitation of nonautonomous rogue waves in an inhomogeneous nonlinear medium. It is identifiedfrom the results frequency modulation on a wavefield induces modulation instability as a result ofrogue waves. We analyze the dependence of parameters coefficient of group velocity dispersion(GVD)and nonlinearity (α(z)) and non - autonomous coefficient (β(z)) and the instability of rogue waves.Our work suggests that the presence of non-autonomous coefficients can have a significant impacton the emergence of extreme events, particularly in relation to their self - steepening nature.

“…So far, several methods have been derived to obtain rogue waves on periodic background, such as DT method, algebraic geometry method, and PINN deep learning method [29][30][31][32]. The periodic background of elliptic function and traveling wave have been widely studied, and periodic rogue wave solutions of many nonlinear evolution equations have been constructed, such as sine-Gordon equation [33], Gerdjikov-Ivanov equation [34], Hirota equation [35] and higher-order NLS equations [30,[36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Solitons, breathers and periodic rogue waves for the variable-coefficient seventh-order nonlinear Schrödinger equation

Jiang¹,

Zhaqilao

2023

Self Cite

Through Darboux transformation (DT) method, Several nonlinear wave solutions of seventh-order variable-coefficient nonlinear Schrödinger (vcNLS) equation are obtained, including solitons, breathers and rogue periodic waves. When the coefficients are linear, parabolic and periodic functions, the parabolic, cubic and quasi-periodic solitons and breathers can be constructed. Then we investigatetheir effects on the solutions, the variation of the coefficients affects the shapes of the solutions. On this basis, the interactions between two solitons are studied and the interactions between two types of breathers are constructed. Next, through the approaches of the nonlinearization of spectral problem and DT method, rogue waves on the background of the Jacobi elliptic functions dn and cn for a seventh-order vcNLS equation are constructed. When the coefficients are selected as linear, exponential and periodic functions, the nonlinear dynamics of two kinds of rogue periodic waves are analysed.