Revealing the Bright Soliton Solution of the Perturbed Fourth-Order SchröDinger-Hirota Equation Having Cubic-Quintic-Septic Laws and Modulation Instability

Abstract

This paper aims to explore the bright soliton solutions of the perturbed fourth-order Schr & ouml;dinger-Hirota equation with cubic-quintic-septic nonlinearities, a model relevant for understanding complex wave dynamics in nonlinear media. Bright solitons are indispensable in optics, particularly for applications that benefit from stable, long-distance transmission with minimal signal degradation. Their usage spans from telecommunications to medical imaging and sensing, where their ability to maintain shape and resist interference makes them highly valuable. Using the addendum to the Kudryashov's method, we systematically derive the bright soliton solutions, including both bright and singular forms, and conduct a modulation instability analysis to identify stability regions and parameter conditions that influence soliton persistence. The methodology involves addendum to the Kudryashov's method through additional analytical steps, tailored to accommodate the equation's complex terms, enabling the extraction of explicit solutions. Findings reveal that the parameters significantly impact soliton amplitude, displacement and stability, as illustrated through 2D and 3D plots. The modulation instability analysis further clarifies the effect of these parameters on soliton stability, underscoring the balance between dispersive and nonlinear influences. This study's limitations include focusing on analytical solutions within specific parameter ranges, which may not fully capture the behavior under extreme conditions. Nonetheless, the originality lies in the integration of Kudryashov-based techniques with a high-order nonlinear model, offering benchmark solutions and insights that support future theoretical and experimental research in nonlinear optics and other wave-propagating media.