Dynamical behavior of the higher-order cubic-quintic nonlinear Schrödinger equation with stability analysis

In this paper, we explore the intricate dynamics of ultrashort light pulses using a sophisticated higher-order nonlinear Schrödinger equation (NLSE). This equation incorporates third-order dispersion (3OD), fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms, providing a nuanced perspective on how light pulses propagate through optical media. The nonlinear Schrödinger equation (NLSE) is a fundamental equation describing light pulse dynamics in optical fibers and finds diverse applications including soliton trains in layered waveguides, dual-mode NLSE with Kerr-law nonlinearity, solitary waves in magnet chains, dark optical soliton theory, dual-wave multiplicative processes, and harmonic generation in left-handed nonlinear transmission lines. The objective of this research is to analyze the dynamics of ultrashort and to gain insights into their propagation in optical media and contribute to advancements in nonlinear optics. Employing analytical techniques such as the unified method and the Sardar subequation method, we uncover a diverse range of soliton solutions for this novel equation. These solutions encompass bright, dark, periodic, kinks, and singular solitons representing a significant milestone as these solutions have never been obtained for this equation before. Furthermore, we delve into the linear stability mechanism to assess the effects of different factors on soliton dynamics, such as group velocity dispersion and self-steepening coefficient terms. Through meticulous calculations and analysis, we gain valuable insights into the physical implications of our proposed model. To aid in visualizing these effects, we present 3D, 2D, and contour graphs, providing a comprehensive depiction of the behavior of light pulses in nonlinear optical systems. Mathematica software is used for all caculation. Overall, our study underscores the effectiveness of our methodologies in deriving analytical solutions for the NLSE, thereby advancing our understanding of pulse dynamics and contributing to the ongoing progress in nonlinear optics research.