Soliton dynamics and nonlinear wave theory: Understanding the interplay of thermal effects in plasma waves.

The regularized long-wave (ℝ ) equation is a fundamental model in shallow water wave theory, with extended relevance to other physical systems, such as plasma physics. The (ℝ ) equation governs key nonlinear wave phenomena, including the propagation and interaction of solitons, or localized solitary waves. In plasma physics, it describes ion-acoustic waves, which are low-frequency compressional waves arising from the interaction between ions and electrons. While the ion-acoustic wave exhibits Korteweg–de Vries soliton behavior in the cold plasma limit, the (ℝ ) equation offers a more accurate representation of ion-acoustic solitons and peaked structures in warm plasmas by accounting for thermal effects. This research presents novel analytical approximations for solitary wave solutions within the (ℝ ) equation, which is critical for modeling nonlinear shallow water and plasma waves. The equation's ability to describe peaked solitary waves, known as "peakons", represents wave-breaking phenomena. By employing the improved Riccati expansion and modified Fan expansion techniques, this study derives periodic peakon solutions, offering new insights into the equation's behavior. A thorough analysis of the (ℝ ) equation is provided, emphasizing its importance for nonlinear wave modeling in both fluid and plasma systems. This paper also includes numerical validation using He's variational iteration method, which enhances the transparency of the findings by addressing the underlying assumptions and limitations. The principal findings contribute to the broader understanding of nonlinear solitary wave theory, with practical implications for nonlinear wave dynamics across multiple physical domains. Suggested extensions are outlined for further investigation within this established theoretical framework. This study maintains consistent notation and terminology to facilitate clear communication of ideas in adherence to academic standards. [ABSTRACT FROM AUTHOR]