Exploring fourth-order nonlinear integrable equation: Solitons, interactions and modulation instability

Abstract

This article examines the dynamics of a fourth-order integrable nonlinear evolution equation using exact soliton solutions, soliton interactions, and modulation instability analysis. Through the application of bilinear transformations, the nonlinear evolution equation is reformulated using a logarithmic transformation, together with analytical techniques including Hirota’s bilinear method, the (G?/G)?expansion method, and the (m+1/G?)?expansion method, in order to construct explicit solutions. The various types of solutions obtained include single, breather, and double solitons. The analysis of soliton interactions suggests that they interact elastically, as indicated by phase shifts and the preservation of soliton profiles. The modulation instability analysis indicates that periodic and coherent waveforms become localized under specific parameter conditions. The results of the study show that different parameter regimes produce different types of waveforms, such as hyperbolic, periodic, and rational forms. This research has implications for understanding the dynamics of nonlinear waves and their stability in higher-order integrable systems, with potential applications in nonlinear optics, fluid dynamics, plasma physics, and Bose–Einstein condensation. © 2026 The Authors.