Conservation laws, modulation instability and analysis of the Rosenau regularized long wave equation via Lie symmetry

In this paper, we investigate the Rosenau Regularized Long Wave (RRLW) equation, an improved model of the Korteweg–de Vries equation that describes nonlinear dispersive long-wave phenomena. Owing to its significance in nonlinear wave propagation, the RRLW equation has attracted considerable attention, particularly through numerical approaches. Despite this, its analytical structure remains insufficiently explored. To bridge this gap, we use the Lie symmetry analysis method to systematically study the equation. The associated infinitesimal generators are derived, yielding similarity reductions that transform the governing partial differential equation into the nonlinear ordinary differential equations. Exact analytical solutions are obtained using similarity reductions and the power series method, and the convergence of the series solutions is rigorously examined. The obtained solutions are new and more general than those reported in the existing literature. Furthermore, conservation laws for the RRLW equation are constructed using a recently developed conservation theorem based on self-adjointness. In addition, a linear stability analysis is performed to examine the modulational behavior of small perturbations around constant background solutions. The results provide deeper insight into the analytical properties, stability, and physical relevance of the RRLW equation.