Traveling wave reductions and adaptive moving mesh computations for the improved Boussinesq equation

This paper is concerned with the analytical and numerical study of the improved Boussinesq (IB) equation, a nonlinear dispersive model for applications in fluid dynamics, elasticity, geophysics, and nonlinear optics. Two systematic symbolic algorithms, i.e., the generalized tanh method and the $(1/{\mathrm{\Theta }}^{\prime })$-expansion method, are used for the recovery of analytical traveling-wave solutions of the IB equation. These solutions reveal a vast taxonomy of nonlinear waveforms corresponding to solitary, rational, and periodic profiles, governed by parameter combinations that regulate dispersion, wave amplitude, and phase. As a complement to the analytical study, we use an $r$-adaptive numerical method built from the Parabolic Monge-Ampère (PMA) moving mesh method and discretized by central differences in space and BDF2 in time. An adaptive algorithm automatically relocates the mesh nodes toward locations where sharp gradients are present, thereby ensuring accuracy and efficiency and preventing unnecessary computational cost. Numerical experiments evidence second-order convergence and stability and demonstrate the ability of the method to resolve sharp wave interaction without spurious oscillations. In total, the combination of exact benchmarks and adaptive simulation provides a practical framework for simulating nonlinear dispersive waves with impact in applications such as tsunami simulation, earthquake wave propagation, and optical signal pulse transmission.