From solitons to chaos: Nonlinear dynamics of the time-fractional modified regularized long-wave equation

This study investigates the nonlinear wave structures and complex dynamical behavior of the time-fractional modified regularized long-wave (mRLW) equation through an integrated analytical, numerical, and dynamical framework, with particular emphasis on connecting localized wave formation with perturbation-induced nonlinear dynamics. Within the adopted fractional framework, a Hirota-type bilinear approach is employed to construct one-wave, two-wave, three-wave, lump, one-lump–one-soliton, and one-lump–one-stripe structures, whose numerical consistency is subsequently examined using the Modified Variational Iteration (MVI) method through graphical comparisons and and error norms. The periodically perturbed mRLW system is further investigated using phase portraits, Lyapunov exponents, bifurcation diagrams, Poincaré sections, return maps, and fractal basin analysis, revealing parameter-dependent transitions from regular wave dynamics to complex and chaotic responses, with a box-counting fractal dimension of ( ) for the investigated basin structure. These findings demonstrate the usefulness of combining analytical wave construction, numerical verification, and nonlinear dynamical diagnostics for characterizing localized structures and complex behavior in fractional nonlinear evolution models.