The Periodic Wave Solution and Riemann Problem for the Modified Benjamin–Bona–Mahony Equation

In this paper, we investigate periodic wave solution and the complete classifications of solutions for the modified Benjamin–Bona–Mahony (mBBM) equation, also commonly referred to as the modified regularized long wave (mRLW) equation with initial discontinuous Riemann problem. The nonlinear dispersive mBBM equation serves as an alternative model to the modified Korteweg–de Vries (mKdV) equation in physical phenomena such as long‐crested waves in near‐shore regions and unidirectional wave propagation in water channels. By employing hyperbolic theory, asymptotic analysis, Whitham modulation theory, and the dispersive shock wave (DSW) fitting method, along with direct numerical simulations, we systematically analyze the fundamental wave structures in the Riemann problem. These basic waves include linear wavetrains composed of one‐phase and two‐phase solutions, RWs, DSWs, and the two‐phase modulation structures for DSW implosion. Due to the non‐integrability of the mBBM equation and the non‐convexity of its linear dispersion relation, its Riemann problem exhibits a richer variety of solutions compared to the mKdV equation. The most notable differences include the emergence of two‐phase linear wavetrains and DSW implosion. Finally, based on the differences in wave structures, we classify the initial parameter space into 10 distinct regions.