Numerical Computation of Smooth and Peaked Decaying Traveling-Wave Solutions of the Degasperis–Procesi Equation: A Benchmark Study

This work develops and benchmarks a numerical framework for computing decaying traveling-wave solutions of the Degasperis–Procesi equation. The approach provides a unified treatment of smooth solitary waves and peaked wave solutions across the relevant parameter regimes. Cusped and anticusped profiles are excluded, as their stronger singular behavior is not compatible with the regularity assumptions underlying the numerical formulation. The method addresses the challenges associated with the asymptotic behavior of the solutions and the non-smoothness of peaked profiles. Its performance is systematically assessed through convergence studies and comparisons with exact solutions. For smooth traveling waves, the numerical results exhibit the expected second-order convergence, while the computations for peaked waves demonstrate the method’s ability to resolve the non-smooth profile. The results also illustrate the dependence of wave profiles and amplitudes on wave speed and other model parameters, providing a benchmark for numerical computation of these solutions.