Nonexistence of traveling wave solutions in the fractional Rosenau-Hyman equation via homotopy perturbation method

TL;DR

This paper uses the homotopy perturbation method to analyze the fractional Rosenau-Hyman equation, revealing the nonexistence of traveling wave solutions due to spatial nonlocality and exploring bifurcation and blow-up phenomena.

Contribution

It demonstrates the application of HPM to fractional PDEs, showing nonexistence of compactons and analyzing wave bifurcation and blow-up behaviors.

Findings

Spatial nonlocality prevents compacton solutions.

Bifurcation occurs with respect to fractional order.

Potential finite time blow-up in cubic case.

Abstract

We apply the homotopy perturbation method to construct series solutions for the fractional Rosenau-Hyman (fRH) equation and study their dynamics. Unlike the classical RH equation where compactons arise from truncated periodic solutions, we show that spatial nonlocality prevents the existence of compactons, and therefore periodic traveling waves are considered. By asymptotic analyses involving the Mittag-Leffler function, it is shown that the quadratic fRH equation exhibits bifurcation with respect to the order of the temporal fractional derivative, leading to the eventual pinning of wave propagation. Additionally, numerical results suggest potential finite time blow-up in the cubic fRH. While HPM proves effective in constructing analytic solutions, we identify cases of divergence, underscoring the need for further research into its convergence properties and broader applicability.