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

Brian Choi

TL;DR

This work investigates the fractional Rosenau-Hyman equation using the homotopy perturbation method to construct analytic series for fractional K(n,n) with n=2,3. It shows that spatial nonlocality, modeled by the Riesz derivative, eliminates compactons and favors spatially periodic traveling waves, with a β-dependent bifurcation in wave propagation for the time-fractional case. For the fractional K(2,2) equation, the authors obtain an explicit solution in terms of Mittag-Leffler functions and demonstrate sublinear phase evolution and a phase-locking phenomenon, including a critical value β_c≈0.672 that separates monotone from oscillatory long-time behavior; at β=1 the classical traveling wave is recovered. In contrast, the fractional K(3,3) problem exhibits numerical evidence of finite-time blow-up, and the associated series have finite radii of convergence, signaling the absence of a global smooth traveling-wave solution. The study also reports instances of divergence in HPM, highlighting limitations and motivating future work to improve convergence analyses and develop hybrid methods. Overall, the paper provides insight into how nonlocality and fractional time dynamics alter wave propagation in nonlinear dispersive systems and offers diagnostic tools for the applicability of HPM to fractional PDEs.

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.

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

TL;DR

This work investigates the fractional Rosenau-Hyman equation using the homotopy perturbation method to construct analytic series for fractional K(n,n) with n=2,3. It shows that spatial nonlocality, modeled by the Riesz derivative, eliminates compactons and favors spatially periodic traveling waves, with a β-dependent bifurcation in wave propagation for the time-fractional case. For the fractional K(2,2) equation, the authors obtain an explicit solution in terms of Mittag-Leffler functions and demonstrate sublinear phase evolution and a phase-locking phenomenon, including a critical value β_c≈0.672 that separates monotone from oscillatory long-time behavior; at β=1 the classical traveling wave is recovered. In contrast, the fractional K(3,3) problem exhibits numerical evidence of finite-time blow-up, and the associated series have finite radii of convergence, signaling the absence of a global smooth traveling-wave solution. The study also reports instances of divergence in HPM, highlighting limitations and motivating future work to improve convergence analyses and develop hybrid methods. Overall, the paper provides insight into how nonlocality and fractional time dynamics alter wave propagation in nonlinear dispersive systems and offers diagnostic tools for the applicability of HPM to fractional PDEs.

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.

Paper Structure

This paper contains 15 sections, 5 theorems, 42 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Fractional calculus
  4. Nonexistence of traveling compactons
  5. Homotopy perturbation method
  6. Method
  7. Divergence of HPM
  8. Fractional K(2,2)
  9. Dynamics of fractional $K(2,2)$
  10. Decay of amplitude
  11. Phase-locking
  12. Fractional K(3,3)
  13. Dynamics of fractional $K(3,3)$
  14. Conclusion
  15. Introduction

Key Result

Proposition 2.1

For $m,n \in \mathbb{N}$ and $\alpha \in (0,2)$, consider If there exists $f \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$ that is compactly supported, and $u(x,t) = f(x-ct)$ where $c \in \mathbb{R}$, then $f = 0$.

Figures (7)

  • Figure 1: Wave profiles approximated by $\sum\limits_{n=0}^{9} u_n(x,t)$ for multiple values of $\beta$ at $t=1$.
  • Figure 2: Left: 3D plot of $\sum\limits_{n=0}^{4} u_n(x,t)$. Right: wave profiles approximated by $\sum\limits_{n=0}^{N} u_n(x,1)$ for $N=3,4$. Both plots use $\beta = 0.5$.
  • Figure 3: Evolution of $A(t)$: whether $A(t)$ decays to zero monotonically depends on $\beta$ relative to $\beta_c$.
  • Figure 4: Left: monotonic growth of $\phi(t)$ for $\beta \leq \frac{1}{2}$. Right: Growth of $|\log (\pi - \phi(t))|$.
  • Figure 5: Contour plots of $u(x,t)$ with $\beta = 0.8$ (Left) and $\beta = 0.6$ (Right) with $\alpha = 2,\ c = 1$. The evolution of phase $x = \phi(t)$ is marked by the round dots. The sinusoidal waves change the direction at times marked by the dotted horizontal lines.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2