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Discrete equations and auto-traveling kinks of the $φ^6$ model

H. Susanto, N. Karjanto

TL;DR

This paper investigates discrete higher-order φ^6 Klein–Gordon models and whether lattice discretizations can preserve stationary kink solutions. Using a one-dimensional map approach $φ_{n+1}=F(φ_n)$ and discretized first-integral ideas, the authors construct two classes of exceptional discretizations that admit static, translationally invariant kinks centered at arbitrary lattice positions, including a cubic-p_3/quintic family and a hexic-p_6 family (with $p_6 = p_3^2$ in a special case). Numerical results show absence of internal modes between the zero eigenvalue and the band edge for the static kinks in these exceptional models; non-exceptional discretizations, by contrast, yield auto-traveling kinks that accelerate to the sonic limit and radiate energy. The study clarifies how discretization structure governs the persistence and dynamics of kinks in higher-order field theories and suggests directions for analyzing sliding velocities, radiation spectra, and extensions to other models.

Abstract

We study the $φ^{6}$ model and derive two broad classes of lattice discretizations that admit static, translationally invariant kinks; that is, stationary kink profiles that can be centered at an arbitrary position relative to the lattice. These discretizations are constructed using a one-dimensional map, $φ_{n+1}=F(φ_{n})$, which provides a direct and systematic algorithm for generating such models. Numerical computations for two representative cases show that the discrete kinks do not possess internal modes, consistent with the continuum theory, although an additional high-frequency mode may appear above the phonon band. We also show that generic discretizations of the $φ^{6}$ model do not support static kink solutions. Instead, the resulting dynamics produce auto-traveling and self-accelerating kinks that propagate at the maximal group velocity while continuously emitting radiation.

Discrete equations and auto-traveling kinks of the $φ^6$ model

TL;DR

This paper investigates discrete higher-order φ^6 Klein–Gordon models and whether lattice discretizations can preserve stationary kink solutions. Using a one-dimensional map approach and discretized first-integral ideas, the authors construct two classes of exceptional discretizations that admit static, translationally invariant kinks centered at arbitrary lattice positions, including a cubic-p_3/quintic family and a hexic-p_6 family (with in a special case). Numerical results show absence of internal modes between the zero eigenvalue and the band edge for the static kinks in these exceptional models; non-exceptional discretizations, by contrast, yield auto-traveling kinks that accelerate to the sonic limit and radiate energy. The study clarifies how discretization structure governs the persistence and dynamics of kinks in higher-order field theories and suggests directions for analyzing sliding velocities, radiation spectra, and extensions to other models.

Abstract

We study the model and derive two broad classes of lattice discretizations that admit static, translationally invariant kinks; that is, stationary kink profiles that can be centered at an arbitrary position relative to the lattice. These discretizations are constructed using a one-dimensional map, , which provides a direct and systematic algorithm for generating such models. Numerical computations for two representative cases show that the discrete kinks do not possess internal modes, consistent with the continuum theory, although an additional high-frequency mode may appear above the phonon band. We also show that generic discretizations of the model do not support static kink solutions. Instead, the resulting dynamics produce auto-traveling and self-accelerating kinks that propagate at the maximal group velocity while continuously emitting radiation.
Paper Structure (9 sections, 33 equations, 2 figures)

This paper contains 9 sections, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Cubic Polynomial
  4. Hexic Polynomial
  5. Numerical Illustrations
  6. Auto-Traveling Kinks
  7. Conclusion
  8. Coefficients in Eq. \ref{['comsq']}
  9. Driven Frenkel-Kontorova Model

Figures (2)

  • Figure 1: Linear spectrum of a kink solution from the discrete models \ref{['ex1']} (top panel) and \ref{['ex2']} (bottom panel) as a function of the discretization $h$.
  • Figure 2: Top view of $\phi_n$ for $h = 0.1$ in the $(x = nh, t)$-plane. The dashed curve represents \ref{['post']}, while the dash-dotted line corresponds to the critical speed \ref{['maxv']}. The inset shows $\phi_n$ at $t = 200$.