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Kinks of fractional $φ^4$ models: existence, uniqueness, monotonicity, stability, and sharp asymptotics

Atanas G. Stefanov, P. G. Kevrekidis

Abstract

In the present work we construct kink solutions for different (parabolic and wave) variants of the fractional $φ^4$ model, in both the sub-Laplacian and super-Laplacian setting. We establish existence and monotonicity results (for the sub - Laplacian case), along with sharp asymptotics which are corroborated through numerical computations. Importantly, in the sub-Laplacian regime, we provide the explicit and numerically verifiable spectral condition, which guarantees uniqueness for odd kinks. We check numerically the relevant condition to confirm the uniqueness of such solutions. In addition, we show asymptotic stability for the stationary kinks in the parabolic setting and also, the spectral stability for the traveling kinks in the corresponding wave equation.

Kinks of fractional $φ^4$ models: existence, uniqueness, monotonicity, stability, and sharp asymptotics

Abstract

In the present work we construct kink solutions for different (parabolic and wave) variants of the fractional model, in both the sub-Laplacian and super-Laplacian setting. We establish existence and monotonicity results (for the sub - Laplacian case), along with sharp asymptotics which are corroborated through numerical computations. Importantly, in the sub-Laplacian regime, we provide the explicit and numerically verifiable spectral condition, which guarantees uniqueness for odd kinks. We check numerically the relevant condition to confirm the uniqueness of such solutions. In addition, we show asymptotic stability for the stationary kinks in the parabolic setting and also, the spectral stability for the traveling kinks in the corresponding wave equation.

Paper Structure

This paper contains 35 sections, 22 theorems, 212 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical Setup and Main Results
  3. Main results
  4. Existence of kinks: sub-Laplacian case
  5. (Conditional) Uniqueness of the kinks
  6. Asymptotic stability with phase
  7. Kinks for the fractional wave equation
  8. Existence of kinks: super-Laplacian case
  9. Preliminaries
  10. Homogeneous tempered distributions
  11. Littlewood-Paley partitions and various function spaces
  12. Estimates on fractional derivatives on nonlinear functionals
  13. Variational construction and consequences: Proof of Theorem \ref{['theo:10']}
  14. Approximate variational problem
  15. Uniform in $\epsilon$ a priori estimates
  16. ...and 20 more sections

Key Result

Theorem 1

Let $\alpha\in (0,2)$. Let $u\in C^{1,1}_{loc.}(\mathbb R)\cap L^\infty(\mathbb R)$ be a solution to where $\lim_{x\to \pm\infty} u(x)=\pm 1$ and $|u(x)|\leq 1$. Assume that $f$ is continuous on $[-1,1]$, non-increasing on $[-1, -1+\delta]\cup [1-\delta, 1]$ for some small $\delta>0$. Then, $u$ is strictly monotonically increasing.

Figures (3)

  • Figure 1: In the top panel we showcase a prototypical example of the decay of the stationary $\phi^4$ kink to the asymptotic state of $\phi=1$ for the case of $\alpha=1.5$ (the asymptotic value is denoted by the black line). The bottom panel shows the distance from the asymptotic value in a loglog plot. The prediction of Theorem \ref{['theo:10']} is shown by the dashed red line, while the stationary kink of the model is shown by the blue solid line. It can be clearly discerned (indeed, especially so in the loglog plot) that the kink tail aligns itself sharply with the proposed asymptotics of the theorem at sufficiently large distances.
  • Figure 2: The present figure effectively encompasses the results of our continuation for the kink waveform of the fractional wave PDE with $\alpha \in [0.9,2.5]$. What is shown is the 2nd eigenvalue of the (opposite of the) linearization operator, i.e., $\lambda_1({\mathcal{L}}_\phi)$ as a function of $\alpha$. It can be clearly discerned that thoughout the relevant interval and indeed for all values of $\alpha$ higher than those shown, this eigenvalue will satisfy the conditions of Theorem \ref{['theo:un']}, ensuring not only the spectral stability, but also the uniqueness of the relevant kinks.
  • Figure 3: Similarly to Fig. \ref{['fig:atan_frac_f1']}, in the top panel we showcase a prototypical example of the decay of the stationary $\phi^4$ kink to the asymptotic state of $\phi=1$ for the case of $\alpha=2.5$ (the asymptotic value again denoted by the black line). The bottom panel shows the distance from the asymptotic value in a loglog plot. The prediction of Theorem \ref{['theo:40']} is shown by the dashed red line, while the stationary kink of the model is shown by the blue solid line. Both the top and bottom panels clearly illustrate the (single, in this case) crossing of $\phi=1$. In the loglog panel it appears as a "dip". Here, too, again both the linear and especially the loglog plot exemplify the sharp alignment of the kink tail with the proposed asymptotics of the theorem at sufficiently large distances.

Theorems & Definitions (31)

  • Theorem 1: Wu-Chen, chen
  • Theorem 2: Existence of kinks in sub-Laplacian regime
  • Theorem 3: Conditional Uniqueness for the kinks
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • Proposition 1: Theorem 1.2, Li
  • Proposition 2
  • Lemma 2
  • ...and 21 more