Pinning and Unpinning in Nonlocal Systems

Abstract

We investigate pinning regions and unpinning asymptotics in nonlocal equations. We show that phenomena are related to but different from pinning in discrete and inhomogeneous media. We establish unpinning asymptotics using geometric singular perturbation theory in several examples. We also present numerical evidence for the dependence of unpinning asymptotics on regularity of the nonlocal convolution kernel.

Appendix: Passage Through an Inflection Point

In this appendix, we prove some results stated in Sect. 3.2 for the system of differential equations

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}w&=-1, \end{aligned}$$

(6.1a)

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}u&=u^3-w. \end{aligned}$$

(6.1b)

This system was obtained after some scalings in the study of the slow passage through an inflection point.

Proposition 6.1

For the system (6.1), the following results hold.

• (i) There exists a unique trajectory $\gamma _0$ in the (u, w)-plane such that $u(t)^3-w(t)\rightarrow 0$ for $t\rightarrow \pm \infty$.

• (ii) The Cauchy Principal Value $C_0$ of u exists and

  $$\begin{aligned} C_0 := P.V. \int _\mathbb {R}u(t)\mathrm {d}t<0. \end{aligned}$$

Proof

We first start by setting $w_{1,-}:=w^{-\frac{1}{3}}$ and $u_{1,-}:=uw^{-\frac{1}{3}}$ for $w>0$ (respectively $w_{1,+}$ and $u_{1,+}$ for $w<0$) such that system (6.1) is transformed into two systems

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}w_{1,\pm }&=\frac{1}{3}w_{1,\pm }^4, \end{aligned}$$

(6.2a)

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}u_{1,\pm }&=\frac{u_{1,\pm }^3-1}{w_{1,\pm }^2}+\frac{1}{3}u_{1,\pm }w_{1,\pm }^3. \end{aligned}$$

(6.2b)

Now, rescaling time such that $w_{1,\pm }^2 \frac{\mathrm {d}}{\mathrm {d}t}:=\frac{\mathrm {d}}{\mathrm {d}s}$, we obtain

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}w_{1,\pm }&=\frac{1}{3}w_{1,\pm }^6, \end{aligned}$$

(6.3a)

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}u_{1,\pm }&=u_{1,\pm }^3-1+\frac{1}{3}u_{1,\pm }w_{1,\pm }^5. \end{aligned}$$

(6.3b)

Note that each system possesses a unique equilibrium given by $(w_{1,\pm },u_{1,\pm })=(0,1)$, corresponding to the “equilibrium” $(w,u)=(\pm \infty ,\pm \infty )$ in system (6.1). The linearization at (0, 1) for each system is given by the Jacobian matrix

$$\begin{aligned} \mathbf {J}=\left( \begin{matrix} 0 &{}\quad 0\\ 0 &{}\quad 3 \end{matrix}\right) . \end{aligned}$$

As a consequence, for any $k\ge 2$, there exists a $\mathcal {C}^k$ center manifold $\mathcal {M}_{1,\pm }$, given locally, as a graph of form

$$\begin{aligned} \mathcal {M}_{1,\pm }=\left\{ \left( \Psi _\pm (w_{1,\pm }),w_{1,\pm }\right) ,~w_{1,\pm } \in \mathcal {V}_\pm \right\} , \end{aligned}$$

(6.4)

where $\mathcal {V}_\pm$ is a neighborhood of the origin in $\mathbb {R}^\mp$, and $\Psi _\pm$ is $\mathcal {C}^k$, with Taylor expansion

$$\begin{aligned} \Psi _\pm (w_{1,\pm })=1-\frac{1}{9}w_{1,\pm }^5+\mathcal {O}\left( w_{1,\pm }^6\right) , \text { as } w_{1,\pm } \longrightarrow 0. \end{aligned}$$

(6.5)

$\square$

We also note that $u_{1,-}\frac{\mathrm {d}}{\mathrm {d}s}u_{1,-}>0$ provided that $u_{1,-}$ is large enough and $w_{1,-}>0$. As a consequence, $u_{1,-}$ stays bounded as we solve backward in time and using Poincaré-Bendixson Theorem we obtain the following result and its corollary.

Lemma 6.2

Fix $0<\delta \ll 1$. Any trajectory of system (6.3) with initial condition $(w_{1,-},u_{1,-})=(\delta ,u_{1,-}^0)$, $u_{1,-}^0$ arbitrary, converges backward in time to (0, 1).

Corollary 6.3

The asymptotics for $s\longrightarrow -\infty$ are given by the center manifold (6.4) and (6.5), up to exponential corrections.

Finally, we remark that $u_{1,+} \ge 1$ is locally backward invariant since

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}u_{1,+}=\frac{1}{3}w_{1,+}^5 < 0, \text { at } w_{1,+}<0, u_{1,+}=1. \end{aligned}$$

And for $-S \le s \le S$, $S>0$ fixed, solving backward in time, we have that solutions exist ($u_{1,\pm }^2$ decreases when large).

As a consequence, we have the existence of a unique trajectory (w, u) for system (6.1) with

$$\begin{aligned} \frac{u^3}{w}\longrightarrow 1 \text { as } t \longrightarrow \pm \infty , \end{aligned}$$

and asymptotics

$$\begin{aligned} u(t)=-t^{\frac{1}{3}}\left( 1+\frac{1}{9}t^{-\frac{5}{3}}+\mathcal {O}\left( t^{-2}\right) \right) , \end{aligned}$$

as $t \longrightarrow \pm \infty$. This further ensures that the Cauchy principal value $C_0$ of u(t) exists. One easily checks that $C_0<0$ and this concludes the proof of the proposition.$\square$