Exponential Asymptotics using Numerical Rational Approximation in Linear Differential Equations

Abstract

Singularly-perturbed ordinary differential equations often exhibit Stokes' phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes curves. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a rational approximation based on a numerical leading-order solution using the adaptive Antoulas-Anderson (AAA) method. We show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. We explore the limitations of this approach, and show that for sufficiently small values of the asymptotic parameter, this approach breaks down; however, the range of validity may be extended by increasing the number of poles in the rational approximation. We finish by presenting a related nonlinear problem and discussing the challenges that arise when attempting to apply this method to nonlinear problems.

Figures (8)

• Figure 1: Magnitude of series terms from \ref{['eq.series']} evaluated at $x = 0$ for $\epsilon = 0.1$. As $n$ increases, the terms become smaller until a minimum value is reached at $n = 5$, after which the terms increase in size due to the factorial contribution to the numerator of $u_n$ in \ref{['eq.seriesterms']} The series \ref{['eq.series']} must therefore be divergent, and the optimal truncation point occurs at the minimum value.
• Figure 2: Stokes' phenomenon in the exact solution to \ref{['eq.ode']} with boundary conditions \ref{['eq.bc']}. The leading-order solution contains branch points at $x = \pm \mathrm{i}$, with the branch cuts extending vertically away from the real axis. The branch points generate Stokes curves, which connect the two points. On the left-hand side of the Stokes curves, there are no exponential contributions. On the right-hand side of the Stokes curves, the solution contains exponentially small oscillations of the form given in \ref{['e.uexp']}.
• Figure 3: The real and imaginary parts of the true leading-order solution $u_0$ and the approximated leading-order solution $\hat{u}_0$ described in Table \ref{['PoleTable']}. The two expressions are visually indistinguishable except on the imaginary axis. The function $u_0$ contains vertical branch cuts. The function $\hat{u}_0$ is a rational function which can only contain simple poles; these poles are arranged in such a way that they approximate the effect of a branch cut in the solution.
• Figure 4: This figure shows the error $|u_0 - \hat{u}_0|$, using $u_0$ and $\hat{u}_0$ from Figure \ref{['fig:complexplots']}. Note that the error is extremely small except in a region near the imaginary axis, where the true branch cut lies.
• Figure 5: Stokes structure in the solution to \ref{['eq.AAAode']}, which uses $\hat{u}_0$ as the leading-order solution. The solution contains 7 pairs of poles, with locations given in Table \ref{['PoleTable']}. Each pair of poles generates Stokes curves that extend vertically from the poles, intersecting the real axis. As each Stokes curve is crossed from left to right, an exponentially small asymptotic contribution appears in the solution. Note that the poles accumulate near the true branch points of $u_0$ at $x =\pm\mathrm{i}$. We will later find that the largest exponential contributions arise from the poles that are nearest to $x = \pm\mathrm{i}$.