Transient waves in linear dispersive media with dissipation: an approach based on the steepest descent path

TL;DR

This work addresses the inverse Laplace transform problem for impulse responses in linear dispersive media with dissipation by introducing a steepest descent path (SPD) deformation of the Bromwich contour. The authors derive the SPD geometry for the Klein-Gordon equation with dissipation, distinguishing the cases $\Delta<0$ (closed elliptical SPD) and $\Delta>0$ (two symmetric SPD branches), and replace the Bromwich integral with real-line quadratures along the SPD to compute $r_{\delta}(x,t)$ and $r_{n}(x,t)$. Numerical results show that SPD-based inversions reproduce the exact time-domain solutions across parameter regimes, highlighting a robust, non-asymptotic alternative to standard Laplace inversion methods. The approach reduces computational difficulty by leveraging real-line integration and path parametrizations, with potential broad applicability to dissipative dispersive media and other Laplace-inversion problems.

Abstract

In the study of linear dispersive media it is of primary interest to gain knowledge of the impulse response of the material. The standard approach to compute the response involves a Laplace transform inversion, i.e., the solution of a Bromwich integral, which can be a notoriously troublesome problem. In this paper we propose a novel approach to the calculation of the impulse response, based on the well assessed method of the steepest descent path, which results in the replacement of the Bromwich integral with a real line integral along the steepest descent path. In this exploratory investigation, the method is explained and applied to the case study of the Klein- Gordon equation with dissipation, for which analytical solutions of the Bromwich integral are available, as to compare the numerical solutions obtained by the newly proposed method to exact ones. Since the newly proposed method, at its core, consists in replacing a Laplace transform inverse with a potentially much less demanding real line integral, the method presented here could be of general interest in the study of linear dispersive waves in presence of dissipation, as well as in other fields in which Laplace transform inversion come into play.

Figures (9)

• Figure 1: Steepest descent path for the case $\Delta < 0$ (blue curve). The red and black dots represent, respectively, the saddle points and the branch points; the grey line represent the branch cut (values of the parameters: $a=1$, $b=0$, $c=1$, $\Delta=-1/4$; $\mu=1/2$).
• Figure 2: Steepest descent path for the case $\Delta > 0$ (blue curve). The red and black dots represent, respectively, the saddle points and the branch points; the grey line represent the branch cut (values of the parameters: $a=1$, $b=5/4$, $c=1$, $\Delta=1$; $\mu=1/2$).
• Figure 3: Comparison between exact and numerical solutions of $r_\delta(x,t)$ for $\Delta < 0$, for several values of $t$ (values of the parameters: $a=1$, $b=0$, $c=1$, $\Delta=-1/4$).
• Figure 4: Comparison between exact and numerical results of $r_\delta(x,t)$ for $\Delta > 0$ (values of the parameters: $a=1$, $b=5/4$, $c=1$, $\Delta=1$, $t=64$).
• Figure 5: Comparison between exact and numerical results of $r_\delta(x,t)$ for $\Delta > 0$ (values of the parameters: $a=10^{-4}$, $b=5$, $c=2$, $\Delta=4.9999999975$, $t=100$).