arXiv subjects

David Amundsen

Publications and source records attributed to David Amundsen.

4 recordsLinked to original sources

arXiv2026

A resonance in phonons scattering off a kink in the absence of a Peierls-Nabarro potential

We investigate the interaction of small-amplitude waves called phonons, with an initially static kink in an exceptional discretization of the $\phi^4$ model that is free of the Peierls-Nabarro potential. Phonons are generated by a localized harmonic source and scattered from one side of the kink. By computing the transmission and reflection coefficients over the entire phonon band, we demonstrate that the scattering properties depend strongly on the lattice spacing. In the weak-discreteness regime ($h<1$), the kink is nearly transparent and phonons are transmitted through it over most of the phonon spectrum. In contrast, for strong discreteness ($h>1$), significant reflection emerges even though the corresponding continuum $\phi^4$ kink is reflectionless. We further show that depending on the frequency of the incoming phonons, the kink experiences negative radiation pressure and is accelerated toward the incoming phonons for all lattice spacings considered, and this effect is much stronger for the strong discretness. The frequency dependence of the kink velocity and energy transfer is explained in terms of resonances associated with Doppler-shifted phonon frequencies and extrema of the phonon group velocity. Our results reveal that strong lattice discreteness can qualitatively modify phonon-kink interactions even in systems where the static Peierls-Nabarro potential is absent.

nlin.PS↗

arXiv2023

Radial positive solutions for mixed local and nonlocal supercritical Neumann problem

In this paper, we establish the existence of positive non-decreasing radial solutions for a nonlinear mixed local and nonlocal Neumann problem in the ball. No growth assumption on the nonlinearity is required. We also provide a criterion for the existence of non-constant solutions provided the problem possesses a trivial constant solution.

math.AP↗

arXiv2023

A mixed local and nonlocal supercritical Dirichlet problems

In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem \begin{equation} Lu=|u|^{p-2}u+\mu|u|^{q-2}u~~\text{in}~~\Omega,~~~~~ u=0~~\text{in}~~\mathbb{R}^N\setminus\Omega,~~~ (0.1) \end{equation} where $L=-\Delta+(-\Delta)^s$ for $s\in(0,1)$ and $\Omega\subset\mathbb{R}^N$ is a bounded domain. Precisely, we show that problem (0.1) for $1<q<2<p$ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $\mu$. Here $u$ can be either a scalar function, or a vector valued function so that (0.1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system \begin{equation} Lu=|v|^{p-2}v,~~~~~~~ Lv=|u|^{d-2}u+\mu |u|^{q-2}u \end{equation} with the Dirichlet boundary condition on $\Omega$ where $1<q<2<p, d$. Our method is variational, and in both problems the lack of compactness for the supercritical problem is recovered by working on a closed convex subset of an appropriate function space.

math.AP↗

arXiv2017

A Step-by-Step Procedure for Local Analysis of Differential Equations

This note provides a detailed algorithm to the application of local (perturbation) analysis of differential equations which is normally taught at graduate math courses. Exercise books often present more abstract and simplified versions of equations for the application of perturbation techniques. The equation we study comes from the theory of competing mechanisms and describes the behavior of rational buyers in a certain environment. While in the latter literature similar equations were solved numerically, an analytical solution adds both theoretical and practical value for students and researchers.

math.CA↗