# Localized heat perturbation in harmonic 1D crystals. Solutions for an equation of anomalous heat conduction

## TL;DR

This paper derives exact solutions for anomalous heat propagation in 1D harmonic lattices, revealing how different initial temperature perturbations decay over time and highlighting observable peaks near the wavefront.

## Contribution

It provides explicit solutions for various initial perturbations in 1D harmonic crystals, advancing understanding of anomalous heat conduction mechanisms.

## Key findings

- Decay near wavefront proportional to 1/√t
- Decay at center proportional to 1/t
- Peaks remain visible near the wavefront over time

## Abstract

In this work exact solutions for the equation that describes anomalous heat propagation in 1D harmonic lattices are obtained. Rectangular, triangular, and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the solution near the wavefront is proportional to $1/ \sqrt{t}$. In the center of the perturbation zone the decay is proportional to $1/t$. Thus the solution decays slower near the wavefront, leaving clearly visible peaks that can be detected experimentally.