Emergence of peaked singularities in the Euler-Poisson system

TL;DR

This paper analyzes the asymptotic behavior of peaked solitary waves in the one-dimensional Euler-Poisson system, revealing their singularities and blow-up profiles through theoretical and numerical methods.

Contribution

It provides the first detailed asymptotic analysis of peaked solitary waves in the Euler-Poisson system and explores their blow-up behavior in the pressureless limit.

Findings

Exact asymptotic behavior of peaked solitary waves near the peak

Numerical evidence of $C^1$ blow-up solutions with similar profiles

Different blow-up mechanism compared to Burgers-type shocks

Abstract

We consider the one-dimensional Euler-Poisson system equipped with the Boltzmann relation and provide the exact asymptotic behavior of the peaked solitary wave solutions near the peak. This enables us to study the cold ion limit of the peaked solitary waves with the sharp range of H\"older exponents. Furthermore, we provide numerical evidence for $C^1$ blow-up solutions to the pressureless Euler-Poisson system, whose blow-up profiles are asymptotically similar to its peaked solitary waves and exhibit a different form of blow-up compared to the Burgers-type (shock-like) blow-up.