Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Traveling waves and effective mass for the regularized Landau-Pekar equations

Simone Rademacher

Abstract

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.

Traveling waves and effective mass for the regularized Landau-Pekar equations

Abstract

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.
Paper Structure (13 sections, 13 theorems, 226 equations)

This paper contains 13 sections, 13 theorems, 226 equations.

Table of Contents

  1. Introduction and main results
  2. Properties of the ground state and traveling waves
  3. Properties of the ground state
  4. Traveling waves
  5. Properties of the energy functional $\mathcal{E}_\alpha$
  6. Existence
  7. Approximation
  8. Properties of the Hessian
  9. Proof of Propositions \ref{['prop:existence-and-unique']},\ref{['prop:gs']}
  10. Proof for traveling waves
  11. Proofs for definitions effective mass
  12. Effective mass through traveling waves
  13. Effective mass through energy-momentum expansion

Key Result

Theorem 1

Let $\varepsilon, v$ satisfy Ass. ass:reg and Ass. ass:super and $\vert {\rm v} \vert < {\rm v}_{\rm crit}$. Then there exists a traveling wave solution of the form def:tw with ${\rm v} \not= 0$.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2.1: Existence
  • Proposition 2.2: Approximation of the ground state
  • Lemma 2.1
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 14 more