Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bifurcation from the Kurth solution in galactic dynamics

Markus Kunze, Rafael Ortega

TL;DR

The paper analyzes the nonlinear gravitational Vlasov-Poisson system near the Kurth equilibrium by constructing an explicit, infinite-dimensional family of weak static states $f_\Gamma$ that share the Kurth density $\rho_{\mathrm{Kurth}}$. It recasts the uniform-density constraint as an Abel-type integral equation for a profile $\varphi$, solving it via a one-dimensional operator framework involving $I^{1/2}$, $K_\Gamma$, and $S_\Gamma$ and obtaining a corresponding $\Phi_\Gamma$. The authors prove $f_\Gamma\to f_{\mathrm{Kurth}}$ in $L^1$ as $\Gamma\to1$, and each static state is surrounded by time-periodic weak solutions, revealing a rich bifurcation structure. By convexity, these equilibria lead to an infinite-dimensional continuum of steady states, all with the same density and with associated periodic perturbations, highlighting a large abundance of equilibria near the Kurth state and offering potential links to Osipkov-Merritt models.

Abstract

It will be shown that there exists an infinite-dimensional continuum ${\cal C}$ of weak static solutions of the Vlasov-Poisson system that bifurcates from the Kurth solution. Each $f_\ast\in {\cal C}$ has the charge density $ρ_{f_\ast}=ρ_{\rm Kurth}$, and (like the Kurth solution itself) each $f_\ast$ is surrounded by time-periodic weak solutions.

Bifurcation from the Kurth solution in galactic dynamics

TL;DR

The paper analyzes the nonlinear gravitational Vlasov-Poisson system near the Kurth equilibrium by constructing an explicit, infinite-dimensional family of weak static states that share the Kurth density . It recasts the uniform-density constraint as an Abel-type integral equation for a profile , solving it via a one-dimensional operator framework involving , , and and obtaining a corresponding . The authors prove in as , and each static state is surrounded by time-periodic weak solutions, revealing a rich bifurcation structure. By convexity, these equilibria lead to an infinite-dimensional continuum of steady states, all with the same density and with associated periodic perturbations, highlighting a large abundance of equilibria near the Kurth state and offering potential links to Osipkov-Merritt models.

Abstract

It will be shown that there exists an infinite-dimensional continuum of weak static solutions of the Vlasov-Poisson system that bifurcates from the Kurth solution. Each has the charge density , and (like the Kurth solution itself) each is surrounded by time-periodic weak solutions.

Paper Structure

This paper contains 12 sections, 19 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The support of the steady states
  3. Construction of steady states
  4. Reformulation of the condition on uniform distribution
  5. Some integral operators
  6. Solving the functional equation $(({\rm id}-S_\Gamma)\psi)(s)=s$
  7. Solving the integral equation $(({\rm id}-S_\Gamma)I^{1/2}\varphi)(s)=\frac{s}{\pi}$
  8. The function $\Phi_\Gamma$ and end of the proof of Theorem \ref{['ex_fGam']}
  9. The limit $\Gamma\to 1$: Proof of Theorem \ref{['Gamto1']}
  10. Periodic solutions $f_{\Gamma, \varepsilon}$ close to $f_\Gamma$: Proof of Theorem \ref{['per_gam']}
  11. An infinite-dimensional continuum of equilibria
  12. Appendix

Key Result

Theorem 1.2

For every $\Gamma\in [1-2^{-12}, 1[$, there exists a weak static solution $f_\Gamma\in L^1(\mathbb R^3\times\mathbb R^3)$ whose essential support is and which satisfies

Figures (1)

  • Figure 1: $r^2\le\Gamma$ (left) and $r^2\ge\Gamma$ (right)

Theorems & Definitions (25)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7: weak solution
  • Remark 1.8
  • Lemma 2.1
  • Lemma 3.1
  • ...and 15 more