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Three self-similar solutions of Yang-Mills equations in high odd dimensions

Piotr Bizoń, Irfan Glogić, Arthur Wasserman

Abstract

We consider spherically symmetric Yang-Mills equations with gauge group $SO(d)$ in $d+1$ dimensional Minkowski spacetime. For any given odd $d\geq 11$, we establish existence and uniqueness (modulo reflection symmetry) of exactly $N$ smooth self-similar solutions, where $N$ is the number of zeros of an explicit polynomial $P_m(z)$ of degree $m=(d-5)/2$ in the interval $0<z<1$. The number $N$ can be determined algorithmically by an explicit computation. We find that $N=3$ for all integer $m$ from $3$ to $15$, the upper bound being merely limited by the extent of our computations. A proof that $N=3$ for all odd $d\ge 11$ remains an open problem.

Three self-similar solutions of Yang-Mills equations in high odd dimensions

Abstract

We consider spherically symmetric Yang-Mills equations with gauge group in dimensional Minkowski spacetime. For any given odd , we establish existence and uniqueness (modulo reflection symmetry) of exactly smooth self-similar solutions, where is the number of zeros of an explicit polynomial of degree in the interval . The number can be determined algorithmically by an explicit computation. We find that for all integer from to , the upper bound being merely limited by the extent of our computations. A proof that for all odd remains an open problem.
Paper Structure (7 sections, 2 theorems, 23 equations)

This paper contains 7 sections, 2 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Local solutions near the endpoints
  3. Shooting argument
  4. Behavior outside the past light cone
  5. Outlook
  6. Appendix

Key Result

Lemma 1

If $d\geq 10$, then $u(y)$ is monotone decreasing from $u(0)=1$ to $u(1)>0$.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3