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Existence of Multistring Solutions of the Self-Gravitating Massive $W-$Boson

Dongho Chae

TL;DR

This work addresses the existence of multistring solutions for a semilinear elliptic system arising from self-gravitating, massive $W$-boson cosmic strings, including nonradial configurations with distinct string locations. The authors decompose solutions into Liouville-type leading profiles $\rho^I$ and $\rho^{II}$ plus small perturbations, derive radial auxiliary corrections $w_1,w_2$ with precise logarithmic decays, and formulate a fixed-point problem in weighted Banach spaces. They establish nondegeneracy of the linearized operator by proving surjectivity and identifying its kernel, and apply the Implicit Function Theorem to obtain a 2-parameter family of small-$\varepsilon$ solutions with explicit asymptotics for $u$ and $\eta$. The results yield sharp decay rates (involving constants $C_1,C_2$ and a Beta function) and solve an open problem on nonradial multistring solutions, with relevance to the physical model where certain positivity and non-smallness conditions hold. Overall, the paper advances the mathematical existence theory for planar Bogomol’nyi-type systems with multiple vortices.

Abstract

We consider a semilinear elliptic system which include the model system of the $W-$strings in the cosmology as a special case. We prove existence of multi-string solutions and obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open problem posed in \cite{yan}

Existence of Multistring Solutions of the Self-Gravitating Massive $W-$Boson

TL;DR

This work addresses the existence of multistring solutions for a semilinear elliptic system arising from self-gravitating, massive -boson cosmic strings, including nonradial configurations with distinct string locations. The authors decompose solutions into Liouville-type leading profiles and plus small perturbations, derive radial auxiliary corrections with precise logarithmic decays, and formulate a fixed-point problem in weighted Banach spaces. They establish nondegeneracy of the linearized operator by proving surjectivity and identifying its kernel, and apply the Implicit Function Theorem to obtain a 2-parameter family of small- solutions with explicit asymptotics for and . The results yield sharp decay rates (involving constants and a Beta function) and solve an open problem on nonradial multistring solutions, with relevance to the physical model where certain positivity and non-smallness conditions hold. Overall, the paper advances the mathematical existence theory for planar Bogomol’nyi-type systems with multiple vortices.

Abstract

We consider a semilinear elliptic system which include the model system of the strings in the cosmology as a special case. We prove existence of multi-string solutions and obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open problem posed in \cite{yan}

Paper Structure

This paper contains 2 sections, 6 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.1

Key Result

Theorem 1.1

Let N \in \Bbb N \cup \{0\}, and \mathcal{Z}=\{ z_{j}\}_{j=1} ^{N} be given in \Bbb R^2 allowing multiplicities. Suppose the coefficients, \lambda_1, \lambda_2,\lambda_3,\lambda_4 satisfy one of the conditions; either or Then, there exists a constant \varepsilon_1 >0 such that for any \varepsilon \in (0, \varepsilon_1 ) and any c_0 > there exists a family of solutions to (11)-(13), (u,\eta ). Mo

Theorems & Definitions (6)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3