[Paper Review] Study of Stability of a Charged Topological Soliton in the System of Two Interacting Scalar Fields

Vakhid A. Gani, N.B. Konyukhova+2 more|arXiv (Cornell University)|Oct 16, 2007

Numerical methods for differential equations6 references5 citations

TL;DR

This paper analyzes the dynamic stability of a charged topological soliton (topological Q-ball) in a (1+1)-dimensional model of two interacting scalar fields— a complex charged field and a real neutral Higgs-like field. Using analytical and numerical methods on a singular self-adjoint spectral problem with a quadratic operator pencil, it demonstrates that the soliton is dynamically stable under small perturbations for $\kappa \in [1.23, \sqrt{2})$, with no complex eigenvalues found, indicating no instability from oscillatory modes.

ABSTRACT

An analytical-numerical analysis of the singular self-adjoint spectral problem for a system of three linear ordinary second-order differential equations defined on the entire real exis is presented. This problem comes to existence in the nonlinear field theory. The dependence of the differential equations on the spectral parameter is nonlinear, which results in a quadratic operator Hermitian pencil.

Motivation & Objective

• To investigate the dynamic stability of a previously found exact solution representing a topological Q-ball in a (1+1)-dimensional model of two interacting scalar fields.
• To analyze the spectral properties of the linearized stability problem around this solution, which leads to a singular self-adjoint spectral problem with a nonlinear dependence on the spectral parameter.
• To determine whether the system exhibits unstable modes (complex eigenvalues) that would lead to dynamical instability of the soliton.
• To clarify the behavior of the eigenvalue $\lambda = 0$—its multiplicity and structure—as a function of the coupling parameter $\kappa$.
• To assess whether the topological Q-ball can decay into non-localized oscillations or split into other configurations under perturbations.

Proposed method

• Formulation of the stability problem via linearization of the field equations around the exact topological Q-ball solution, resulting in a system of three coupled second-order ODEs with a quadratic operator pencil in the spectral parameter $\lambda$.
• Transformation of the original Lagrangian to dimensionless variables to simplify the analysis and reduce the number of parameters, with $\kappa = h/m$ as the key dimensionless coupling parameter.
• Application of analytical techniques to study the structure of the eigenvalue $\lambda = 0$, including algebraic and geometric multiplicity analysis, particularly at $\kappa = \sqrt{2}$.
• Numerical computation of eigenvalues using multiple contours in the complex $\lambda$-plane, including circles, sectors, and intervals on real/imaginary axes, to detect complex eigenvalues.
• Use of symmetry properties of eigenvalues and eigenfunctions to validate results and control for spurious or missed eigenvalues.
• Employment of high-precision numerical solvers with relative accuracy up to $10^{-4}$, even for stiff problems at small $\kappa \approx 0.05$, to ensure robustness of the spectral analysis.

Experimental results

Research questions

• RQ1 Does the topological Q-ball solution $\phi_0(x) = \tanh(x)$, $\xi_0(x) = \sqrt{1/\kappa^2 - 1} \, \text{sech}(x)$ remain dynamically stable under small perturbations for $\kappa < \sqrt{2}$?
• RQ2 What is the nature of the eigenvalue $\lambda = 0$ in the linearized stability problem, particularly its algebraic and geometric multiplicities, and how do they change with $\kappa$?
• RQ3 Are there any complex eigenvalues in the admissible domain of the spectral parameter $\lambda$, indicating potential dynamical instability?
• RQ4 Can the topological Q-ball decay into non-localized oscillations over the kink solution $\phi_w(x) = \tanh(\sqrt{2}x/\kappa)$, and is such a decay energetically favorable?
• RQ5 Is the soliton absolutely stable in the sector of topological charge $P_0 = 1$ and U(1)-charge $Q_0$?

Key findings

• The singular boundary value problem for the linearized stability equations has only one eigenvalue, $\lambda = 0$, with algebraic multiplicity 4 and geometric multiplicity 2 for $\kappa \in [1.23, \sqrt{2})$.
• No complex eigenvalues were found in the numerical experiments within the domain $|\lambda| \leq \sqrt{-\mu_{\text{min}}}$, indicating the absence of exponentially growing modes.
• The eigenvalue $\lambda = 0$ remains degenerate with multiplicity 4 as $\kappa$ varies from $\sqrt{2}$ to 0, and its splitting into distinct eigenvalues does not occur.
• The geometric multiplicity of $\lambda = 0$ reduces from 3 to 2 as $\kappa$ moves away from $\sqrt{2}$, a typical behavior in quadratic pencils due to the $\lambda^2$ dependence in the operator.
• The topological Q-ball solution is dynamically stable under small perturbations for $\kappa \in [1.23, \sqrt{2})$, with the highest confidence in this range.
• The solution does not decay into non-localized oscillations over the kink, as such a process would not be energetically favorable.