Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model

Abstract

We investigate different types of complex soliton solutions with regard to their stability against linear pertubations. In the Bullough-Dodd scalar field theory we find linearly stable complex $\mathcal{P}\mathcal{T}$-symmetric solutions and linearly unstable solutions for which the $\mathcal{P}\mathcal{T}$-symmetry is broken. Both types of solutions have real energies. The auxiliary Sturm-Liouville eigenvalue equation in the stability analysis for the $\mathcal{P}\mathcal{T}$-symmetric solutions can be solved exactly by supersymmetrically mapping it to an isospectral partner system involving a shifted and scaled inverse $\cosh$-squared potential. We identify exactly one shape mode in form of a bound state solution and scattering states which when used as linear perturbations leave the solutions stable. The auxiliary problem for the solutions with broken $\mathcal{P}\mathcal{T}$-symmetry involves a complex shifted and scaled inverse $\sin$-squared potential. The corresponding bound and scattering state solutions have complex eigenvalues, such that when used as linear perturbations for the corresponding soliton solutions lead to their decay or blow up as time evolves.

15 pages, 4 figures

References in corpus (6)

• Collective Coordinates in One-Dimensional Soliton Models Revisited
• Spectral singularities in PT-symmetric periodic finite-gap systems
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• The Bullough-Dodd model coupled to matter fields
• Complex BPS Skyrmions with real energy
• Nonlinear Classical and Quantum Integrable Systems with PT-symmetries

Cited by in corpus (2)

• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• Moduli spaces for PT-regularized solitons