Multiple Self-Similar Solitons to the Higgs Field in the de Sitter Space 1

Abstract

This paper deals with singular boundary value problems (BVPs) for the second order nonlinear ordinary differential equations arising from inflationary cosmology models with a scalar field. For the Higgs-field nonlinear wave equation in the de Sitter space, we look for automodeling solutions: one-dimensional ones as the domain walls and billows, cylindrically symmetrical (CS) solutions as the tubes, and spherically symmetrical (SS) as the spherical bubbles. The singular BVPs are posed in terms of the obtained automodeling variables and the results of their analytical-numerical investigations are formulated. In particular, we obtained that these singular BVPs are solvable, their nontrivial solutions are continuable with no limit when the independent variable tends to infinity and there occurs a multiplicity of the indicated soliton-type solutions. The number of solutions increases with the growth of the problem parameter. Moreover, the eigenvalues of a relevant linear singular BVP are the points of a global bifurcation to initial nonlinear singular BVP. The SS (CS) solution describes the bubble (the tube) which collapses in an infinite time. As the perturbation parameter increases indefinitely, the radius of the self-similar bubble (tube) tends to an exact solution of a thin-wall approximation. Thus, the nontrivial effects of the expansion of the early Universe on some classical physical objects are obtained.