The nonsingular brane solutions via the Darboux transformation

TL;DR

This paper develops a method using Darboux transformations to construct exact, nonsingular three-dimensional brane solutions interacting with five-dimensional gravity and scalar fields, ensuring finite Ricci scalar and small cosmological constants.

Contribution

It introduces a novel application of Darboux transformations to generate nonsingular brane solutions with finite curvature and small cosmological constants, extending previous models.

Findings

Constructed exact nonsingular brane solutions with finite Ricci scalar.

Demonstrated the formalism's applicability to cosmologically expanding branes.

Produced models with exponentially small cosmological constant on the visible brane.

Abstract

We consider the Darboux transformation as a method of construction of exact nonsingular solutions describing the three-dimensional brane that interacts with five-dimensional gravity and the bulk scalar field. To make it work, the five-dimensional Einstein's equations and the Israel's conditions are being reduced to the Schr\"odinger equation with the jump-like potential and the wave functions sewing conditions in jump point correspondingly. We show further that it is always possible to choose the functions in Crum's determinants in such way, that the five-dimensional Ricci scalar $R$ will always be finite both on brane and in bulk. The new exact solutions being the generalizations of the model with the odd superpotential are presented. Described formalism is also appliable to the cases of more realistic branes with cosmological expansion. As an example, via the usage of the simple orbifold model ($S_1/{\Bbb Z}_2$) and one-time Darboux transformation we construct the models where the cosmological constant on the visible brane is exponentially small.