Cosmology and the Korteweg-de Vries Equation

Abstract

The Korteweg-de Vries (KdV) equation is a non-linear wave equation that has played a fundamental role in diverse branches of mathematical and theoretical physics. In the present paper, we consider its significance to cosmology. It is found that the KdV equation arises in a number of important scenarios, including inflationary cosmology, the cyclic universe, loop quantum cosmology and braneworld models. Analogies can be drawn between cosmic dynamics and the propagation of the solitonic wave solution to the equation, whereby quantities such as the speed and amplitude profile of the wave can be identified with cosmological parameters such as the spectral index of the density perturbation spectrum and the energy density of the universe. The unique mathematical properties of the Schwarzian derivative operator are important to the analysis. A connection with dark solitons in Bose-Einstein condensates is briefly discussed.