Cosmology and the Korteweg-de Vries equation The Korteweg-de
Vries equation is a nonlinear 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 Korteweg-de
Vries 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.