Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 66, pp. 1-10.
Title: Special solutions of the Riccati equation with applications
to the Gross-Pitaevskii nonlinear PDE
Authors: Anas Al Bastami (Texas A&M Univ., Qatar)
Milivoj R. Belic (Texas A&M Univ., Qatar)
Nikola Z. Petrovic (Texas A&M Univ., Qatar)
Abstract:
A method for finding solutions of the Riccati differential equation
$y' = P(x) + Q(x)y + R(x)y^2$ is introduced. Provided that certain
relations exist between the coefficient $P(x)$, $Q(x)$ and
$R(x)$, the above equation can be solved in closed form.
We determine the required relations and find the general solutions
to the aforementioned equation. The method is then applied to the
Riccati equation arising in the solution of the multidimensional
Gross-Pitaevskii equation of Bose-Einstein condensates by the
F-expansion and the balance principle techniques.
Submitted April 6, 2010. Published May 08, 2010.
Math Subject Classifications: 34A34, 34A05.
Key Words: Riccati equation; solutions to Gross-Pitaevskii equation;
chirp function.