International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 6, Pages 371-377

Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation

Kenneth L. Jones, Xiaogui He, and Yunkai Chen

Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville 28301-4298, North Carolina, USA

Received 15 October 1999

Copyright © 2000 Kenneth L. Jones et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries equation in the form of (uη+u/(2η)+[f(u)]ξ+uξξξ)ξ+uθθ/η2=h0. The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation, (ut+[f(u)]x+uxxx)x+uyy=h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.