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
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 . The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation, , 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.