Existence of periodic traveling wave solution to the forced generalized nearly concentric Korteweg-de Vries equation
Summary: This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries (KdV) equation in the form of \[ (u_\eta+ u/(2\eta)+ [f(u)]_\xi+ u_{\xi\xi\xi})_\xi+ u_{\theta\theta}/\eta^2= h_0. \] The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili (KP) equation, \[ (u_t+ [f(u)]_x+ u_{xxx})_x+ u_{yy}= h_0, \] 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.
generalized Kadomtsev-Petviashvili equation
nonlinear ordinary differential equation
existence of periodic traveling wave solutions
scientific article; zbMATH DE number 1548199