Nonexistence of small periodic traveling wave solutions to the power Kadomtsev-Petviashvili equation (Q1374839)

This paper is concerned with the nonexistence of small periodic traveling wave solutions to the power Kadomtsev-Petviashvili equation of the form \[ [u_t+\gamma u^n u_x+ \alpha u_{xxx}]_x+ \beta u_{yy}= 0. \] This equation is converted to a periodic boundary value problem for a nonlinear fourth-order ordinary differential equation, which is shown to be equivalent to a nonlinear integral equation with kernel of Green function type. The authors define a compact operator \(A\) on the Banach space \(C_{2T}\) of real-valued continuous periodic functions with a given period \(2T\). They prove that for any small \(T>0\), there exists an \(r>0\) such that \(A\) has no nontrivial fixed point in \(B(0,r)\subset C_{2T}\).