Remarks on Existence/Nonexistence of Analytic Solutions to Higher Order KdV Equations

In this note, we discuss the existence of analytic solutions to the nonlinear wave equations of the higher order than the ubiquitous Korteweg-de Vries equation. First, we recall our recent results which show that the extended Korteweg-de Vries equation, that is, the equation obtained within second-order perturbation approach possesses three kinds of analytic solutions. These solutions have the same functional form as the corresponding Korteweg-de Vries solutions. We show, however, that the most intriguing multi-soliton solutions, known for the Korteweg-de Vries equation, do not exist for extended Korteweg-de Vries equation. Moreover, we show that for the equations obtained in the third order perturbation approach (and then in any higher order) analytic solutions in the forms known from Korteweg-de Vries theory do not exist.