Solution of ODE u″ + p(u)(u′)2 + q(u) = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Abstract

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc, are reduced to an integrable ODE expressed by u″ + p(u)(u′)2 + q(u) = 0 whose general solution can be given. Furthermore, combining complete discrimination system for polynomial, the classifications of all single travelling wave solutions to these equations are obtained. The equation u″ + p(u)(u′)2 + q(u) = 0 includes the equation (u′)2 = f(u) as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.