Non-existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary-gravity waves on the free surface of a two-dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth-order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non-existence of truly solitary waves is obtained by using a contradiction on this constant.