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 s hown that there are no solitary-wave solutions of the exact governing equat ions of the flow that decay to zero exponentially at infinity if the surfac e tension coefficient is less than its critical value and lies in some inte rvals. 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 equ ation when the solution is extended to the complex plane. Then the non-exis tence of truly solitary waves is obtained by using a contradiction on this constant.