Nonlinear Capillary Waves on the Surface of Liquid Column

Weakly nonlinear capillary waves on a liquid column of circular cross-section are investigated on the basis of the derivative expansion method. It is found that the complex amplitude of a quasi-monochro-matic travelling wave can be described by a nonlinear Schrodinger equation in a frame of reference moving with the group velocity. The known property of this equation reveals that wave trains of constant amplitude are modulationally unstable, which suggests a new possibility of the break-up of column. On the other hand, the complex amplitude of a quasi-monochromatic standing wave near the cut-off is governed by another type of nonlinear Schrodinger equation in which the role of the time and that of the space are interchanged. This equation makes it possible to estimate the nonlinear effect on the cut-off wave-number.