liquid is proposed. This model is the natural extension of the pioneering work by Benjamin

et al. [1]. The model considered here relates to the initial-value problem for the equation

where is nondimensional kinematic viscosity of the liquid, whose solution is

considered in a class of real nonperiodic functions defined for

This equation is an approximation derived for moderately long waves of small but finite

amplitude on a viscous liquid, which has the same formal justification as the Korteweg–de

Vries equation

when the effect of liquid viscosity is taken into account. The emphasis of this paper is on

rigorous mathematics which is used to prove the existence of the solution, and following