1. I. Introduction Purposes & Applications
Certain physical phenomena cannot be described using Linear Wave Theory (LWT), but can be described by Non-linear Wave Theory (NWT), such as mass transport, nonlinear energy transfer among waves of different frequencies, and bound waves.
The prediction based on LWT sometimes is not accuracy. By using NWT, the accuracy of predictions can be greatly improved, for example, the wave kinematics near steep ocean wave crests.

2. Nonlinear Periodic Wave Train
NWT based on periodic waves lays a solid foundations for future studies of NWT for irregular waves.
Governing Equations and Boundary Conditions

3. Governing equation and boundary conditions are:
C(t) is Bernoulli constant, and chosen so that the still water is at
z = 0. The Characteristics of a regular wave train are sketched in
the figure below.