Details

The KdV equation, applied in a special form with variable coefficients, is:

,

with and with these choices for the variable coefficient function being:

,

,

,

and where , and so on. The solution is a solitary wave described by:

.

From the continuity equation it follows for the velocity field, which is derived from . As long as is integrable, there exists an analytical solution for the time evolution of a idealized particle:

with the initial position . The starting positions of the particles are linearly distributed around the peak of the wave at , where the maximum of the wave depends only on and : The solution of the KdV equation is based on [1].

Reference

[1] R. Sabry, M. A. Zahran, and E. Fan, "A New Generalized Expansion Method and Its Application in Finding Explicit Exact Solutions for a Generalized Variable Coefficients KdV Equation," Physics Letters A 326, 2004 pp. 93–101.