Korteweg—de Vries Equation and Solitons

Abstract

In the previous chapter, we have seen that nonlinear dispersive systems can have characteristically different types of wave solutions compared to linear dispersive systems. In particular, the former can admit solitary wave solutions, and as an example we have demonstrated this for the case of the KdV equation. How does the KdV equation itself arise typically, how stable is the solitary wave and how do the solitary waves interact mutually? We will analyse these aspects in this chapter and bring out the fact that the KdV equation indeed describes the Scott Russel phenomenon. We also point out how the KdV equation recurs in an entirely different physical context, namely wave propagation in the famous Fermi—Pasta—Ulam (FPU) anharmonic lattice. Further we explain how the results on nonenergy sharing among the modes of the FPU lattice led Norman Zabusky and Martin Kruskal to carry out the now famous numerical experiments, which ultimately lead to the notion of soliton. Finally, we also point out how the explicit soliton expressions can be obtained through an algorithmic procedure called the bilinearization method, introduced by Hirota, from which one can easily understand the basic soliton properties.