ABSTRACT
To describe the lowest order nonlinear corrections to the basic linear wave analysis, it is an advantage to classify wave-types according to their dispersion relation. The important observation will be that many nonlinear wave phenomena can be “classified” into either Korteweg-deVries types or nonlinear Schrödinger types, as explained. There are many physically relevant phenomena which are described by this simple equation, but one would expect that in more general situations the pulses would be distorted and damped as they propagate. Physically, the wave steepening is due to harmonic generation by the nonlinearity being resonant since all harmonics are on the linear dispersion relation for the waves. For a boundary value problem with a harmonic wave excitation, we have at the point of excitation a well-defined frequency, and find higher harmonics develop at larger distances. This latter phenomenon is best illustrated by what is known as Fubini’s solution.
