Intercomparison of Truncated Series Solutions for Shallow Water Waves

Abstract

The relationship between truncated Fourier series solutions of several different long-wave evolution equations is explored. The models are all designed to represent the same physics in the asymptote of small nonlinearity and frequency dispersion, and yet give different numerical solutions for given values of dimensionless parameters. It is found that the problem of obtaining a steady solution in a coupled-mode model of shallow-water wave evolution is related more to the problem of properly choosing the corresponding time-dependent evolution equation than to the problem of truncating the infinite series representation of the solution to that equation. In the process, a particular lowest-order coupled-mode model is related to a particular modified form of the Kortweg–de Vries equation. It is shown that the existing lowest-order model may be inherently inaccurate in the case of relatively high waves, due to the nonphysical behavior of the form of the nonlinear term inherent in that model. Finally, solitary and cnoidal wave solutions are given for the entire family of modified Kortweg–de Vries equations, and their properties are compared.