Approximate asymptotic integration of a higher order water-wave equation using the inverse scattering transform

Abstract. The complete mathematical and physical characterization of nonlinear water wave dynamics has been an important goal since the fundamental partial differential equations were discovered by Euler over 200 years ago. Here I study a subset of the full solutions by considering the irrotational, unidirectional multiscale expansion of these equations in shallow-water. I seek to integrate the first higher-order wave equation, beyond the order of the Korteweg- deVries equation, using the inverse scattering transform. While I am unable to integrate this equation directly, I am instead able to integrate an analogous equation in a closely related hierarchy. This new integrable wave equation is tested for physical validity by comparing its linear dispersion relation and solitary wave solution with those of the full water wave equations and with laboratory data. The comparison is remarkably close and thus supports the physical applicability of the new equation. These results are surprising because the inverse scattering transform, long thought to be useful for solving only very special, low-order nonlinear wave equations, can now be thought of as a useful tool for approximately integrating a wide variety of physical systems to higher order. I give a simple scenario for adapting these results to the nonlinear Fourier analysis of experimentally measured wave trains.