Riemann-Hilbert Formulation and Solution of Nonlinear Shallow-Water Wave Equations: Nonlocal Dbar Problem as a Unified Approach to Computing Exact Solutions in the Time Domain

Abstract

Abstract Linear wave theories and stochastic linear wave spectra are foundational to ocean and coastal engineering. However, there are some important limitations with linear wave theories, and some of which were originally observed experimentally more than a century ago by John Scott Russell. Nonlinearities will cause the linear wave spectra to evolve in time and space as waves move across a basin, or as waves evolve in the ocean and coastal waters, and in coastal channels. Using a nonlinear wave theory can therefore allow researchers and engineers to better capture the dynamic nature of the ocean waves. Common classical nonlinear wave theories include the (2+1)D KP equation and the (1+1)D Korteweg-de Fries (KdV) equation. As described by Osborne [2010], the invariance nature of these nonlinear wave equations can be characterized in an exact manner by their corresponding nonlinear frequency or wave-number spectrum parallel to the linear case. A nonlocal dbar model was used by Ablowitz, Bar-Yaacov and Fokas [1983], and Zakharov and Manakov [1984] to solve the KP equation by the inverse scattering transform (IST). In this study we discuss the use of the nonlocal dbar problem (NDP) to compute the soliton solutions by the inverse scattering transform, and finite-genus solutions to the KdV equation, and undular bores (dispersive shockwaves). We also discuss the use of NDP to produce soliton solutions and the calculation of periodic finite-genus solutions to the KP equation using Riemann theta functions and complex algebraic geometry.