Interplay among dispersion, gain and nonlinearity for nonlinear Schrödinger equations in an inhomogeneous medium: applications to coastal engineering

Abstract

The evolution of a weakly nonlinear, narrow-banded wave packet in an inhomogeneous medium is studied in the context of coastal engineering. Analytically the model is a variable coefficient nonlinear Schrödinger equation (vcNLS) with a gain/loss term, arising from the waves running up a sloping beach. The crucial distinction from previous works is the choice of a ‘forcing’ boundary condition. Energy then enters the computational domain. This dynamics serves as a crude model for storm surge waves approaching shore. The competition among dispersion, gain and nonlinearity dictates the amplification process. In shallow water (or defocusing regime of vcNLS), waves moving up a sloping beach experience the largest steepening on a convex seafloor, followed by a straight-line seafloor, and then concave seafloor. For a larger spatial domain, a wave packet moving from the open ocean to the shoreline will pass through a critical depth where vcNLS changes from focusing to defocusing. The presence of modulation instability, breather and Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) further complicates the dynamics. Wave amplification in the focusing regime is then most pronounced on a concave seafloor, followed by a straight-line seafloor, and finally a convex seafloor. The growth rate of modulation instability and the magnitude of eigenvalues obtained from Floquet analysis provide quantitative confirmation for these predictions on the dynamics under different seafloor bathymetries. Compared with growth rates of modulation instability, the eigenvalues of the Jacobian matrix in computational analysis provide reasonably accurate predictions of the critical depths. Implications in terms of nonlinear science as well as actual coastal defense mechanisms are discussed.