Abbreviated Journal Title
Nat. Hazards Earth Syst. Sci.
NONLINEAR SCHRODINGER-EQUATION; DEEP-WATER WAVES; FREQUENCY DOWNSHIFT; MODULATIONAL INSTABILITY; MELNIKOV ANALYSIS; SURFACE-WAVES; EVOLUTION; MECHANISM; DYNAMICS; SYSTEMS; Geosciences, Multidisciplinary; Meteorology & Atmospheric Sciences; Water Resources
Recently Gramstad and Trulsen derived a new higher order nonlinear Schrodinger (HONLS) equation which is Hamiltonian (Gramstad and Trulsen, 2011). We investigate the effects of dissipation on the development of rogue waves and downshifting by adding an additonal nonlinear damping term and a uniform linear damping term to this new HONLS equation. We find irreversible downshifting occurs when the nonlinear damping is the dominant damping effect. In particular, when only nonlinear damping is present, permanent downshifting occurs for all values of the nonlinear damping parameter beta. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. We examine the generation of rogue waves in the presence of damping for sea states characterized by JONSWAP spectrum. Using the inverse spectral theory of the NLS equation, simulations of the NLS and damped HONLS equations using JONSWAP initial data consistently show that rogue wave events are well predicted by proximity to homoclinic data, as measured by the spectral splitting distance delta. We define delta(cutoff) by requiring that 95% of the rogue waves occur for delta < delta(cutoff). We find that delta(cutoff) decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur. As a result when damping is present the proximity to homoclinic data and instabilities is more crucial for the development of rogue waves.
Natural Hazards and Earth System Sciences
Islas, A. and Schober, C. M., "Rogue waves and downshifting in the presence of damping" (2011). Faculty Bibliography 2010s. 1420.