Evolution of solitons over a randomly rough seabed

Abstract

For long waves propagating over a randomly uneven seabed, we derive a modified Korteweg-de Vries (KdV) equation including new terms representing the effects of disorder on amplitude attenuation and wave phase. Analytical and numerical results are described for the evolution of a soliton entering a semi-infinite region of disorder, and the fission of new solitons after passing over a finite region of disorder.

Solitary wave dynamics in shallow water over periodic topography

“…IV. Usually, this problem has been analyzed in the literature for solitary waves when the "random" effect is weak ͑Benilov, 1992; Pelinovsky, 1996;Mei and Li, 2004;Fouque et al, 2004͒. The results of our analytical calculations for a periodic seabed are in reasonable agreement with experimental data, and with the known formulas for the random seabed.…”

“…This leads to the localization of wave energy, and this process has been analyzed in the framework of various mathematical models of water waves ͑Stepaniants, 2001; Ardhuin andHerbers, 2002͒ andexperimentally ͑Bel-zons et al, 1988͒. A similar approach also can be applied for weakly nonlinear waves above a random seabed ͑Kawahara, 1976; Rosales and Papanicolaou, 1983;Benilov, 1992;Pelinovsky, 1996;Mei and Li, 2004;Fouque et al, 2004͒. The case of large and stepped variations of the bottom topographic profile is more difficult and the number of solved problems is then very limited ͑Nachbin and Papanicolaou, 1992;Nachbin, 1995͒. The main problem here is the complicated structure of the scattered field which contains both waves and evanescent modes.…”

Are rogue waves robust against perturbations?

“…For fibres, the influence of loss was investigated in [35,36], and some other perturbations have been studied in [37,38]. For the ocean, a loss term has been taken into account in [39,40]. The initial conditions for the creation of rogue waves can be part of natural chaotic small amplitude waves in the ocean.…”

Propagation of Water Waves over Uneven Bottom under the Effect of Surface Tension

“…In case of constant depth, the coefficient ( ) is identically one and system (5) reduces to a system derived by Quintero and Montes in [4]. Wave-topography interaction has been the subject of considerable mathematical research [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The physical applications range from coastal surface waves [19] to atmospheric flows over mountain ranges [20,21].…”