Transformation of envelope solitons on a bottom step

TL;DR

This study investigates how surface envelope solitons transform when crossing a bottom step in water, deriving formulas for wave amplitude and confirming results with numerical simulations showing significant wave amplification.

Contribution

The paper introduces analytical formulas for soliton transformation over a bottom step and validates them with numerical simulations, revealing potential wave amplification effects.

Findings

Wave amplification exceeds double in simulations

Analytical formulas match numerical results

Constructive interference enhances soliton amplitude

Abstract

In this paper we study the transformation of surface envelope solitons travelling over a bottom step in water of a finite depth. Using the transformation coefficients earlier derived in the linear approximation, we find the parameters of transmitted pulses and subsequent evolution of the pulses in the course of propagation. Relying on the weakly nonlinear theory, the analytic formulae are derived which describe the maximum attainable wave amplitude in the neighbourhood of the step and in the far zone. Solitary waves may be greatly amplified (within the weakly nonlinear theory formally even without a limit) when propagating from relatively shallow water to the deeper domain due to the constructive interference between the newly emerging envelope solitons and the residual quasi-linear waves. The theoretical results are in a good agreement with the data of direct numerical modelling of soliton transformation. In particular, more than double wave amplification is demonstrated in the performed simulations.