A numerical and experimental study of Galilei-transformed nonlinear wave groups

The Galilei transformation (GT) is a universal operation connecting the fixed and translated co-ordinates of a dynamical system. When considering exact wave envelope solutions of the nonlinear Schrödinger equation (NLSE) propagating with a relative Galilei speed (GS), the GT imposes a frequency shift to satisfy the symmetry. This limits the applicability of the GT to nonlinear dispersive waves. We show that the Galilei-transformed envelope solitons and Peregrine breathers generated in a wave tank clearly deviate from their respective pure NLSE hydrodynamics. The type of discrepancies depends on the sign of the GS while these can be still quantified by the modified NLSE or solving the Euler equations. Furthermore, Galilei-transformed envelope solitons with positive GSs exhibit self-modulation. With designated GS and steepness values, such solitons can be transformed to follow the exact dynamics of higher-order solitons, which under specific circumstances are responsible for the generation of supercontinua.