  The Galilean transformation is a universal operation connecting the
coordinates of a dynamical system, which move relative to each other with a
constant speed. In the context of exact solutions of the universal nonlinear
Schr\"odinger equation (NLSE), inducing a Galilean velocity (GV) to the pulse
involves a frequency shift to satisfy the symmetry of the wave equation. As
such, the Galilean transformation has been deemed to be not applicable to wave
groups in nonlinear dispersive media. In this paper, we demonstrate that in a
wave tank generated Galilean transformed envelope and Peregrine solitons show
clear variations from their respective pure dynamics on the water surface. The
type of deviations depends on the sign of the GV and can be captured by the
modified NLSE or the Euler equations. Moreover, we show that positive
Galilean-translated envelope soliton pulses exhibit self-modulation. While
designated GS and wave steepness values expedite multi-soliton dynamics, the
strong focusing of such higher-order coherent waves inevitably lead to the
generation of supercontinua as a result of soliton fission. We anticipate that
kindred experimental and numerical studies might be implemented in other
dispersive wave guides governed by nonlinearity.
