Spatiotemporal properties of solitons excited on the surface of shallow water in a hydrodynamic resonator

Abstract

We have investigated the spatiotemporal properties of solitons generated on the shallow water surface over a background of a large-scale mode in a hydrodynamic resonator when it is forced near the second frequency mode. We have used the space-time diagrams to highlight the spatiotemporal dynamics of nonlinear fields for two solitons colliding in a resonator and compared them to those of solitons occurring in an unbounded system. A state diagram of experimentally observed modes for different values of the excitation parameters has been obtained. In particular, we have evidenced period doubling and the multistability of nonlinear waves excited in the resonator. For a theoretical description of these experimental results, we have developed a phenomenological model, which leads to amplitude and phase equations of a soliton propagating over the background of a harmonic wave. To reproduce experimental results on the multistability, we have supplemented our analysis with a numerical simulation of a modified system of Boussinesq equations for shallow water, taking into account the dissipation effect

Formation dynamics of sand bedforms under solitons and bound states of solitons in a wave flume used in resonant mode

“…For values of the amplitude of horizontal displacement of the oscillating paddle averaged over depth a h lower than 2 cm, only standing harmonic waves are generated in the flume. For values of a h greater than 2 cm, pulses propagating from one end of the channel to the other end are excited on the background of the standing harmonic wave [16]. The generation of solitons in the flume results from the excitation of higher harmonics.…”

“…In the case of the generation of one pulse propagating in each direction of the flume on the time period of the flow, the shape of this pulse is well represented by the theoretical sech-squared profile η S1 of soliton [16], where V S1 is the propagation velocity of soliton: V S1 = V 0 (1 + A S1 /2d). The profile of the soliton (Eq.…”

“…This method is used in the present study and contra-propagative solitary waves are excited on the background of a standing harmonic wave. The spatiotemporal properties of these solitary waves were investigated in [16]. From one to four pulses propagating in each direction of the flume may be generated on the time period of the flow, depending on the frequency and the amplitude of horizontal displacement of the oscillating paddle.…”

“…From one to four pulses propagating in each direction of the flume may be generated on the time period of the flow, depending on the frequency and the amplitude of horizontal displacement of the oscillating paddle. These pulses were identified as solitons (one pulse) and bound states of solitons (several pulses) [16]. To the authors' knowledge, the only study on ripple formation induced by solitons was carried out by Marin et al [17].…”

Frequency divider by underdamped Josephson transmission line

“…Considered effect is not restricted only to the sineGordon equation but generally speaking it can characterize any nonlinear system which permits the existence of moving localized solutions, particularly, the resent experimental observation of the similar stationary regime as a motion forth and back of solitary shallow water waves could be quoted as an example [26].…”