Observation of accelerating solitary wavepackets

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Observation of accelerating solitary wavepackets

Georgi Gary Rozenman, Lev Shemer, and Ady Arie

Phys. Rev. E 101, 050201(R) – Published 20 May 2020

Abstract

We study theoretically and observe experimentally the evolution of solitary surface gravity water wavepackets propagating in homogeneous and time-dependent flow created by a computer-controlled water pump, resulting in an effective linear potential. Unlike a potential free soliton, in this case the wavepacket envelope accelerates, while its phase is proportional to the cubic power of the position in the water tank. For increased wave steepness, we observe the emergence of asymmetry in the envelope, and hence it no longer retains its soliton shape. Furthermore, we study a case of ballistic dynamics of solitary surface gravity water wavepackets with initial nonzero momentum and demonstrate that their trajectory is similar to that of a projectile pulled by gravity. Nevertheless, their envelope shape is preserved during propagation, and the envelope phase is identical to that measured without an initial momentum.

Received 3 October 2019
Accepted 1 May 2020

DOI:https://doi.org/10.1103/PhysRevE.101.050201

Physics Subject Headings (PhySH)

Hydrodynamic waves Plasma waves Solitons Surface gravity waves

Nonlinear waves

Schroedinger equation

Fluid DynamicsInterdisciplinary PhysicsGeneral PhysicsNonlinear Dynamics

Authors & Affiliations

Georgi Gary Rozenman^1,2, Lev Shemer ³, and Ady Arie²

¹Raymond and Beverly Sackler School of Physics & Astronomy, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
²School of Electrical Engineering, Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
³School of Mechanical Engineering, Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

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Issue

Vol. 101, Iss. 5 — May 2020

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Images

Figure 1
Experimental setup for generating hyperbolic-secant surface gravity water wavepackets moving in a time-dependent homogeneous flow created by a water pump.
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Figure 2
Evolutions of hyperbolic-secant envelopes [right, obtained from the measurements by the Hilbert transform; left, simulated based on Eq. (1) for (a), (c), and (e) and based on (7) for (g), with $t_{0} = 0.33$ s, $η = 0.3$ in a frame of reference at moving velocity $c_{g}]$ . Simulations (left column) and measurements (right column) were performed at (a), (b) $a_{0} = 1.5$ mm, $ɛ = 0.03$ , (c), (d) $a_{0} = 4$ mm, $ɛ = 0.09$ , (e), (f) $a_{0} = 5.50$ mm, $ɛ = 0.13$ , and (g), (h) $a_{0} = 8.0$ mm, $ɛ = 0.18$ . (i) Experimental (point) and theoretical widths (solid lines), calculated using Eq. (6) for different values of steepness. While at low steepness ( $ɛ = 0.03$ ) the width is dominated by dispersion, for the higher steepness ( $ɛ = 0.13$ ), $t_{w} (x)$ exhibits soliton behavior, remaining constant throughout the entire measurement range. For the case of a highly nonlinear steepness ( $ɛ = 0.18$ ), the width increases slowly with $x$ .
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Figure 3
Global phase change and accumulated cubic phase for accelerating solitary water wavepackets. (a) The observed surface elevation of wavepackets in different locations along the water tank; $ζ^{So}$ (black line) and predicted by Eqs. (4) envelopes (red lines) for $k_{0} = 23 m^{- 1}, a_{0} = 5.50 mm$ ( $ɛ = 0.13$ ), $c_{g} = 0.33 m / s$ , and $t_{0} = 0.33 s$ . Quadratic fits were performed for 20 measuring points, determining $c_{g} = 1 / a_{1}$ together with $F \equiv - 0.69$ . (b) The phase offset of the accelerating solitary water wavepacket at the maximum of the mean value of every elevation. (c) The cubic phase term, obtained by subtracting the linear terms from the global phase change shown in (b); for both (b) and (c) the red markers are the phase values given by the Hilbert transform from the surface elevations (after the removal of the carrier phase) and black solid lines are analytical results. (d) The slope $c$ of the linear time-dependent function of the phase; each point is given by extracting $cos (ϕ)$ vs $t$ . (e) The extracted linear phase term $ϕ = F ξ τ$ in the moving frame.
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Figure 4
Evolutions of hyperbolic-secant envelopes with an initial momentum kick [right, obtained from the measurements by the Hilbert transform; left, simulated based on Eqs. (18) with $t_{0} = 0.66$ s], in a frame of reference at a moving velocity $c_{g}$ (a), (c), and (e). Simulations (left column) and measurements (right column) for (a), (b) $a_{0} = 10.5$ mm, $ɛ = 0.18, p_{0} = - 2$ rad/s, and (c), (d) $a_{0} = 6.2$ mm, $ɛ = 0.18, p_{0} = 2$ rad/s, and the effective force $F \equiv - 0.69$ . The global phase is measured by using the Hilbert transform for both (e) and (f). For both (e) and (f) the black markers are the measured global phase values (after the removal of the carrier phase) and black solid lines are analytical results given by Eq. (3). The red markers are the cubic phase terms, obtained by subtracting the linear terms from the global phase, i.e., an analytical subtraction of the linear terms $F τ ξ - 4 η^{2} ξ$ , and red solid lines are analytical results given by $φ_{A}^{(Cub)} (τ, ξ) = \frac{F^{2}}{3} ξ^{3}$ . (e) presents the extracted phases from (b) and (f) corresponding to the measurements in (f), i.e., (e) corresponds to $p_{0} = - 2$ rad/s and (f) corresponds to $p_{0} = 2$ rad/s.
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