Resonant radiation shed by dispersive shock waves

Matteo Conforti, Fabio Baronio, and Stefano Trillo

Phys. Rev. A 89, 013807 – Published 9 January 2014

Abstract

We analyze resonant radiation emitted by dispersive shock waves owing to higher-order dispersive corrections of the leading term in the defocusing nonlinear Schrödinger equation. We give criteria for calculating the radiated frequency based on an analytical estimate of the shock velocity and reveal a diversity of scenarios controllable via the corrections, ranging from the radiation-induced transition of the dispersive shock into a classical-type shock to the qualitative modification of the underlying gradient catastrophe or the competition between different breaking mechanisms.

Received 5 May 2013

DOI:https://doi.org/10.1103/PhysRevA.89.013807

Authors & Affiliations

Matteo Conforti¹, Fabio Baronio¹, and Stefano Trillo²

¹Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy
²Dipartimento di Ingegneria, Università di Ferrara, Via Saragat 1, 44122 Ferrara, Italy

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Vol. 89, Iss. 1 — January 2014

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Images

Figure 1
Radiating DSW from dNLSE (2) with $ɛ = 0.03$ , input step $ρ_{l}, ρ_{r} = 1, 0.5$ , and 3-HOD $β_{3} = - 0.35$ : (a) Color-level plot of density $ρ (t, z)$ (the dashed line gives the DSW leading edge velocity $V_{l}$ ); (b) corresponding spectral evolution; (c) snapshots at $z = 2$ of $ρ, u$ of unperturbed case $β_{3} = 0$ (in dashed red is the corresponding classical SW); (d) comparison of output spectra with (thick solid blue line) and without (dashed red) 3-HOD (input thin dashed line). Inset shows graphical solution of Eq. (3).
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Figure 2
As in Fig. 1 with larger 3-HOD $β_{3} = - 1$ . The dashed vertical line in panels (b) and (d) gives $ω_{R R}$ from Eq. (9) with velocity $V_{s} = V_{c}$ [corresponding to dashed line in panel (a)]. Here $V_{c} = 0.69$ and $u_{r} = - 0.543$ are the parameters of the classical SW [shown by dashed red line in panel (c)] from Eq. (8), see also Fig. 6 in Appendix pp1. Solid line in panel (a) indicates the velocity $V_{R R}$ of the RR.
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Figure 3
Radiating DSW from Gaussian pulses with small background $ν^{2} = 0.09$ : (a) $β_{3} = - 0.35$ . (b) $β_{3} = - 0.6$ . (c) Parameters determining the leading-edge velocity $V_{s} = \sqrt{ρ_{\min}} + u_{l} = \sqrt{10^{- 4}} + 0.76 = 0.77$ [dashed line in panel (a)]; snapshot at $z = 3$ . (d) Output spectra (thick blue line, $β_{3} = - 0.35$ ; thin red line, $β_{3} = - 0.6$ ). The dashed lines give $ɛ ω_{R R}$ calculated from Eq. (9). Here $ɛ = 0.03$ and blue curves in panels (a) and (b) depict output snapshots.
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Figure 4
(a), (b) Temporal and spectral evolution of a Gaussian pulse without background emitting RR, for $β_{3} = - 0.35$ . (c), (d) Mixed hyperbolic-elliptic type of wave breaking for large 3-HOD $β_{3} = - 2$ , $ν^{2} = 0.01$ . Panel (d) reports a snapshot of $ρ$ and $u$ near the gradient catastrophe occurring at $z \approx 0.9$ . Here $ɛ = 0.03$ .
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Figure 5
RR ruled by 4-HOD ( $β_{3} = 0$ , $β_{4} = - 0.5$ ). (a) Dispersion curves from Eq. (3) for positive ( $V_{s}^{+}$ , solid blue line) and negative ( $V_{s}^{-}$ , dashed green line) velocity crossing $k_{n l}$ (horizontal red line) in $\pm ω_{R R 1, R R 2}$ ; vertical stripes indicate MI gain bands. (b) Spectral evolution. (c) Temporal dynamics (blue curve depicts output snapshot). (d) Output spectrum [central red dashed line is peak MI gain, blue and green dashed lines correspond to RR frequencies obtained from panel (a)]. Here $ɛ = 0.05$ , input Gaussian pulse with background $ν^{2} = 0.09$ .
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Figure 6
Right velocity $u_{r}$ and velocity $V_{c}$ of a weak SW solution for fixed left values $ρ_{l} = 1$ , $u_{l} = 0$ and variable right density $ρ_{r}$ (i.e., variable amplitude of the jump), obtained from the RH conditions. We compare the case $β_{3} = 0$ (solid curves) with $β_{3} = - 1$ (open circles). $V_{c}$ satisfies the entropy criterion $V_{l}^{+} > V_{c} > V_{r}^{+}$ , where $V_{l}^{+}$ and $V_{r}^{+}$ (dashed curves) stand for the eigenvelocity $V^{+}$ calculated on the left and on the right of the jump for $β_{3} \neq 0$ .
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