Bright traveling breathers in media with long-range nonconvex dispersion

The existence and properties of envelope solitary waves on a periodic traveling-wave background, called traveling breathers, are investigated numerically in representative nonlocal dispersive media. Using a fixed-point computational scheme, a space-time boundary-value problem for bright traveling breather solutions is solved for the weakly nonlinear Benjamin-Bona-Mahony equation, a nonlocal, regularized shallow water wave model, and the strongly nonlinear conduit equation, a nonlocal model of viscous core-annular flows. Curves of unit-mean traveling breather solutions within a three-dimensional parameter space are obtained. Resonance due to nonconvex, rational linear dispersion leads to a nonzero oscillatory background upon which traveling breathers propagate. These solutions exhibit a topological phase jump and so act as defects within the periodic background. For small amplitudes, traveling breathers are well approximated by bright soliton solutions of the nonlinear Schrödinger equation with a negligibly small periodic background. These solutions are numerically continued into the large-amplitude regime as elevation defects on cnoidal or cnoidal-like periodic traveling-wave backgrounds. This study of bright traveling breathers provides insight into systems with nonconvex, nonlocal dispersion that occur in a variety of media such as internal oceanic waves subject to rotation and short, intense optical pulses.

Figure 1

(a) Traveling breather in the weakly nonlinear regime (blue solid line) of family 1 with $T\approx 18.180$ and $c\approx -0.031$ on a spatial domain with $L=400$. The numerical envelope (black dashed line) is compared to the NLS bright soliton envelope (red solid line). (b) Evaluation of the traveling breather at $\chi =0$ over one period (blue solid line) compared with the leading-order NLS approximation (16) (red solid line).

Figure 2

Traveling breather frequency shift from computation (black dotted line) and predicted (red solid line) traveling breathers.

Figure 3

Family 1 of traveling breather solutions of the BBM equation with (a) $(T,c)\approx 16.917,-0.036$, (b) $(T,c)\approx 11.969,-0.076$, (c) $(T,c)\approx 11.942,-0.079$, (d) $(T,c)\approx 11.906,-0.082$, (e) $(T,c)\approx 11.316,-0.099$, and (f) $(T,c)\approx 10.682,-0.129$.

Figure 4

Variation of the (a) periodic background amplitude (23), (b) breather amplitude (24), (c) periodic background wave number, (d) phase shift $\sigma$ (26), and (e) time period $T$ with breather velocity $c$ for family 1 of BBM breathers.

Figure 5

Family 5 of BBM traveling breathers with (a) $(\tilde{a},c,T)\approx 0.162,-0.110,10.142$, (b) $(\tilde{a},c,T)\approx 0.306,-0.124,9.708$, and (c) $(\tilde{a},c,T)\approx 0.623,-0.133,9.486$.

Figure 6

(a) Projections in amplitude of the velocity ($c$)-frequency ($\mathrm{\Omega }$) relation of the five traveling breather branches. (b) Computed unit-mean traveling breathers from the five families (closed circles) and the weakly nonlinear prediction $T=\mathcal{T}$ (28) (colored surface).

Figure 7

Family 1 of traveling breathers of the conduit equation with (a) $(T,c)\approx 7.172,-0.101$, (b) $(T,c)\approx 7.093,-0.106$, (c) $(T,c)\approx 7.079,-0.107$, (d) $(T,c)\approx 7.073,-0.109$, (e) $(T,c)\approx 7.072,-0.111$, and (f) $(T,c)\approx 7.116,-0.113$.

Figure 8

Variation of the (a) periodic background amplitude (23), (b) breather amplitude (24), (c) periodic background wave number, (d) phase shift $\sigma$ (26), and (e) time period $T$ with breather velocity $c$ for family 1 of conduit breathers.

Figure 9

Computed unit-mean traveling breathers from the families (closed circles) and the weakly nonlinear prediction $T=\mathcal{T}$ (28) (colored surface).

Figure 10

Family 2 of conduit weakly nonlinear wave packets near the zero-dispersion line, characterized by (a) carrier wave number $\tilde{k}\approx 2$ and $(\tilde{a},c,T)\approx 0.166,-0.240,4.907$ and (b) $\tilde{k}\approx 1.8$ and $(\tilde{a},c,T)\approx 0.210,-0.243,4.886$.

Figure 11

Evolution of a perturbed strongly nonlinear conduit traveling breather from family 1. (a) Contour plot describing the spatiotemporal structure of the perturbed waveform when evolved to $t=150T\approx 1067$. (b) Time snapshots of the perturbed (red solid line) and unperturbed waveforms (blue dashed line) at the end time, pointing to the coherence of the breather.

Figure 12

Continuation path (black dots) marked in the $c\text{−}\mathrm{\Omega }$ phase plane; in the weakly nonlinear regime the path approaches the linear dispersion curve [magenta curve, ${\mathrm{\Omega }}_0\left(c_0\right)$] as $c\to -3$. The inset shows slices at $\tau =0$ of (a) the computed weakly nonlinear mKdV breather and (b) a strongly nonlinear breather.