Effective dispersion in the focusing nonlinear Schrödinger equation

For waves described by the focusing nonlinear Schrödinger equation (FNLS), we present an effective dispersion relation (EDR) that arises dynamically from the interplay between the linear dispersion and the nonlinearity. The form of this EDR is parabolic for a robust family of “generic” FNLS waves and equals the linear dispersion relation less twice the total wave action of the wave in question multiplied by the square of the nonlinearity parameter. We derive an approximate form of this EDR explicitly in the limit of small nonlinearity and confirm it using the wave-number-frequency spectral (WFS) analysis, a Fourier-transform based method used for determining dispersion relations of observed waves. We also show that it extends to the FNLS the universal EDR formula for the defocusing Majda-McLaughlin-Tabak (MMT) model of weak turbulence. In addition, unexpectedly, even for some spatially periodic versions of multisolitonlike waves, the EDR is still a downward shifted linear-dispersion parabola, but the shift does not have a clear relation to the total wave action. Using WFS analysis and heuristic derivations, we present examples of parabolic and nonparabolic EDRs for FNLS waves and also waves for which no EDR exists.

Figure 1

FNLS wave generated from an initial perturbed standing wave with amplitude $A=72$, using ${\alpha }^2=1/16$, leading to approximately 50 unstable modes. The computational domain uses $L=2\pi$ with $N_x=2^{10}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=6\pi$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. To avoid diagnosing initial transient, the wave is depicted from time $t=\pi /2$ to $t=\pi$. (b) The initial spatial wave profile of a perturbed standing wave in Eq. (12) at $t=0$ (red), and an example of the evolved wave profile at a $t=\pi$ (blue). (c) The wave-number dependence of the wave's Fourier amplitude, $|\widehat{q}(k,t)|$ defined in Eq. (3), in logarithmic scale at $t=0$ (red) and at $t=\pi$ (blue). According to Eq. (12), the initial wave-number dependence of the wave consists of the standing wave at $k=0$ and plane-wave perturbations within the $k$-interval of width $\sqrt{2}\left|A\right|\alpha$. [cf. Eq. (11)]. Note that the initial random perturbation is masked in the logarithmic scale. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color axis. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The resulting EDR (blue) is compared with the theoretically predicted parabolic EDR in Eq. (9) (red). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 24 windows of width $T_{\text{win}}=\pi /4$.

Figure 2

FNLS wave generated from an initial perturbed standing wave with amplitude $A=72$, using ${\alpha }^2=256$, leading to approximately 3200 unstable modes. The computational domain uses $L=2\pi$ with $N_x=2^{16}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=3\pi /2^{11}$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. To avoid diagnosing initial transient, the wave is depicted from time $t=\pi /2^{16}$ to $t=\pi /2^{15}$. Note that in this figure, the spatial domain has been restricted to $x\in [-\pi ,-7\pi /8]$ to more clearly indicate the increased density of wave crests, in comparison with Fig. 1. (b) The initial spatial wave profile of a perturbed standing wave in Eq. (12) at $t=0$ (red), and an example of the evolved wave profile at a $t=\pi /2^{12}$ (blue). (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), in logarithmic scale at $t=0$ (red) and at $t=\pi /2^{12}$ (blue). According to Eq. (12), the initial wave-number dependence of the wave consists of the standing wave at $k=0$ and plane-wave perturbations within the $k$-interval of width $\sqrt{2}\left|A\right|\alpha$. [cf. Eq. (11).] Note that the initial random perturbation is masked in the logarithmic scale. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color axis. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The resulting EDR (blue) is compared with the theoretically predicted parabolic EDR in Eq. (9) (red). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 24 windows of width $T_{\text{win}}=\pi /2^{14}$.

Figure 3

FNLS wave generated from an initial condition of spatial “white noise.” The nonlinearity parameter is ${\alpha }^2=1/16$, the computational domain uses $L=2\pi , N_x=2^{12}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=\pi /4$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. The wave is depicted from time $t=\pi /8$ to $t=\pi /4$. (b) The initial spatial wave profile of “white noise” at $t=0$, generated as described in Appendix pp5. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. Note that neither the spatial profile or Fourier spectrum of the evolved wave are shown since they appear similar to the initial profile (in contrast to Figs. 1 and 2). (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color axis. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The theoretically predicted parabolic EDR in Eq. (9) is nearly indistinguishable from the observed EDR for this case. Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /16$.

Figure 4

Single “soliton” wave of Eq. (14), with parameters: $A=32, V=2, \delta =0, \psi =0$, and $\alpha =1$. The computational domain was $L=2\pi$ with $N_x=2^9, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and final time $T=\pi /2$. (a) Color contour plot of the wave where the color indicates the magnitude of the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted using a logarithmic color scale. (c) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (b). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. A least-square fit line to this data is $\omega =-4k-1028$, which corresponds to the expected EDR from Eq. (18). Note that the PSD along the faint parabola visible in panel (b) is four orders of magnitude smaller that that along the line near $k=0$. Panels (b) and (c) were computed using the argument in Eq. (4) over a single time window of length $T_{\text{win}}=T$.

Figure 5

Multi-“breather”-type wave generated from Eq. (19) with $M=4, A=4$, and $\alpha =1$. The computational domain was $L=2\pi$, with $N_x=2^{12}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=\pi /2$. (a) Surface plot of the wave, displayed here at an angle to more clearly highlight the behavior of the breather, with color indicating the magnitude of the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted using a logarithmic color scale. (c) Peak-frequency $\omega$ values for each wave number $k$ of the PSD in panel (b). Here, all the peaks of the PSD within 90% of the global peak value at every $k$ were marked with a blue dot. These peaks (blue) are compared with the relevant theoretically predicted parabolic EDR in Eq. (9) (red). Panels (b) and (c) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /8$.

Figure 6

FNLS wave consisting of five distinct pulses, with parameters $(A,V,\delta )=(20,-4,-\pi /2), (24,5,-\pi /4)$, (16, 16, 0), $(32,-2,\pi /4), (28,-8,\pi /2)$, and with $\alpha =1$. The computational domain was $L=2\pi$ with $N_x=2^{11}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=\pi$. (a) Color contour plot of the solution with color indicating the magnitude of the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (c) Peak-frequency $\omega$ values for each wave number $k$ of the PSD in panel (b). In contrast to previous examples, here we have used a blue dot to mark each peak of the PSD which is at least 10 times as high as its nearest local minimum value at every $k$. This is done so that all five pulses are diagnosed. In this example, the pulses, respectively, each contribute a line to the spectrum, where fitting these lines to $\omega =-(A^2+V^2)-2Vk$ result in $(A,V)=(20.06,-3.92), (24.02,5.004), (16.42,16.26), (31.98,-2), (27.99,-8.005)$, respectively. Panels (b) and (c) were computed using the argument Eq. (4) and employed averaging over 8 windows of width $T_{\text{win}}=\pi /8$.

Figure 7

FNLS wave consisting of 15 pulses, which are evenly distributed and well-separated across the spatial domain, as well as widely distributed in wave-number space. In particular, we sum 15 pulses of the form Eq. (14), with parameters $A=32, \psi =0, \alpha =\sqrt{2}A, {\delta }_j=\pm 13Lj/210, V_{\ell }=\pm 32\ell$. Pulse velocities were sorted randomly so $V_j$ need not correspond to ${\delta }_j$. The computational domain was $L=8\pi$ with $N_x=2^{12}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=\pi$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The initial spatial wave profile at $t=0$. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The resulting EDR (blue) is compared with the theoretically predicted parabolic EDR in Eq. (9) (red). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /4$.

Figure 8

FNLS wave consisting of 15 pulses, which are evenly distributed and well-separated across the spatial domain, but now narrowly distributed in wave-number space. In particular, we sum 15 pulses of the form Eq. (14), with parameters $A=32, \psi =0, \alpha =\sqrt{2}A, {\delta }_j=\pm 13Lj/210, V_{\ell }=\pm 2\ell$. Pulse velocities were sorted randomly so $V_j$ need not correspond to ${\delta }_j$. The computational domain was $L=8\pi$ with $N_x=2^{12}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$, and a final time $T=\pi$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. Notice the difference in timescales between panel (a) of this figure and panel (a) of Fig. 7. (b) The initial spatial wave profile at $t=0$. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The resulting EDR (blue) is compared with the theoretically predicted parabolic EDR in Eq. (9) (red). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /4$.

Figure 9

FNLS wave consisting of 15 pulses, which are evenly distributed and narrowly separated across the spatial domain, but widely distributed in wave-number space. In particular, we sum 15 pulses of the form Eq. (14), with parameters $A=32, \psi =0, \alpha =\sqrt{2}A, {\delta }_j=\pm 13Lj/210, V_{\ell }=\pm 256\ell$. Pulse velocities were sorted randomly so $V_j$ need not correspond to ${\delta }_j$. The computational domain was $L=\pi /4$ with $N_x=2^{10}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$ and a final time $T=\pi /16$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The initial spatial wave profile at $t=0$. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. Individual lines contained in the blue data were fit to equations of the type $\omega \left(k\right)={\omega }_{j,0}-2{\tilde{V}}_{j}k$ using least squares. A least-squares fit to the tangency points $(k_j=-{\tilde{V}}_j,{\omega }_j={\omega }_{j,0}+2{\tilde{V}}_j^2)$, as discussed in the text in Sec. 3b3, yields the parabolic EDR fit $\omega =k^2-5521$, which is compared with the theoretically predicted parabolic EDR in Eq. (9) $\omega =k^2-4857$ (shown in red). These two curves are indistinguishable on the scale of the plots in panels (d) and (e). See discussion in Sec. 3b3 for additional comments about panel (e). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /64$.

Figure 10

FNLS wave consisting of 15 pulses, which are evenly distributed and narrowly separated across the spatial domain, and also narrowly distributed in wave-number space. In particular, we sum 15 pulses of the form Eq. (14), with parameters $A=32, \psi =0, \alpha =\sqrt{2}A, {\delta }_j=\pm 13Lj/210, V_{\ell }=\pm 32\ell$. Pulse velocities were sorted randomly so $V_j$ need not correspond to ${\delta }_j$. The computational domain was $L=\pi /4$ with $N_x=2^{10}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$ and a final time $T=\pi /16$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The initial spatial wave profile at $t=0$. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. A least-squares fit to the blue data yields $\omega =k^2-5227$, which is compared with the theoretically predicted parabolic EDR in Eq. (9) $\omega =k^2-4762$ (red). These two curves are indistinguishable on the scale of the plots in panels (d) and (e). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /64$.

Figure 11

FNLS wave generated from an initial condition of 15 equal size, identically spaced solitonlike pulses plus one tall, narrow solitonlike pulse. The parameters of the 15 pulses are the same as those used to compute Fig. 9. The additional pulse has the form given in Eq. (14), with parameters $A=128, \psi =0, \alpha =32\sqrt{2}, \delta =\pm 13\pi /1680, V=128$. The computational domain was $L=\pi /4$ with $N_x=2^{10}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$ and a final time $T=\pi /16$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The initial spatial wave profile at $t=0$. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The resulting EDR (blue) is compared with the theoretically predicted parabolic EDR in Eq. (9) (red). The EDR predicted by Eq. (9) is $\omega =k^2-6189$. The least-squares parabolic fit for small $k$, not including the line segments, gives the EDR of $\omega =k^2-8205$. The parabolic fit for large $k$ [of the periodically repeated parabola pieces on either side of panels (d) and (e)] gives EDR of $\omega =k^2-8740$. The fit for the dominant straight line, using all three line segments in the blue data, gives $A=147$ and $V=128$ in Eq. (18). Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /64$.

Figure 12

FNLS wave generated from an initial condition of a (smoothed) periodic spatial random walk, as described in Appendix pp5. The nonlinearity parameter was $\alpha =8$ and the computational domain was $L=2\pi$ with $N_x=2^{12}, \mathrm{\Delta }t=\mathrm{\Delta }{x}^2/2\pi$ and a final time $T=\pi$. (a) Three-dimensional space-time profile of the wave at constant-time slices, where color indicates the wave amplitude $\left|q\right(x,t\left)\right|$. (b) The initial spatial wave profile at $t=0$. (c) The wave-number dependence of the wave's Fourier amplitude $|\widehat{q}(k,t)|$, defined in Eq. (3), at $t=0$. (d) The spatiotemporal PSD of the wave, i.e., the argument on the right-hand side of Eq. (2), used in the WFS analysis, plotted with a logarithmic color scale. (e) EDR obtained according to the WFS analysis by finding the peak frequency $\omega$ values for each wave number $k$ of the PSD in panel (d). Here, all the peaks of the PSD within 1% of the global peak value at every $k$ were marked with a blue dot. The resulting EDR (blue) is compared with the theoretically predicted parabolic EDR in Eq. (9) (red). The EDR predicted by Eq. (9) is $\omega =k^2-489$; the fit of the parabolic part of the blue data for $\left|k\right|>60$ gives EDR of $k^2-487$. Panels (d) and (e) were computed using the argument in Eq. (4) and employed averaging over 4 windows of width $T_{\text{win}}=\pi /4$.