Solitonic model of the condensate

We consider a spatially extended box-shaped wave field that consists of a plane wave (the condensate) in the middle and equals zero at the edges, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. Within the inverse scattering transform theory, the scattering data for this wave field is presented by the continuous spectrum of the nonlinear radiation and the soliton eigenvalues together with their norming constants; the number of solitons $N$ is proportional to the box width. We remove the continuous spectrum from the scattering data and find analytically the specific corrections to the soliton norming constants that arise due to the removal procedure. The corrected soliton parameters correspond to symmetric in space $N$-soliton solution, as we demonstrate analytically in the paper. Generating this solution numerically for $N$ up to 1024, we observe that, at large $N$, it converges asymptotically to the condensate, representing its solitonic model. Our methods can be generalized for other strongly nonlinear wave fields, as we demonstrate for the hyperbolic secant potential, building its solitonic model as well.

Figure 1

Qualitative illustration of the difference between the observed $x_n^{\left(O\right)}$ and IST $x_n^{\left(\text{IST}\right)}$ positions of soliton in presence of localized nonlinear radiation. The solid black line shows the soliton-and-radiation potential, in which the soliton has the observed position $x_n^{\left(O\right)}$ and the IST position $x_n^{\left(\text{IST}\right)}$, while the dashed red line indicates single soliton reconstructed at $x_n^{\left(\text{IST}\right)}$.

Figure 2

(a)–(c) Wave field of the $N$-soliton solution ($N$-SS) for $N=32$ (red), 64 (green), 128 (blue), 256 (purple), and 1024 (brown): (a) in linear scales, (b) in logarithmic horizontal scale, (c) in linear scales at the center of the box. Each N-SS curve is marked with the corresponding number of solitons. (d) Amplitude of the residual oscillations at the center $|{\psi }_{\left(N\right)}\left(0\right)-1|$ as a function of $N$ (blue points). The dashed black line indicates the fit $\propto N^{-1/2}$. The $N$-SS for all the figures are constructed from the scattering data (29) and (43) with $\mathrm{\Delta }L=\pi /4$ in Eq. (30). The inset in panel (b) shows dependency of $(1-L_s/L)$ (red circles) and $(1-L_c/L)$ (blue squares) on $N$ in double logarithmic scales, where $L_s$ and $L_c$ are defined according to Eqs. (50) and (51), respectively. The short-dashed red line indicates the fit $(1-L_s/L)\propto N^{-{\alpha }_s}$ with ${\alpha }_s\approx 0.37$, while the long-dashed blue line—the fit $(1-L_c/L)\propto N^{-{\alpha }_c}$ with ${\alpha }_c\approx 0.89$. The inset in panel (d) represents the same plot as in the main figure, but in double logarithmic scales.

Figure 3

Amplitude of the oscillations $\epsilon \left(x\right)$ at different sections of the box $x=\varkappa L/2$ versus the number of solitons $N$, for $\varkappa =0$ (red triangles), 0.2 (magenta squares), 0.4 (green dots), 0.6 (blue diamonds), and 0.8 (black crosses); note the double logarithmic scales. The dashed lines show the fits $\propto N^{-1/2}$. The amplitude $\epsilon \left(x\right)$ is calculated for the $N$-SS constructed from the scattering data (29) and (43) with $\mathrm{\Delta }L=\pi /4$ in Eq. (30).

Figure 4

Wave fields of $N$-SS for $N=32$ (red), 64 (green), and 128 (blue), constructed from (a) the box eigenvalues ${\lambda }_n$ (29) and the corrected DM norming constants ${\tilde{C}}_n$ (43), (b) the semiclassical box eigenvalues ${\lambda }_n^{\left(sc\right)}$ (31) and the corrected DM norming constants ${\tilde{C}}_n$, and (c) the box eigenvalues ${\lambda }_n$ and the noncorrected DM norming constants $C_n$ (52). Each N-SS curve is marked with the corresponding number of solitons. The parameter $\mathrm{\Delta }L$ in Eq. (30) is the same for all cases, $\mathrm{\Delta }L=\pi /4$. The panel (a) coincides with Fig. 2. The inset in panel (b) shows two 128-SS at the center of the box: constructed from the box eigenvalues ${\lambda }_n$ (dashed line) and from the semiclassical eigenvalues ${\lambda }_n^{\left(sc\right)}$ (solid line).

Figure 5

The $N$-SS constructed from the scattering data (29) and (43) for different values of the parameter $\mathrm{\Delta }L$ in Eq. (30). (a) The general form of the 32-SS, for $\mathrm{\Delta }L=0$ (solid blue), $\pi /4$ (dash-dotted red), and $\pi /2$ (dashed green). (b) The same solutions at the center of the box. (c) Amplitude of the oscillations at the center, $|{\psi }_{\left(N\right)}\left(0\right)-1|$, versus the number of solitons $N$ (points), in double logarithmic scales. The blue circles mark the case $\mathrm{\Delta }L=0$, the red triangles – $\pi /4$, and the green squares – $\pi /2$. The dashed lines indicate the fits $\propto N^{-1/2}$.

Figure 6

(a), (b) Amplitude of the oscillations (a) at the edges ${\epsilon }_{max}$ and (b) at the center ${\epsilon }_{min}$ versus the parameter $\mathrm{\Delta }L$ in Eq. (30), for 32-SS (solid red), 64-SS (dash-dotted green), and 128-SS (dashed blue). The short-dashed vertical lines indicate $\mathrm{\Delta }L=\pi /4$, while the long-dashed vertical lines mark the minimums of the 128-SS curves. The inset in panel (b) shows zoom near the minimum. (c) Ratio ${\mathcal{I}}_1^{\left(\text{nr}\right)}/{\mathcal{I}}_1$ between wave actions of the nonlinear radiation ${\mathcal{I}}_1^{\left(\text{nr}\right)}$ (A2) and the box potential ${\mathcal{I}}_1$, as a function of $L$ in semilogarithmic scales. The inset shows zoom near $N=64$; the short-dashed red and dashed black lines indicate $\mathrm{\Delta }L=0$ and $\pi /4$, while the long-dashed green line marks the local minimum of the presented curve. The $N$-SS for all the figures are constructed from the scattering data (29) and (43).

Figure 7

(a) Wave field of $N$-SS (solid red line) constructed from the scattering data (C4), (C9), for $N=32$ and $\mathrm{\Delta }L=0$ in Eq. (C5). The original sech-potential (C1) is shown with the dashed black line. The inset shows zoom near the symmetry point $x=0$. (b) Difference $\delta \left(x\right)=\mathrm{sech}(x/L)-{\psi }_{\left(N\right)}\left(x\right)$ between the sech potential and its solitonic model, for $N=32$ (red, reaches the largest maximum amplitude in the figure), 64 (green, the intermediate maximum amplitude), and 128 (blue, the smallest maximum amplitude); the parameter $\mathrm{\Delta }L=0$ corresponds to the maximum oscillations. (c) Zoom of the above panel demonstrating varying oscillation period.

Figure 8

(a) Ratio ${\mathcal{I}}_1^{\left(\text{nr}\right)}/{\mathcal{I}}_1$ between wave actions of the nonlinear radiation ${\mathcal{I}}_1^{\left(\text{nr}\right)}$ (A2) and the sech-potential ${\mathcal{I}}_1$ as a function of $L$ in semilogarithmic scales. The inset shows zoom near $N=64$; the dashed red lines indicate $\mathrm{\Delta }L=0$. (b) Wave field of $N$-SS (solid red line) constructed from the eigenvalues (C4) and the noncorrected IST norming constants (C6) using transition to the DM formalism via Eq. (21), for $N=32$ and $\mathrm{\Delta }L=0$ in Eq. (C5). The original sech-potential (C1) is shown with the dashed black line. The inset shows the nonsymmetry of the difference $\delta \left(x\right)=\mathrm{sech}(x/L)-{\psi }_{\left(N\right)}\left(x\right)$ with respect to the mirror transformation $x\to -x$.