Bi-Solitons on the Surface of a Deep Fluid: An Inverse Scattering Transform Perspective Based on Perturbation Theory

We investigate theoretically and numerically the dynamics of long-living oscillating coherent structures—bi-solitons—in the exact and approximate models for waves on the free surface of deep water. We generate numerically the bi-solitons of the approximate Dyachenko-Zakharov equation and fully nonlinear equations propagating without significant loss of energy for hundreds of the structure oscillation periods, which is hundreds of thousands of characteristic periods of the surface waves. To elucidate the long-living bi-soliton complex nature we apply an analytical-numerical approach based on the perturbation theory and the inverse scattering transform (IST) for the one-dimensional focusing nonlinear Schrödinger equation model. We observe a periodic energy and momentum exchange between solitons and continuous spectrum radiation resulting in repetitive oscillations of the coherent structure. We find that soliton eigenvalues oscillate on stable trajectories experiencing a slight drift on a scale of hundreds of the structure oscillation periods so that the eigenvalue dynamics is in good agreement with predictions of the IST perturbation theory. Based on the obtained results, we conclude that the IST perturbation theory justifies the existence of the long-living bi-solitons on the surface of deep water that emerge as a result of a balance between their dominant solitonic part and a portion of continuous spectrum radiation.

Figure 1

Nonlinear behavior of the DZE bi-soliton. (a) Spatiotemporal diagram of the wave field envelope $|q_{(\mathrm{DZE})}|$ with characteristic oscillation period $T\approx 24.5$. (b) Stable trajectories of soliton eigenvalues obtained as a union of the computed time series of $\lambda (t_j)$. Blue and red lines correspond to two complete cycles of the bi-soliton oscillations separated by $200T$.

Figure 2

Further IST analysis of the DZE bi-soliton. (a) Spatiotemporal dynamics of the two-soliton model (13). (b) Time evolution of $I$ for each component of the IST spectrum: red and blue lines correspond to ${\lambda }_1$ and ${\lambda }_2$, respectively, while the green line shows the impact of continuous spectrum radiation. Panels (c) and (d) show the DZE wave field (red solid lines) at minimum (c) and maximum (d) amplitude, together with the model (13) (black dashed lines). Insets in (c) and (d) show real and imaginary parts of the wave fields and also indicate the eigenvalue locations by black squares and dots.

Figure 3

Changes of discrete eigenvalues $\mathrm{\Delta }{\lambda }_j=\mathrm{\Delta }{\xi }_j+i\mathrm{\Delta }{\chi }_j$ at each time step $t_j$ within one oscillation period $T$ of the DZE bi-soliton. Panels (a) and (b) correspond to $\mathrm{\Delta }\xi$ and $\mathrm{\Delta }\chi$ respectively. Blue dots show the direct numerical computation of soliton eigenvalues changes for each subsequent wavefield $q(t_j)$ and $q(t_{j+1})$. Red circles correspond to predictions of the perturbation theory provided by Eq. (12).

Figure 4

Nonlinear behavior of the RVE bi-soliton. (a),(b) Surface profiles $\eta (x)$ at minimum and maximum amplitude, respectively. (c) Stable trajectories of soliton eigenvalues with $T\approx 30.0$. Blue and red lines correspond to two complete cycles of the bi-soliton oscillations separated by $200T$.