Integrable perturbation theory for dark solitons of the defocusing nonlinear Schrödinger equation

TL;DR

This work develops a rigorous integrable perturbation framework for the defocusing NLS on a nonzero background by proving the completeness of squared eigenfunctions with careful treatment of branch-point singularities. It derives a robust 1-soliton closure relation, enabling correct slow-time evolution for all dark-soliton parameters and a quantitative description of the radiation shelf formed by perturbations. A key result is that the first-order correction contains a pole at the branch points, which explains shelf formation and allows explicit predictions of shelf height, velocity, and phase gradients, all corroborated by numerical simulations. The approach reconciles discrepancies from previous IST-based perturbation theories, clarifies the role of shelf dynamics, and outlines extensions to higher-order effects and other integrable systems on nonzero backgrounds.

Abstract

The goal of this work is to revisit the squared-eigenfunctions-based perturbation theory of the scalar defocusing nonlinear Schrödinger equation on a nonzero background, and develop it so that it can correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the sides of the soliton. Proof of the completeness of the squared eigenfunctions is provided. Our closure relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions of the closure relation. Using the one-soliton closure relation and its discrete eigenmodes, the slow-time evolution equations of the soliton parameters are determined. Moreover, the first-order correction integral to the dark soliton is shown to contain a pole due to singularities of the scattering data at the branch points. Analysis of this first-order correction integral leads to predictions for the height, velocity and phase gradient of the shelves, as well as a formula for the slow time evolution of the soliton's phase, which in turn allows one to determine the slow-time dependence of the soliton center. All the results are corroborated by direct numerical simulations, and compared with earlier results.

Figures (14)

• Figure 1: Top Row: Comparison of the modulus (left) and phase (right) of the predicted (dashed red) and numerical (solid black) solutions for a dark soliton under the influence of the linear damping perturbation $F[q]=-iq$ with $\epsilon=0.02$ at time $t=10$. The initial soliton parameters are $q_{0}(0)=2$, $k_{1}(0)=-1/2$, $x_{1}(0)=\sigma_{1}(0)=0$. Bottom Row: The same at $t=20$, in which a slight deviation in our prediction for the phase from the numerics is already visible.
• Figure 2: The predicted evolution of the parameters $\sigma_{1}$ (left) and $x_{1}$ (right) with time (dashed red), compared with numerical measurements (dotted black). The parameters are the same as in Figure \ref{['f:LD_Core']}. Note that both predictions are accurate for a moderate time interval, but start to deviate as time increases. The blue dashed line in the right panel shows the prediction of the method of FrantzPRSA2011 for the soliton center, which is seen to be more accurate for long time. A similar deviation in the prediction for the soliton phase for larger times can be seen in Fig. 7 in FrantzPRSA2011.
• Figure 3: A space-time plot showing the development of a raised shelf around the soliton under linear damping. The blue dashed lines denote the boundaries of the shelf region, and the red dashed line corresponds to the predicted path of the soliton core, accounting for the slow evolution of the velocity. For visualization purposes, the evolution of the background has been artificially removed. In the right panel, a comparison of the predicted decrease in the background $q_{0}$ (top) and trough amplitude $|k_{1}|$ (bottom) with numerical measurements are shown up to $t=1/\epsilon=50$. The initial parameters are the same as in Figure \ref{['f:LD_Core']}.
• Figure 4: The modulus (left) and phase (right) of a numerical simulation of a dark soliton under the influence of the linear damping perturbation $F[q]=-iq$ with $\epsilon=0.02$ at $t=10$. The blue dashed lines denote the predicted boundaries of the shelf region. The red (resp., green) dashed lines represent the predictions for the shelf height and phase gradient on the right (resp., on the left) of the soliton. The initial parameters are the same as in Figure \ref{['f:LD_Core']}.
• Figure 5: Top Row: Comparison of the modulus (left) and phase (right) of the predicted (dashed red) and numerical (solid black) solutions for a dark soliton under the influence of the nonlinear damping perturbation $F[q]=-i|q|^{2}q$ with $\epsilon=0.02$ at time $t=5$. The initial soliton parameters are $q_{0}(0)=2$, $k_{1}(0)=-1/2$, $x_{1}(0)=\sigma_{1}(0)=0$. Bottom Row: The same at $t=10$, from which a deviation in the phase can be seen.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3