Varied Branches of Nondegenerate Vector Solitons

TL;DR

This work uncovers a rich branch structure for nondegenerate vector solitons in multi-component BECs, starting with the three-component Manakov model in which a dark and two bright components yield four distinct branches at fixed velocity, with two positive-mass and two negative-mass solutions that form two disjoint energy-velocity loops. It provides explicit nondegenerate dark-bright-bright soliton solutions, derives the relations between bright-component particle numbers and soliton widths, and shows that the maximum velocity is governed by the larger bright component $\max\{N_{B2},N_{B3}\}$. A Bogoliubov–de Gennes stability analysis confirms linear stability for all branches, and dynamics under weak linear forcing reveal mass-sign–dependent motion and branch-specific fission at the maximal velocity. The framework generalizes to an $N$-component BEC, predicting $2^{N-1}$ branches split evenly into $2^{N-2}$ positive- and $2^{N-2}$ negative-mass solitons, yielding $2^{N-2}$ disjoint dispersion loops and highlighting the intricate dispersion landscapes possible in nondegenerate multicomponent vector solitons.

Abstract

Our study on nondegenerate dark-bright-bright solitons in a three-component Manakov model with repulsive interactions reveals the existence of diverse branches of nondegenerate vector solitons. For fixed bright component particle numbers and a given soliton velocity, the nondegenerate dark-bright-bright solitons exhibit four distinct branches with different density profiles and phase distributions, comprising two positive mass branches and two negative mass branches. The energy-velocity dispersion relation of each pair of positive- and one negative-mass branches form a closed loop, resulting in two mutually independent loops for the soliton's overall dispersion. All soliton branches share a common maximal velocity, which is determined by the larger bright soliton particle number. Linear stability analysis shows that all these branches are stable against weak perturbations. Extending to an $N$-component Manakov system, the nondegenerate solitons have $2^{N-1}$ distinct branches, of which $2^{N-2}$ branches solitons is positive mass and $2^{N-2}$ branches solitons is negative mass. Each pair of positive- and negative-mass branches form a closed dispersion relation loop, so that the vector solitons have $2^{N-2}$ disjoint loops. These results uncover the rich branches and interesting dispersion relations of nondegenerate vector solitons in multi-component models.

Figures (6)

• Figure 1: (Top) The energy-velocity dispersion relation of nondegenerate dark-bright-bright solitons with particle number for bright soliton $N_{B2}=1$ and $N_{B3}=3$. The nondegenerate soliton maximum velocity $v_{\max} = 0.2835$. The red and blue curves correspond to the positive mass branches, while the black and green curves correspond to the negative mass branches. (Bottom) The density and phase distributions of the nondegenerate soliton with velocity $v=0.1$. The first component forms a double-valley dark soliton, while the second and third components are asymmetric bright solitons. Insets display the weakly localized tails on an expanded horizontal scale. Insets display the weakly localized solitons on an expanded horizontal scale.
• Figure 2: The dispersion relations $E_s(v)$ of nondegenerate dark-bright-bright solitons for varying $N_{B2}$. The black, red, green, and blue correspond to the branches $E_{s1}$, $E_{s2}$, $E_{s3}$, and $E_{s4}$, respectively. (a) Case $N_{B2}<N_{B3}$: red and blue denote positive inertial mass branches, whereas black and green denote negative inertial mass branches. The common maximal velocity is $v_{\max}=0.2835$. (b) Case $N_{B2}=N_{B3}$: the red and green curves coincide, indicating energy degeneracy of $E_{s2}$ and $E_{s3}$. The dotted curves correspond to the limiting dispersions of $E_{s1}$ and $E_{s4}$ branches as $N_{B2}=N_{B3}+10^{-5}$. The maximal velocity remains $v_{\max}=0.2835$. (c) Case $N_{B2}>N_{B3}$: green and blue correspond to positive inertial mass branches, while black and red correspond to negative inertial mass branches. Here $v_{\max}=0.1865$. Other parameters: (a) $N_{B2}=1$; (b) $N_{B2}=3$; (c) $N_{B2}=5$ (with the stated $N_{B3}=3$ in each panel).
• Figure 3: The maximum velocity of degenerate (a) and nondegenerate (b) dark-bright-bright solitons with varying $N_{B}$ in the region [0,4]. The black solid curves denote iso-velocity lines. For degenerate solitons, the maximum velocity is governed by the total bright component particle number, whereas for nondegenerate solitons it is controlled by the larger bright component particle number.
• Figure 4: Bogoliubov-de Gennes excitation spectra of four nondegenerate dark-bright-bright solitons with $v=0.1$. They exhibit a spectrum without imaginary part. The parameters are the same as symbols in Fig. (\ref{['fig1']}).
• Figure 5: Density evolutions of nondegenerate dark-bright-bright solitons with different branches driven by $F_2=0.003$ and $F_3=0.001$. (a1)-(a3) The nondegenerate soliton of $E_{s1}$ with negative mass is accelerated and diffuses two negative mass dark-bright solitons. (b1)-(b3) The nondegenerate soliton of $E_{s2}$ with positive mass is accelerated and diffuses a negative and a positive mass dark-bright solitons. (c1)-(c3) The nondegenerate soliton of $E_{s3}$ with negative mass is accelerated and diffuses a negative and a positive mass dark-bright solitons. (d1)-(d3) The nondegenerate soliton of $E_{s4}$ with positive mass is accelerated and diffuses two positive mass dark-bright solitons. The parameters are the same as the symbols in the Fig. (\ref{['fig1']}), respectively.