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Project supported by the National Natural Science Foundation of China (Grant No. 11771151), the Guangdong Natural Science Foundation of China (Grant No. 2017A030313008), the Guangzhou Science and Technology Program of China (Grant No. 201904010362), and the Fundamental Research Funds for the Central Universities of China (Grant No. 2019MS110).
We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrödinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrödinger systems, and require more in-depth studies in the future.
Research of multi-component nonlinear Schrödinger (MCNLS) equations has attracted a great deal of attention since they can be used to describe a variety of complex physical phenomena (Bose–Einstein condensates,[1] nonlinear optics,[2] superfluid,[3] and alpha helical protein,[4] etc.), and they possess more abundant dynamical patterns of localized wave solutions (soliton,[5] rogue wave[6–8]) than ones in scalar models. Particularly, soliton is an important nonlinear phenomenon in nonlinear Schrödinger systems. Various solitons and other defects have been experimentally observed and theoretically investigated, such as dark soliton,[9] bright soliton,[10] vector soliton,[11–13] kink,[14] and magnon,[15] etc. Among these solitons, dark soliton has been found to appear in de-focusing nonlinear systems arising in nonlinear optics[16] and Bose–Einstein condensates,[17] etc.
For research of soliton on the integrable nonlinear Schrödinger system, most of the previous studies focus on the zero background or non-vanishing background.[18–22] Meanwhile, there are lots of results for the soliton solutions on the nonlinear Schrödinger-typed models with variable coefficient or high dimension.[23–29] More than a decade ago, Shin found dark soliton solutions on a cnoidal wave background in a defocusing medium.[30] Recently, Lan et al. presented different profiles of dark-solitons on stripe phase background in a two-component Bose–Einstein condensate.[31] Motivated by the above-mentioned two works, it is natural to ask whether there exist some dark solitons on the general nonzero background for the nonlinear Schrödinger system.
To the best of our knowledge, there is almost no work on discussing the dark soliton for the three-component nonlinear Schrödinger (TCNLS) equations on the general nonzero background. In this work, we study dynamics of the single dark soliton and two dark solitons for the TCNLS equations on the general nonzero background, and obtain various general nonzero backgrounds by adjusting parameters. Furthermore, the collision of two dark solitons also shows the elastic interaction between solitons on the general nonzero background. These new phenomena enrich the study of the dark solitons on various backgrounds and provide the theoretical basis for physical experiment.
This paper is organized as follows. In Section
In this section, we mainly study the general nonzero background solutions and modulational instability for the TCNLS equations
In this subsection, we discuss the general nonzero background solutions, which are totally different from the original plane wave solutions. The general nonzero background solutions can be written as
In what follows, we would like to analyze the dynamics for general nonzero background solution in detail. Based on the zero elements in the matrix
From the expression of Eqs. (
Based on the special setting of parameter matrix
In this subsection, we study the modulational instability (MI) for the general nonzero background solution through the linear stability analysis. Considering the complexity of matrix
According to the knowledge of linear differential equations with constant coefficients, we can obtain the fundamental matrix solution
In this section, we mainly intend to study the behavior of dark solitons on the general nonzero background and find some new phenomena to further enrich the nonlinear system.
The single dark soliton solutions on the plane wave background can be derived directly by the Darboux transformation.[21,34] After the original single-soliton solutions go through the SU(3) transformation, the new one-dark-soliton solutions on the general nonzero background can be written as
The multi-dark-soliton solutions on the plane wave background can also be derived by the Darboux transformation.[21,34] It has shown that the interaction among the dark solitons on plane wave background is elastic and dark solitons hold their shape and intensity or cause a slight oscillation about the phase, which will be a big effect in the quantum transmission.[35] With the number of the dark solitons increasing, the interaction among solitons will be more complex, so we just discuss the situation of two dark solitons. Comparing with the plane wave background, we study the phenomena on the general nonzero background. We can use the two-dark-soliton solutions to model the BECs and to simulate the collision on the general nonzero background. After the original two-dark-soliton solutions together with unitary transformation SU(3), the new two-dark-soliton solutions can be written as
Compared with the previous backgrounds (plane wave background, stripe phase background, etc.), the general nonzero background has more colorful dynamics than the previous ones. Under this background, we can add the dark solitons and observe many fascinating dynamic behaviors which have never been reported before. In the following, we mainly study and discuss case III (the parameters (ai, i = 1,2,3) are unequal with each other) in Section
The above four figures all demonstrate that the dark soliton can propagate stably and continuously on the general nonzero background, the collision between two dark solitons is elastic. The stability of collision on the plane wave background has been already proved before, but on the general nonzero background is the first time, which not only further strengthens the theory of the invariance of solitons collision, but also promotes the further study of soliton stability. The four figures are the representational phenomena from general nonzero background. The other cases can be derived in a similar method.
In the above subsection, we exhibit several interesting kinds of different dynamics of single dark soliton and two dark soliton solutions on the general nonzero background. For a physical system, the stability of solution is an important criterion to observability in experiments. Thus we use the numeric method to study the stability of these dark solitons. For convenience, we just test one dark soliton on the fluctuated background.
In Fig.
In summary, we have studied the dynamics of a single dark soliton and two dark solitons on the general nonzero background for the TCNLS equations. The phenomena we obtain in this paper possess much more complicated dynamics. Then we analyze the MI of the fluctuated background, where we find that the background may be modulational stability in certain parameters. The dynamics of the dark solitons and the collision of two dark solitons on the general nonzero background are shown by computer graphics. Furthermore, we use the numerical method and add the white noise to test the numerical stability of dark solitons, which shows that they are robust against the white noise. We believe that the dark soliton on fluctuated background can be observed in the physical experiments in the near future.