Xiong Zhi-Jin, Xu Qing, Ling Liming. Dark and multi-dark solitons in the three-component nonlinear Schrödinger equations on the general nonzero background. Chinese Physics B, 2019, 28(12): 120201
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Dark and multi-dark solitons in the three-component nonlinear Schrödinger equations on the general nonzero background
Xiong Zhi-Jin1, Xu Qing2, Ling Liming2, †
School of Electric Power Engineering, South China University of Technology, Guangzhou 510640, China
School of Mathematics, South China University of Technology, Guangzhou 510640, China
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).
Abstract
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 2, we give the SU(3) transformation for the TCNLS equations and study the modulational instability of the general nonzero background solution by the linear stability analysis. In Section 3, we study the propagation of single-dark soliton and discuss the collision between the two dark solitons on the general nonzero background. Then we test the numerical stability of the single dark soliton on the general nonzero background by numerical method. Section 4 involves some discussions and conclusions.
2. General nonzero background solution and its modulational instability analysis
In this section, we mainly study the general nonzero background solutions and modulational instability for the TCNLS equations
where q = (q1, q2, q3)T with q1, q2, q3 being the functions with respect to space x and time t, the symbols † and T denote the conjugate transpose and transpose, respectively. The TCNLS equations (1) possess the SU(3) symmetry. In other words, the solutions to the TCNLS equations are invariant under the SU(3) transformation, i.e. if q is a solution to the TCNLS equations (1), so does Aq, here A ∈ SU(3), i.e. the matrix A satisfies , is the identity matrix, then we will obtain some solutions possessing the novel dynamic behavior. It is easy to observe that the plane wave solution
satisfies the TCNLS equations (1). Combining the SU(3) transformation and the plane wave solution (2), we can obtain the general nonzero background solution, which is constituting of three different periodic functions. In the following, we study the general nonzero background solution deeply. Here we should stress that general nonzero background solution is different from the algebraic geometry solution by algebraic curve with high genus.[32]
2.1. General nonzero background
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
where , a1, a2, and a3 jointly determine the characteristics of general nonzero background, and the amplitude of the wave background is decided by c1, c2 and c3, the matrix A = (Ai,j)1 ≤ i,j ≤ 3 is an element of unitary group SU(3). We can notice that the new solution possesses the periodic or non-periodic behavior, which depends on the specific values of ak and gk (1 ≤ k ≤ 3). For the two-component case, the background will turn to the periodic background since there are two different phase factors under the SU(2) transformation.[31]
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 A, we classify them into three cases as follows.
From the expression of Eqs. (6), we can realize that the magnitude of fluctuation of general nonzero background is determined by three cosine functions, and the amplitude of fluctuation is approximately along three families of straight lines:
where k1, k2, k3 are integral numbers. The magnitude of fluctuation can be maximized, and the parameters k1, k2, k3 must satisfy the relationship
In the following, we discuss the influence on the fluctuation of general nonzero background by the parameters. By analyzing the expressions, we can obtain that the variation with time and space is determined only by the parameters a1, a2, a3. According to the their values, we can divide the situation into three categories.
Based on the special setting of parameter matrix A, we can obtain different dynamics of background solutions with the aid of numeric method. We will exploit the dark solitons on the general nonzero background and observe its dynamic behavior in the following section, which provides a new researchable background for dark soliton experiments on diverse physical fields.
2.2. Modulational instability analysis
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 A, we choose the case that matrix A equals to the unit matrix to study the MI. The other cases can be studied in a similar procedure. Then adding the perturbation into background solutions (2), we can obtain
where ε is a very small parameter, and u1, u2, u3 are the functions with respect to x and t. Substituting Eq. (9) into the TCNLS Eq. (1), and ignoring the higher terms (ε2, ε3) and keeping the ε0 and ε1 terms, then we can obtain the equations
Then we can assume the solutions of uj (j = 1,2,3) as the Fourier modes
where uj,1 and uj,2 are the periodic functions only about t, and k is the wave number. Then substituting Eq. (11) into Eq. (10), we can obtain a series of equations rewritten as the matrix form
where , and the coefficient matrix H is
where
According to the knowledge of linear differential equations with constant coefficients, we can obtain the fundamental matrix solution
where λi are the eigenvalues satisfying the characteristic equation det(λII−H) = 0, and Vi are corresponding eigenvectors. Then we can study the eigenvalues to judge the linear stability. The characteristic equation det(λII−H) = 0 can reduce into
where λ′ = λ/k. Since the characteristic equation about λ is a hexagonal polynomial, the discriminant is very complex. We ignore to give it. Thus the judgement of MI can be claimed: Once the characteristic root has positive real part, the corresponding solutions must be MI. To visualize the process of MI, we plot Fig. 1 to exhibit the MI. In Fig. 1, with the choice of parameters, the root of the polynomial has positive real part λR for some corresponding k values, so the general nonzero background possesses modulational instability. However, although the general nonzero background is modulationally unstable, not all components of the general nonzero background are modulationally unstable. Modulation stable region still exists here. Then we discuss Eqs. (15) and (20) (in the next section), which represent the modulational stability of the general nonzero background and the existence of dark soliton, respectively. The form of two equations is very similar. On the other hand, it is well known that the modulational stability analysis is an important factor for the generation of Akhmediev breather and rogue waves[33] cannot be used to explain the existence of dark solitons. However, we still conjecture that there is a certain relationship between modulational stability of background and the existence of dark soliton, which would be disclosed in the future work.
Fig. 1. The relation between the maximum real part λR and modulation frequency k, the parameters are c1 = c2 = 0.5, c3 = 0.1, a1 = π, a2 = 0, a3 = π/3.
3. Dark solitons on the general nonzero background
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.
3.1. The single-dark-soliton solutions for TCNLS equations
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
where , λ1 is a complex number (λ1 = λ1R + iλ1I), qn are given by Eq. (3), and h is a parameter, which is connected with λ1. Through the above expressions (16), we know the peak of dark soliton is along , so we can obtain the velocity of dark soliton:
and the amplitudes of the dark soliton in the three-components are different from each other:
In the literature,[21] the authors used the plane wave seed solution to solve the Lax pair equation. From Ref.[21], we can obtain the parameter matrix U0. Then we study the relationship between h and λ1. Consider the characteristic matrix
where λ1 is the eigenvalues. By the mathematical calculation, we obtain the equation between h and λ1 as follows:
Then we have a conclusion on the condition of parameters. The dark solitons exist only if equation (20) possesses at least a pair of complex roots with respect to h. So we plot Fig. 2 to show the relationship between h and λ1, which can be greatly helpful for us to select the parameter h. Similarly, we can judge the effective range of h by the selected parameters in the next research, which can ensure the existence of the dark solitons.
Fig. 2. The relationship between h and λ1I, the parameters are c1 = c2 = 0.5, c3 = 0.1, a1 = π, a2 = 0, a3 = π/3.
3.2. The multi-dark-soliton solutions for TCNLS equations
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
where both N and N[l] (l = 1,2,3) are 2 × 2 matrices and they can be written as
with the symbol * representing the conjugation, and
where αi(i = 1,2) equal to zero. The relations between the parameter hi and λi are as follows:
The situation is the same as the single-dark-soliton solution, we can also find the effective value of parameters hi to ensure the existence of two dark solitons.
3.3. The dynamics for the dark solitons on the general nonzero background
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 2. According to different phenomena, we can roughly classify them into four categories, which are named as cases i, ii, iii, and iv, respectively. Comparing the expression about the amplitude of background and soliton and considering the randomness of matrix A, we can only discuss the values of and c1c2 + c2c3 + c1c3. To make dark solitons visible in the background, the amplitude of dark soliton must be much higher than the amplitude of background, i.e., , or c1 ≫ c2,c3. For the aesthetic effect of background, we must opt appropriate parameters (a1, a2, a3) to prevent background waves from becoming too dense or too sparse. In other words, the spatial period of background () can not be too large or too small. We give the specific classification in the following.
Fig. 3. Dark solitons on the general nonzero background. Here (a), (b), and (c) are the model square map of single dark soliton for q11, q21, and q31, respectively; (d), (e), and (f) are the model square map of two dark solitons for q12, q22, and q32, respectively. The parameters c1 = 5, c2 = 1, c3 = 1, a1 = −0.1, a2 = 0.1, a3 = 0.8.
Fig. 4. Dark solitons on the general nonzero background. Here (a), (b), and (c) are the model square map of single dark soliton for q11, q21, and q31, respectively; (d), (e), and (f) are the model square map of two dark solitons for q12, q22, and q32, respectively. The parameters c1 = 0.5, c2 = 3, c3 = 0.5, a1 = −0.3, a2 = 0.3, a3 = 1.
Fig. 5. Dark solitons on the general nonzero background. Here (a), (b), and (c) are the model square map of single dark soliton for q11, q21, and q31, respectively; (d), (e), and (f) are the model square map of two dark solitons for q12, q22, and q32, respectively. The parameters c1 = 0.5, c2 = 0.5, c3 = 4, a1 = −0.2, a2 = 0.2, a3 = 1.
Fig. 6. Dark solitons on the general nonzero background. Here (a), (b), and (c) are the model square map of single dark soliton for q11, q21, and q31, respectively; (d), (e), and (f) are the model square map of two dark solitons for q12, q22, and q32, respectively. The parameters c1 = 2, c2 = 2, c3 = 0.5, a1 = −0.2, a2 = 0.2, a3 = 3.
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.
3.4. Numerical stability test
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. 1, when k is small, the roots of characteristic equation are not all positive real part, where we know that the general nonzero background solutions may be stable under this condition. In the following, we choose the same parameters as Fig. 1 to discuss the numerical stability of dark soliton solutions. Then we test the numerical stability of one dark soliton on the general nonzero background. Firstly, we use the same parameters as before and set the h = 0 to plot Figs. 7(a)–7(c), which are the exact solutions. Here the choice of the parameters belongs to case iv. Departing from the initial condition about the single-dark soliton at time t = 0 with a small white noise, and combining the Fast Fourier Transform with fourth-order Runge–Kutta method and carrying out numerical evolution, we can obtain the numerical solutions showing in Figs. 7(d)–7(f). The variation of errors E between the exact solution and the numerical solution with respect to time t (Fig. 8) shows that the dark soliton on the fluctuated background is stable against the white noise, which further verifies the observability in experiments.
Fig. 7. Upper: the actual single-dark soliton on the general nonzero background. Here (a), (b), and (c) are the model square map for q11, q21, and q31, respectively, which are the exact solutions, and the parameters are c1 = 0.5, c2 = 0.5, c3 = 0.1, a1 = π, a2 = 0, a3 = π/3. Lower: the numerical simulation for TCNLS equations under an initial condition. Here we use the single soliton solutions from exact solutions as the initial value; (d), (e) and (f) are the model square map for q11, q21, and q31, respectively.
Fig. 8. The error E between the exact solutions and the numerical solutions (Fig. 7) about time t.
4. Conclusion
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.
