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Abstract
We demonstrate the existence of surface gap solitons, a special type of asymmetric solitons, in the one-dimensional nonlinear Schrödinger equation with quintic nonlinearity and a periodic linear potential. The nonlinearity is suddenly switched in a step-like fashion in the middle of the transverse spatial region, while the periodic linear potential is chosen in the form of a simple sin 2 lattice. The asymmetric nonlinearities in this work can be realized by the Feshbach resonance in Bose-Einstein condensates or by the photorefractive effect in optics. The major peaks in the gap soliton families are asymmetric and they are located at the position of the jump in nonlinearity (at x = 0). In addition, the major peaks of the two-peak and multi-peak solitons at the position x = 0 are higher than those after that position, at x > 0. And such phenomena are more obvious when the value of chemical potential is large, or when the difference of nonlinearity values across the jump is big. Along the way, linear stability analysis of the surface gap solitons is performed and the stability domains are identified. It is found that in this model, the solitons in the first band gap are mostly stable (excepting narrow domains of instability at the edges of the gap), while those in the second band gap are mostly unstable (excepting extremely narrow domains of stability for fundamental solitons). These findings are also corroborated by direct numerical simulations.
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1. Introduction
Solitons have attracted lots of attention in the fields of Bose-Einstein condensates (BECs) [1–6] and nonlinear optics [7–16] over the past few decades. Generally speaking, solitons [17–20] are created in the framework of balancing dispersion/diffraction and nonlinear effects [21–27], such as the solitons under periodic [28–32] or anti-Gaussian-shaped [33–38] nonlinear potentials. In addition, with periodic linear potentials (as optical lattices in BECs), the bandgap in energy bands would appear and gap solitons can be generated in that case, like the matter wave gap solitons [39–46] and optical gap solitons [47–53]. Note that the gap solitons, as a special type of solitons, are quite different from the common solitons. The former can only exist in bandgaps (formed with the help of periodic linear potentials) and their profiles are always modulated with minor peaks, while the latter always embrace smooth simple shapes (such as Gaussian or $\rm {sech}^2$). Now, various types of gap solitons have been predicted theoretically; for example, fundamental [54], multi-peaked [55], vector [56], and vortex solitons [57]. In addition, gap solitons have been also observed in many experiments, including the matter wave [39,40] and the optical gap solitons [47–49]. Recently, gap soliton families under moiré lattices have been observed in experiments [58–60], and they are attracting more and more attention nowadays.
Surface solitons are formed at the interfaces, where different physical properties meet [61,62]. Due to asymmetric physical situations, surface solitons are always asymmetric [63]. Note that many types of solitons not only have been predicted theoretically [64–72] but also demonstrated in relevant experiments [73–77]. For example, the surface solitons under asymmetric linear potentials have been both predicted theoretically [63] and observed experimentally [77]. It should also be mentioned that the solitons are most often considered under cubic nonlinearity; the quintic nonlinearity would be added only if enhanced stability is sought for.
The solitons in quadratic nonlinear media have been also widely reported, and they are often seen in the works on the second-harmonic generation [78,79]. In addition, the quadratic nonlinearity can be also seen as a correction term (called the Lee-Huang-Yang correction) to the cubic nonlinearity in the area of quantum droplets, which attracted much attention in recent years [80,81]. The quintic nonlinearity on its own has been considered less often, because it is hard to excite. Note also that the homogeneous isotropic cubic self-focusing nonlinearity cannot support the stable propagation of two-dimensional optical spatial solitons [82], while the stable ones in cubic-quintic nonlinearities have been observed [83]. And the cubic-quintic nonlinear Schrödinger equation, which includes the three-photon absorption caused by the quintic term, is a model that is highly consistent with the experimental results [83].
Further, the quintic nonlinearity attained more attention recently, since the experimental observation of solitons in quintic-septimal media (without the cubic term) has been reported [84]. After that, many types of solitons in quintic nonlinear media have been found [85–87], such as the fundamental [88,89], singular [90], vortex [91], dark [92,93], and multi-peak solitons [94]. However, to the best of our knowledge, the surface gap solitons in pure quintic nonlinearity have not yet been investigated. This task is undertaken in the present paper.
In this work, we pay attention to the generation of surface gap solitons in a step-function-like quintic nonlinearity. We analyze how quintic nonlinearity affects the formation of different surface gap soliton families, including the fundamental, two-peak, three-peak, and four-peak solitons. In addition, the stability of these soliton families is addressed by employing the linear stability analysis and also checked by direct numerical simulations. We arrange the rest of the article as follows. The theoretical model and the method of linear stability analysis are presented in Sec. 2. Numerical results on surface gap solitons families are presented in Sec. 3. Finally, the article is concluded in Sec. 4.
2. Model and the method of treatment
Our model is the well known one-dimensional nonlinear Schrödinger equation (alias, the Gross-Pitaevskii equation), which can be used to describe the propagation of light beams in nonlinear optics or the evolution of BECs under the mean-field approximation, written in a dimensionless form:
where $\psi$ is the slowly-varying field amplitude, $z$ stands for the propagation distance, $V$ is the linear potential, and $\xi >0$ defines the profile of the self-defocusing quintic nonlinearity. $\nabla ^2$ denotes the Laplacian, but here it is simply the second-order derivative with respect to the spatial coordinate $x$. Such a model can be also employed for the evolution of BECs, where $\psi$ and $z$ are replaced by the wave function and time, respectively.The stationary solutions for the above model can be found by selecting the field amplitude as $\psi (x,z)=\phi (x)~{\rm exp}(ibz)$, where $b$ stands for the real propagation constant (in BECs, $b$ should be replaced by the real chemical potential). Then, the differential equation for the stationary wave function $\phi (x)$ is given by:
In this work, the periodic linear potential (an optical lattice) is adopted, to generate the necessary energy band structure and the gap solitons within such a structure, and here it is chosen in a simple form:
where the amplitude $V_0>0$ also stands for the strength of the linear potential [see its profile in Fig. 1(a)]. Concerning the nonlinear potential, an asymmetric structure is employed to form surface solitons; hence, the coefficient $\xi (x)$ is given as a step-like discontinuous function:Here, $\epsilon >1$ is a constant defining the size of the nonlinearity jump and the escarpment of nonlinearity appears at the $x=0$ position [see its profile in Fig. 1(b)].
Fig. 1. (a) The profile of the linear potential $V$ at $V_0=5$. (b) The profiles of the nonlinear potential $\xi$ with $\epsilon =2$ (blue solid line) and $\epsilon =10$ (yellow dashed line). (c) The band spectrum for different values of $V_0$. ${\rm 1st}~BG$ and ${\rm 2nd}~BG$ stand for the first and second bandgap, respectively. (d) The Bloch-wave spectrum at $V_0=5$.
Next, we discuss the method of linear stability analysis for the surface gap solitons. Firstly, we disturb the wave function by
Substituting the above equation into Eq. (1), one easily obtains the relevant eigenvalue problem for the instability growth rate $\lambda$, as follows:
On the basis of eigenvalue Eqs. (6), it can be confirmed that the solutions of surface gap solitons are stable only when all real parts of eigenvalues $\lambda$ are null.
The soliton power for the optical solitons is another important quantity. In this work, the soliton power $P$ is defined as:
Below, we display numerical results for the surface gap solitons, including the power, profiles, and simulated evolutions of fundamental (single peak), two-, three-, and four-peak solitons.
It should be remarked that we adopt such a nomenclature of solitons to distinguish them from the other common nomenclature that mentions dipole, tripole, and quadrupole solitons. That nomenclature surmises the existence of certain phase relations between the peaks (e.g., the dipole soliton surmises an out-of-phase relation between the two peaks). However, in our case there are no definite phase relations between the peaks; the phase differences are quite arbitrary and strongly depend on the values of parameters in the model. A couple of other useful remarks are that the modified squared-operator method [95] is employed to calculate the stationary solutions and the commonly-used finite difference time domain method is employed to simulate numerically the evolution of the wave function in Eq. (1).
3. Numerical results
In this Section, we present our numerical results on the families of surface gap solitons. At first, we report the profile of the periodic linear potential (optical lattice) $V$ [given by Eq. (3)] in Fig. 1(a). Then, the step-function-like nonlinearity coefficient $\xi$ [given by Eq. (4)] is shown in Fig. 1(b), where the nonlinearities with $\epsilon =2$ and $\epsilon =10$ are displayed by blue solid and yellow dashed lines, respectively. This relatively large difference in the values of $\epsilon$ is chosen to better observe the difference in soliton structures obtained. It should be mentioned that the escarpment of the nonlinearity is located at the position $x=0$, which causes the profiles of solitons to be different in the regions of $x\geq 0$ and $x<0$. The linear bandgap spectrum of the periodic potential $V$ for different values of $V_0$ is portrayed in Fig. 1(c), in which the 1st BG and 2nd BG stand for the first and second bandgap, respectively. According to panel 1(c), both the first and second bandgaps will enlarge as $V_0$ increases. The Bloch-wave spectrum with specified $V_0$ is depicted in Fig. 1(d), where the spectrum includes both the first and second bandgaps. We use the spectrum from Fig. 1(d) for numerical simulations in the rest of this work.
The relationship of the soliton power $P$ versus the propagation constant $b$ is displayed in Fig. 2. There, the curves of $P$ versus $b$ for the fundamental (with the number of peaks $N_p=1$), two-peak (with $N_p=2$), three-peak (with $N_p=3$), and four-peak (with $N_p=4$) solitons are presented in Figs. 2(a1–a4). It should be noted that the upper and lower lines in Figs. 2(a1–a4) are the results for $\epsilon =2$ and $\epsilon =10$, respectively. As already mentioned, such a large change in $\epsilon$ is needed to clearly depict the changes in soliton characteristics. Notice also that the blue and red lines in panels (a1–a4) stand for the stable and unstable solutions, respectively. In Fig. 2(a1), $P$ increases with the increase of $|b|$ in the first bandgap, for both the lines with $\epsilon =2$ and $\epsilon =10$, satisfying the anti-Vakhitov-Kolokolov criterion (anti-VK criterion), a necessary but not sufficient condition for the stability of solitons [96]. On the other hand, $P$ versus $b$ for both $\epsilon =2$ and $\epsilon =10$ dissatisfy the anti-VK criterion at the edge of the bandgap (close to the position of the second band). Here, one can also see that the domains of stable solitons in the first bandgap are quite large, while the ones in the second bandgap do exist only for fundamental solitons, but these domains of stability are extremely narrow. The profiles of the fundamental solitons labelled by A1–A4 are reported in Figs. 3(a1–a4), and their simulated evolutions are displayed in Figs. 5(a1–a4).
Fig. 2. Soliton power $P$ versus the propagation constant $b$ for surface gap solitons with $\epsilon =2$ (upper lines) and $\epsilon =10$ (lower lines): (a1) Fundamental soliton; (a2) two-peak soliton; (a3) three-peak soliton; (a4) four-peak soliton. ${\rm N_p}$ denotes the number of peaks. Blue and red lines represent the stable and unstable solutions. The profiles of solitons labelled by A1–A4, B1–B4, C1–C4, and D1–D4 are reported in Figs. 3, 4, 6, and 7, respectively, and their evolutions are presented in Figs. 5 and 8. The amplitude $A$ (the maximum of $\phi$) of surface gap solitons versus $b$ for $\epsilon =2$ (blue solid lines) and $\epsilon =10$ (yellow dashed lines): (b1) Fundamental soliton; (b2) two-peak soliton; (b3) three-peak soliton; (b4) four-peak soliton. $V_0=5$ is used in this and the remaining figures.
Fig. 3. Profiles of the fundamental solitons with different values of $b$ at $\epsilon =2$: (a1) $b=-1.465$; (a2) $b=-2.5$; (a3) $b=-4.35$; (a4) $b=-4.6$. Profiles of the fundamental solitons with different values of $b$ at $\epsilon =10$: (b1) $b=-1.465$; (b2) $b=-2.5$; (b3) $b=-4.35$; (b4) $b=-4.6$. The evolution of solitons labeled by A1–A4 is displayed in Figs. 5(a1–a4).
Next, we focus on the curves of $P$ versus $b$ for two-peak solitons, which are presented in Fig. 2(a2). Here, $P$ increases when $|b|$ increases, in the first bandgaps, for both cases of $\epsilon =2$ and $\epsilon =10$, satisfying the anti-VK criterion. The stability region of two-peak solitons in the first bandgap is quite large; that is, they are unstable only when they are very close to the edges of the bandgap. On the other hand, we could not find stable two-peak solitons in the second bandgap, for both $\epsilon =2$ and $\epsilon =10$ in this model. Here, the profiles of the two-peak solitons labeled by B1–B4 are presented in Figs. 4(a1–a4), and their simulated evolutions are shown in Figs. 5(b1–b4).
Fig. 4. Same as Fig. 3 but for two-peak solitons. The evolution of solitons labeled by B1–B4 is displayed in Figs. 5(b1–b4).
Fig. 5. The perturbed evolution of fundamental solitons: (a1) unstable one with $b=-1.465$; (a2) stable one with $b=-2.5$; (a3) unstable one with $b=-4.35$; (a4) stable one with $b=-4.6$. The perturbed evolution of two-peak solitons: (b1) unstable one with $b=-1.465$; (b2) stable one with $b=-2.5$; (b3) unstable one with $b=-4.35$; (b4) unstable one with $b=-4.6$. The random noise is added by $1{\%}$ of the amplitude.
The relationship of $P$ versus $b$ for three-peak and four-peak solitons is portrayed in Figs. 2(a3,a4). In Fig. 2(a3), $P$ increases with the increase of $|b|$ in the first bandgap, for both cases of $\epsilon =2$ and $\epsilon =10$, satisfying the anti-VK criterion. The stability domains of three-peak solitons are also very large in the first bandgap, while stable three-peak solitons cannot be found in the second bandgap. The results for the four-peak solitons are similar to the cases of two-peak and three-peak solitons; that is, they also satisfy the anti-VK criterion in both the first and second bandgaps, and they are always stable in the central parts of the first bandgap. The profiles of the solitons labelled by C1–C4 and D1–D4 are portrayed in Figs. 6 and 7, and their simulated evolutions are reported in Figs. 8(a1–a4) and 8(b1–b4).
Fig. 6. Same as Fig. 3 but for the three-peak solitons. The evolution of solitons labeled by C1–C4 is displayed in Figs. 8(a1–a4).
Fig. 7. Same as Fig. 3 but for the four-peak solitons. The evolution of solitons labeled by D1–D4 is displayed in Figs. 8(b1–b4).
Fig. 8. The perturbed evolution of three-peak solitons: (a1) Unstable one with $b=-1.465$; (a2) Stable one with $b=-2.5$; (a3) Unstable one with $b=-4.35$; (a4) Unstable one with $b=-4.6$. The perturbed evolution of four-peak solitons: (b1) Unstable one with $b=-1.465$; (b2) Stable one with $b=-2.5$; (b3) Unstable one with $b=-4.35$; (b4) Unstable one with $b=-4.6$. The random noise is added by $1{\%}$ of the amplitude.
The relationship of amplitude $A$ versus the propagation constant $b$ for these surface soliton families is also discussed in this work, as shown in Figs. 2(b1–b4). Here, we portray the curves of $A$ versus $b$ for the fundamental, two-peak, three-peak, and four-peak solitons in Figs. 2(b1–b4). Note that the blue solid and yellow dashed lines in Figs. 2(b1–b4) are the results for $\epsilon =2$ and $\epsilon =10$, respectively. Obviously, the amplitude $A$ always increases with the increase of $|b|$ for both cases of $\epsilon =2$ and $\epsilon =10$ in panels (b1–b4). By comparing the results in panels (b1–b4), one can easily observe that the amplitudes of fundamental and multi-peak solitons are similar, and the amplitudes $A$ for $\epsilon =2$ are larger than those for $\epsilon =10$, for all surface soliton families displayed in panels (b1–b4).
We display the profiles of surface fundamental solitons in Fig. 3, where the ones with $\epsilon =2$ are shown in panels (a1–a4) (the top row) and the ones with $\epsilon =10$ are depicted in panels (b1–b4) (the bottom row). According to the top row of Fig. 3, the amplitude $A$ increases with the increase of $|b|$, and the minor peaks of fundamental solitons close to the edges of bandgaps are more obvious than the ones in the central parts of bandgaps. For example, the minor peaks of the soliton in panel (a1) (close to the edge of the bandgap) are more obvious than the ones in panel (a2) (the central part of the bandgap). Note that the positions of these surface solitons in the bandgap can be referred to in Fig. 2. The solitons in panels (a1–a4) (the top row) are the results with $\epsilon =2$, and the counterparts with $\epsilon =10$ are displayed in panels (b1–b4) (the bottom row). As for the cases with $\epsilon =10$, they are similar to the ones with $\epsilon =2$; that is, the amplitude increases when $|b|$ increases, and the minor peaks of solitons close to the edges of bandgaps are also more obvious than the ones in the central parts of bandgaps. We point out that the amplitudes of solitons with $\epsilon =10$ are always smaller than the ones with $\epsilon =2$. It should be also stressed that the fundamental solitons in the second bandgap are asymmetric, that is, the left part of the major peak is larger than the right part. In other words, it looks as if there is a dip in the right side of the major peak, and this is more obvious for the solitons with $\epsilon =10$, such as the solitons in Figs. 3(b3,b4). The evolution of solitons labelled by A1–A4 in panels (a1–a4) is presented in Fig. 5(a1–a4).
The profiles of surface two-peak solitons are portrayed in Fig. 4, in which the case with $\epsilon =2$ is reported in Figs. 4(a1–a4) (the top row) and the counterpart with $\epsilon =10$ is presented in Figs. 4(b1–b4) (the bottom row). In Fig. 4, one can clearly see that the results for two-peak solitons with $\epsilon =10$ are also similar to the ones with $\epsilon =2$. The amplitude of two-peak solitons increases with the increase of $|b|$, and the minor peaks of these solitons close to the edges of bandgaps are more obvious than the ones in the central parts of bandgaps, for both cases with $\epsilon =2$ and $\epsilon =10$. It should be also pointed out that the two major peaks of the two-peak solitons in this model are not of equal heights; that is, the major peak at the position $x=0$ is higher than the one at position $x>0$. And such a phenomenon is more obvious in the cases with larger values of $|b|$, for example, in the results displayed in panels (a2–a4) and (b2–b4). Note also that the difference between the amplitudes of the two major peaks of two-peak solitons in the case with $\epsilon =10$ is more obvious than for the ones with $\epsilon =2$, which can be clearly seen in panels (a2–a4) and (b2–b4). It should be also stressed that the major peak (the tallest one) of two-peak solitons is asymmetric; that is, its left part is larger than its right part. The phenomenon is more obvious in the case with $\epsilon =10$, like the solitons in Figs. 4(b3,b4). We report the evolution of solitons labelled by B1–B4 [in panels (a1–a4)] in Fig. 5(b1–b4).
The perturbed propagation of surface fundamental and two-peak solitons is reported in Fig. 5, where the results of fundamental solitons are displayed in the top row and the ones of 2P solitons are portrayed in the bottom row. Note that the propagation of both the fundamental and two-peak solitons in Figs. 5(a2,a4,b2) is stable; the solitons keep their shapes and amplitudes during the long-distance propagation. On the other hand, the propagation of the fundamental and two-peak solitons in the rest of Fig. 5 is unstable, and the solitons diverge from the centre to the outside during their propagation.
Next, we pay attention to the numerical results of surface multi-peak solitons, including the profiles and evolution of three-peak and four-peak solitons. The profiles of surface three-peak solitons are displayed in Fig. 6, where the case with $\epsilon =2$ is presented in panels (a1–a4) (the top row) and the case with $\epsilon =10$ is reported in panels (b1–b4) (the bottom row). According to Fig. 6, it is clear that the profiles of surface three-peak solitons with $\epsilon =10$ are similar to the ones with $\epsilon =2$. The amplitude of three-peak solitons increases if $|b|$ increases, and the minor peaks of these solitons close to the edges of bandgaps are more obvious than the ones in the central parts of bandgaps. Interestingly, the three major peaks of three-peak solitons in this model are not of equal heights; that is, the major peak of three-peak solitons at the position $x=0$ is higher than the ones (the two neigboring peaks) at the positions $x>0$. Such a phenomenon is more obvious in the cases with larger values of $|b|$, like the results shown in panels (a3,a4) and (b3,b4). It could be noted that the difference between the amplitudes of the three major peaks of three-peak solitons in the case with $\epsilon =10$ is more obvious than in the case with $\epsilon =2$. Notice also that the major peak (the tallest one, at $x=0$) of three-peak solitons is asymmetric, especially for the solitons with $\epsilon =10$, see Fig. 6(b3,b4). The evolution of solitons labelled by C1–C4 [in panels (a1–a4)] is presented in Figs. 8(a1–a4).
We display the profiles of surface four-peak solitons in Fig. 7, in which the solitons with $\epsilon =2$ are presented in panels (a1–a4) and the ones with $\epsilon =10$ are portrayed in panels (b1–b4). Here, one can easily find that the profiles of these solitons with $\epsilon =10$ are similar to those with $\epsilon =2$. Similar to the results of fundamental, two-peak, and three-peak solitons, the amplitude of four-peak solitons increases with the increase of $|b|$, and the minor peaks are more obvious close to the edge of bandgaps. Similar to the case of three-peak solitons, the major peak of four-peak solitons at the position $x=0$ is higher than the ones (the three neighboring peaks) at the positions $x>0$. And such a phenomenon is more obvious in the cases with larger $|b|$, as is shown in panels (a3,a4) and (b3,b4). Notice that the difference between the amplitudes of four major peaks of the four-peak solitons in the case with $\epsilon =10$ is more obvious than for the case with $\epsilon =2$. Similar to the results obtained for fundamental, two-peak, and three-peak solitons, the major peak (the tallest one, at $x=0$) of the four-peak solitons is asymmetric, especially for the solitons with $\epsilon =10$, see Figs. 7(b3,b4). The evolution of the solitons labelled by D1–D4 [in panels (a1–a4)] is presented in Fig. 8(b1–b4).
Finally, we display the perturbed propagation of surface three-peak and four-peak solitons in Fig. 8, in which the results for three-peak solitons are presented in the top row and the ones for four-peak solitons are displayed in the bottom row. Note that the propagation of these solitons in Figs. 8(a2,b2) is stable; the solitons always keep their shapes and amplitudes during simulation. On the other hand, the propagation of soliton families in the rest of Fig. 8 [except for panels (a2,b2)] is unstable, and obvious distortions occur during their propagation.
The perturbed propagation of solitons for $\epsilon =10$ displays similar behavior as that of solitons for $\epsilon =2$, therefore it will not be presented here. It should be also mentioned that we have found stable asymmetric structures in our model, see also Figs. 5(b2) and 8(a2,b2). Such stable asymmetric structures are important, since they may be helpful for the better understanding of the interaction of solitons. In addition, they may be also helpful for the detection of the interface for the asymmetric nonlinearity in such a model.
4. Conclusion
In conclusion, we have shown that stable surface gap soliton families do exist in the step-function-like quintic nonlinear media, which can be realized by the Feshbach resonance in BEC or by the photorefractive effect in optics. The amplitudes $A$ of these surface gap solitons increase with the increase of the propagation constant $|b|$, and the curves of $A$ versus $b$ for fundamental, two-, three-, and four-peak solitons are similar. Interestingly, we find that the major peaks of these gap solitons are asymmetric and they are located at the position of the escarpment of the quintic nonlinearity. And the minor peaks of these gap soliton families close to the edges of bandgaps are more obvious than the ones in the central parts of bandgaps. As for the two- and multi-peak (three- and four-peak) solitons, the major peaks of the two-peak and multi-peak solitons at the position $x=0$ are higher than the ones at the positions $x>0$. Such a phenomenon is more evident in the cases with larger values of $|b|$ (the propagation constant) and $\epsilon$ (the strength of the quintic nonlinearity at $x>0$).
The stability domains for these surface gap solitons are obtained by the linear stability analysis and are confirmed by direct numerical simulations. We find that the solitons in the first bandgap are mostly stable (excepting those very close to the edge of the gap), while those in the second bandgap are mostly unstable, though a very narrow domain of stability is found only for fundamental solitons. The value of the parameter $\epsilon$ (the size of the nonlinearity jump) does not affect too much the stability regions. Our results may be found useful for the description of asymmetric nonlinear media, and the corresponding physical behavior at the sudden change of nonlinearity.
Funding
National Natural Science Foundation of China (62205224); Guangdong Basic and Applied Basic Research Foundation (2023A1515010865); Guangdong Province Education Department Foundation of China (2018KZDXM044); Meizhou City Social Development Science and Technology Plan Project (2021B127); Romanian Ministry of Research, Innovation, and Digitization (PN 23210101/2023); Qatar National Research Fund (NPRP 13S-00121-200126).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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