Phase Characters of Optical Dark Solitons with the Third-order Dispersion and Delayed Nonlinear Response

We start with one SVDS solution of the Eq. (1), with choosing $n=1$ of Eq. (13),

where $c$ and $a$ are the amplitude and wavenumber of plane wave background, respectively. We set $c=1$ thereinafter. $\gamma_{1}$ is the position of SVDS in distribution direction. $\tilde{w}_{1}=\sin(z_{1})$ is the width-dependent parameter. $v_{1}$ is the velocity of soliton center, calculated as $v_{1}=\beta(4c^{2}\tilde{v}_{1}^{2}-6ac\tilde{v}_{1}+3a^{2}+2c^{2})-c\tilde{v}_{1}+a$ with $\tilde{v}_{1}=\cos(z_{1})$. Importantly, the pure velocity of SVDS should be the relative velocity between the soliton center and plane wave background. Since the units are dimensionless of Eq. (1), therefore the velocity of plane wave background equals to the value of background wavenumber, based on the quantum mechanics theory Landau . Thus, the velocity of SVDS is expressed as

It should be emphasized that the wavenumber $a$ cannot be ignored by the Galilean transformation in HE. Meanwhile, owing to the Eq. (3) is a quadratic function of $\tilde{v}_{1}$, the variations of velocity will be nonlinear. Therefore, the upper and lower speed limits are not only one, as summarized in Table 1. As we can see, the velocity ranges depend on not only the sign of high-order nonlinearity $\beta$ but also the background wavenumber $a$. With fixing the sign of $\beta$, the velocity ranges can be categorized into four classes according to the regions of wavenumber $a$, which is greatly different from that of SVDS in NLSE (i.e. for $\beta=0,|v|<c$). That is because the background wavenumber can be removed by the Galilean transformation in NLSE, so it doesn’t come into the properties of nonlinear modes nop ; Kivshar . In contrast to the dark soliton, the velocity of bright soliton is inversely proportional to the strength of high-order nonlinearity and its velocity range is totally unrestricted in HE Radhakrishnan ; solution7 ; HE1 , similar to that of in NLSE HE1 . For the other nonlinear waves of HE, such as rational solutions and breathers, although the background wavenumber cannot also be neglected by the Galilean transformation, it just affects the velocity value without determining the velocity range solution2 ; CLiu .

Generally, there is a close connection between the velocity and phase jump. For a dark soliton of NLSE, there is a one-to-one correspondence between the velocity and phase jump zhaods . The discussions of the velocity properties imply us both the high-order effects and background wavenumber are crucial to the physical properties of a dark soliton in HE. Then we would like to investigate the phase characters of the SVDS.

We shall first look briefly at the variations of velocity with phase jump by taking $\beta=-0.3$ and $a=0$, as shown in Fig. 1(a) with blue solid line. For comparison, we also show the one-to-one correspondence between phase jump and velocity of an SVDS in NLSE ($\beta=0$) with the black dashed dotted line. It is seen that even if the background wavenumber is not considered, the emergence of the high-order effects also induces the change of velocity with the phase jump to be distinctly different from that of dark solitons in NLSE. Particularly, there exists a critical phase jump $\Delta\phi_{c_{1}}=2\arccos(1/6)$ labeled by the red triangle symbol, which divides the relations between phase jumps and velocity into the two cases mentioned above. It is seen that when $\Delta\phi\in[0,\Delta\phi_{c_{1}})$, the phase jump and velocity of SVDS satisfy the case I relation. Interestingly, when $\Delta\phi\in(-\pi,0)\cap(\Delta\phi_{c_{1}},\pi)$, an SVDS with the same velocity can admit two distinct phase jumps, namely, it meets the case II relation. This phase property has never been reported in pervious literature.

Moreover, the case II includes two subcases as shown in Fig. 1(b), which is distinguished by the second critical phase jump $\Delta\phi_{c_{2}}=-2\arccos(5/6)$ marked as the left red triangle symbol. For the case II-A, $\Delta\phi\in(-\pi,\Delta\phi_{c_{2}})$, the two distinct phase jumps have the identical sign but the different values for an SVDS with the same velocity. While for the case II-B, $\Delta\phi\in(\Delta\phi_{c_{2}},0)\cap(\Delta\phi_{c_{1}},\pi)$, both the values and the signs of two phase jumps are different. Such abundant phenomena greatly deepen our knowledge of the phase properties for the dark solitons in HE.

More interestingly, we find that background wavenumber $a$ and high-order effects coefficient $\beta$ have a significant impact on the existence regions of the case II. For example, we demonstrate the changes of velocity with phase jump and background wavenumber in Fig. 1(c) with fixing $\beta=-0.3$. The existence regions of the case II are surrounded by the white solid lines. It is shown that the case II can emerge when the background wavenumber $a\in(-\frac{1}{9},\frac{11}{9})$ and the phase jump in some ranges. The related critical phase jump $\Delta\phi_{c}$ will be varied with the background wavenumber. For $a\in(-\frac{1}{9},\frac{5}{9})$, the case II is existed when $\Delta\phi\in(-\pi,0]\cap(\Delta\phi_{c},\pi)$ with $\Delta\phi_{c}=2\arctan\Big{[}\frac{\sqrt{(5-9a)(9a+7)}}{9a+1}\Big{]}$. For $a\in[\frac{5}{9},\frac{11}{9})$, the case II can be admitted when $\Delta\phi\in(-\pi,\Delta\phi_{c})\cap[0,\pi)$ with $\Delta\phi_{c}=2\arctan\Big{[}\frac{\sqrt{(17-9a)(9a-5)}}{9a-11}\Big{]}$. The outside of these regions, the phase jump and velocity of the SVDS satisfy the case I, which is the same as the one in NLSE.

We further study the influence of the high-order nonlinearity on the existence areas of the case II, with taking the background wavenumber $a\!=\!0$. The result has been shown in Fig. 1(d). Considering the high-order effects coefficient is a small value generally, we set $\beta\in[-1,1]$ herein. It is seen that there are two discontinuous regions of $\beta$ for the case II. One is $\beta\in[-1,-\frac{1}{4})$, with $\Delta\phi\in(-\pi,0]\cap(\Delta\phi_{c},\pi)$ and $\Delta\phi_{c}=-2\arctan\Big{(}\frac{\sqrt{-1-8\beta}}{1+4\beta}\Big{)}$. The other is $\beta\in(\frac{1}{4},1]$, with $\Delta\phi\in(-\pi,\Delta\phi_{c})\cap[0,\pi)$ and $\Delta\phi_{c}=2\arctan\Big{(}\frac{\sqrt{-1+8\beta}}{1-4\beta}\Big{)}$. Naturally, the parameter spaces except these regions correspond to the case I. It is clear that when the background wavenumber is not considered, there is the case I and only the case I when $|\beta|<\frac{1}{4}$. In this regime, the properties of SVDS are basically the same as the ones in NLSE. The two subcases of case II as mentioned in Fig. 1(b) are also existed in Fig. 1(c) and (d), we won’t elaborate herein.

Above discussions clearly illustrate that an SVDS with the same velocity can admit two distinct phase jumps under certain conditions in HE, in sharp contrast to the dark solitons with only the second-order dispersion and self-phase modulation, which admits a one-to-one match between the velocity and phase jump. Recently, the topological phases of nonlinear waves such as dark solitons zhaods ; Qinds , bight solitons Wu , rogue waves and breathers zhaorw have garnered much attention. It was reported that their phase jump are determined solely by the topological vector potentials, which are defined on the complex plane and are periodic along the imaginary axis. It inspires us to investigate the topological vector potentials underlying the distinct phase jumps to further understand the phase characters of the SVDS in the region of the case II, by utilizing the topological vector potential theory proposed in Refs. zhaorw ; zhaods directly.

We first get integrand function $F[x]=\partial_{x}\varphi$ in the area integral, where $\varphi$ is the phase of wave function $q^{[n]}$ (by expressing the $n$-dark solitons solution Eq. (13) as $q^{[n]}=|q^{[n]}|e^{\mathrm{i}\varphi}$ with leaving out the phase of background). Then, we introduce a function $F[z]$, which is obtained by extending the real coordinate variable $x$ to the complex coordinate space $z=x+\mathrm{i}y$ of $F[x]$. The total phase jump can be described by an integral of effective vector potential $\mathbf{A}$ along the real axis (i.e. the $x$ axis). Such an integral corresponds to a circle integral, $\Re\left(\oint_{C}F[z]\mathrm{d}z\right)=\Re\left(\oint_{C}(u[x,y]+\mathrm{i}w[x,y])(\mathrm{d}x+\mathrm{i}\mathrm{d}y)\right)=\oint\mathbf{A}\cdot\mathrm{d}\mathbf{r}$, where $\Re$ expresses the real part of this integral, $\mathbf{A}=u[x,y]\mathbf{e}_{x}-w[x,y]\mathbf{e}_{y}$ and $\mathrm{d}\mathbf{r}=\mathrm{d}x\mathbf{e}_{x}+\mathrm{d}y\mathbf{e}_{y}$. Then, the corresponding magnetic filed is derived as $\mathbf{B}=\nabla\times\mathbf{A}$. It provides a way to explore density zeros ($|q^{[n]}(z_{N})|^{2}=0$) of dark solitons in the extended complex plane and then understand the topological properties underlying their phase jumps. The density zeros $z_{N}$ are the singularities of $F[z]$ (marked as $z_{N}=x_{N}+\mathrm{i}y_{N}$, $N$ is an integer). Consequently, these singular points constitute monopole fields which appear in pairs. Based on the Cauchy integral formula, the magnetic flux at each of monopoles is calculated as $\Omega=\int_{\Gamma}f[z]\textrm{d}z=2\pi\mathrm{i}\textrm{{Res}}{[f(z_{N})]}$ Hayman , where $\Gamma$ denotes the closed curves encircling the singularity $z_{N}$. The integral is zero in the complex plane $z$ excepted the singularities. The $\textrm{{Res}}{[f(z_{N})]}=\Omega/2\pi\mathrm{i}$ is the residue. This integral gives rise to a quantized magnetic flux of elementary $\pi$ at each of the monopoles. The directions of magnetic flux lead to a positive or negative phase jump. Then, we can explore the topological phases of $n$-dark solitons by performing topological theory on the dark soliton solution $q^{[n]}$ in HE.

Now we characterize the topology of the SVDS. Based on the solution Eq. (2), the integrand function is expressed as $F[z]=\frac{2\tilde{w}^{2}_{1}\tilde{v}_{1}}{1-2\tilde{w}_{1}^{2}+\cosh[2\tilde{w}_{1}z]}$. Then, we can calculate the vector potential $\mathbf{A}$ by determining the singularities in the complex plane,

where $y_{0}=\frac{\mathrm{arccosh}(2\tilde{w}_{1}^{2}-1)}{2\tilde{w}_{1}}$ and the period $T=\frac{\pi}{\tilde{w}_{1}}$. Then, we describe the topological property of an SVDS in the regime of case II. As an example, we exhibit the topological phase of an SVDS with the velocity $v=-0.74$. The related two different topological phases have been presented Fig. 2(a1-a2) and (b1-b2), respectively. The vector potentials $\mathbf{A}$ are shown in the left panel, and the corresponding magnetic monopole fields $\mathbf{B}$ (red symbols) and phase distributions $\phi$ (blue solid line) are depicted in the right panel. The pairs of positive and negative magnetic fluxes emerge periodically along the imaginary axis $y$. The monopoles with $+\pi$ and $-\pi$ magnetic fluxes are indicated by $\bigodot$ and $\bigotimes$, respectively. Obviously, these two topological fields in the extending complex plane $z$ are completely different, such as the position, period and even the direction of the magnetic monopoles. Then, the different topological vector potentials give rise to the distinct phase distributions. Therefore, the corresponding phase jump is different. One is $\Delta\phi=-0.19\pi$, the other is $\Delta\phi=0.92\pi$.

Importantly, the relation of case II between the velocity and phase jump shown in Fig. 1 also indicates that two different SVDSs with the identical velocity are admitted in HE. Namely, two parallel SVDSs can be obtained in the region of case II. Such relation for the SVDS can physically explain the generation mechanism of previously reported two parallel SVDSs in HE Lou ; zhangds ; solution6 . When two valleys of these SVDSs are overlapped, the two parallel SVDSs become a DVDS zhangds . One the other hand, the velocity expression Eq. (3) is a quadratic function of $\tilde{v}_{1}$. Therefore, the two parallel SVDSs can be mathematically constructed with choosing different spectral parameters Lou ; zhangds ; solution6 , which is not admitted in NLSE. Based on the abundant phase characters of the SVDS demonstrated above, we can expect that the phase properties of DVDS will be more interesting in HE.

We start with the inelastic interaction between one DVDS and one SVDS. For an example, we show their spatial-temporal intensity distribution in Fig. 5(a) with $v_{1}=0$ and $v_{3}=0.0125$ (see caption for detailed parameters setting). Before (after) the collision, two solitons, the SVDS and DVDS are marked as “$q^{[3]}$” (“$q^{[3]^{\prime}}$”), “$S_{1}$” (“$S_{1}^{\prime}$”) and “$S_{2}$” (“$S_{2}^{\prime}$”), respectively. The black dashed lines on upper and lower the Fig. 5(a) are used to represent the finial state ($t=300$) and initial state ($t=-300$) of these two solitons. In the following, we will analyze the intensity profiles and phase distributions of solitons at these two states. Figure. 5(b1)-(b2) are the intensity profiles before and after collision, in which the black solid line, cyan dashed line and green dashed line are plotted based on the solution $q^{[3]}$, $q_{s_{1}}^{\pm}$ and $q_{s_{2}}^{\pm}$ at fixed $t$. As it can be observed, the depths of two valleys for DVDS have changed dramatically, while SVDS remain unchanged. In order to analyze and understand the interaction process comprehensively and deeply, the investigations of phase properties of two solitons before and after the collision are also indispensable. Then, we will conduct the complex extending on expressions Eq. (13), Eq. (7) and Eq. (8) to analyze their topological phase, based on the topological vector potential theory mentioned in section II.3.

Before (after) the collision, the phase characteristics of $q^{[3]}$ ($q^{[3]^{\prime}}$), $``S_{1}"$ ($``S_{1}^{\prime}"$) and $``S_{2}"$ ($``S_{2}^{\prime}"$) have been presented in Fig. 5(c1-e1) (Fig. 5(c2-e2)). In these graphs, the phase distributions in the distribution direction $x$ are presented by the blue solid curves, and their corresponding magnetic field distributions in complex space are depicted by the red symbols. We also only show the paired monopoles which are closer to the $x$-axis herein. Interestingly, the phase jumps of two solitons initially presents triple-step structure in one direction, as shown in Fig. 5(c1). Surprisingly, when two solitons accomplish the whole collision process, the total phase distribution turns into a more complex structure composed by three phase jumps in different directions, as exhibited in Fig. 5(c2). This dramatic change in the phase jump structure is derived from the underlying topological potentials. By analyzing the magnetic fields associated with the vector potential, we find there are three pairs of monopoles for both cases. Strikingly, three pairs of magnetic monopoles are distributed in the same direction before the collision, while the direction of one of them is reversed after the collision, compared Fig. 5(c2) with Fig. 5(c1). This motivates us to explore what happened to each of the solitons.

We show the phase distributions and the corresponding magnetic field distributions of SVDS before and after collision in Fig. 5(d1) and (e2). It is seen that SVDS maintains single-step structure with one pair of monopoles. Before the collision, monopoles are located on the line $x=x_{1}$, and the coordinate values in the complex plane is $(-8.31,\pm 0.13)$, the associated phase step is $\Delta_{1}=-0.92\pi$. Then, it moves to the line $x=x_{4}$, and monopoles are located at $(4.74,\pm 0.13)$ after the collision. Obviously, there is only a translation in the $x$-axis for monopoles. Consequently, the phase step is $\Delta_{4}=\Delta_{1}=-0.92\pi$. Therefore, both the intensity profile and phase feature of SVDS can be held well after interacting with DVDS.

However, the topological phase of DVDS undergoes a remarkable change, as shown in Fig. 5(e1) and (d2). In the initial state, the phase distribution of DVDS is double-step type with two pairs of monopoles in the identical direction. The paired monopoles on the line $x=x_{2}$ are located at $(-1.57,\pm 0.13)$, which contributes to the phase jump $\Delta_{2}=-0.91\pi$. Interestingly, the monopoles on the line $x=x_{3}$ are colliding with each other, tending to merge in the real axis $x$, and the corresponding singularities are located at $(1.17,\pm 0.02)$. Then, it leads to a phase jump $\Delta_{3}=-\pi$ approximately. Due to the coordinate values along the imaginary axis $y$ are too small to see them separately, we further plot the insert for them in Fig. 5(e1). Then, we can get the total phase jump of DVDS before the collision is $\Delta_{2}+\Delta_{3}=-1.91\pi$.

Then, we analyze the phase characteristic of DVDS after the collision, which is shown in Fig. 5(d2). Strikingly, the phase distribution of the DVDS is transformed from the double-step type to U-shaped type. Moreover, the collision of paired monopoles is disappeared, in contrast to that of before the collision. Two pairs of monopoles are scattered on two separate lines $x=x_{6}$ and $x=x_{5}$, and their corresponding pointlike magnetic fields are located at $(-4.72,\pm 0.20)$ and $(-1.28,\pm 0.05)$. These give rise to the two phase steps are $\Delta_{6}=-0.87\pi$ and $\Delta_{5}=0.96\pi$ approximately. Thus, the total phase jump of DVDS after the collision is $\Delta_{6}+\Delta_{5}=0.09\pi$. This means that the relation of phase jumps and velocity for the two parallel SVDSs constructed the DVDS are transformed from the case II-A to the case II-B before and after the collision. This is an all-important sign of the inelastic collision.

These results imply that inelastic collision of DVDS can cause a striking transformation between two different types of topological phases. Such intriguing phenomenon has never been observed in previous literature. In our pre-work Ref. zhangds , our understanding of the inelastic collision only depends on the variations of intensity profiles, while important phase properties hidden the change of soliton profiles have been ignored. Then we’d like to know whether such phase transition is always existence for the collision dynamics involving the DVDS. For this purpose, we intend to establish phase diagrams for the phase characters of the DVDS before and after the collision, which can be realized by judging the sign of the phase gradient flow $F[x]$ of the asymptotic expressions $q_{s_{2}}^{\pm}$ Eq. (8).

For convenience, we choose the parameters are identical with the ones in Fig. 5 except parameters $\gamma_{1}$ and $\gamma_{2}$. Due to the velocity $(v_{1})$ of DVDS center is less than the velocity $(v_{3})$ of SVDS center, the asymptotic expressions $q_{s_{2}}^{-}$ expressed by Eq. (8) is actually identical to the DVDS solution Eq. (6). It demonstrates that the asymptotic behavior of DVDS before the collision is independent of the SVDS. Meanwhile, owing to the other related parameters $(z_{1},z_{2},a,c$, and $\beta)$ in Fig. 5 and Fig. 3 are the same, thus the phase diagram of phase characters for the DVDS before the collision is identical to the Fig. 4. The square symbol in Fig. 4 corresponds to the $``S_{2}"$ soliton exhibited in Fig. 5(a), which admits the double-step type phase distribution, as shown in Fig. 5(e1). In a similar way, we further give a phase diagram for the DVDS after the collision, with the aid of the asymptotic expressions $q_{s_{2}}^{+}$ expressed by Eq. (8). The results has been presented in Fig. 6, where the square symbol corresponds to the $``S_{2}^{\prime}"$ soliton depicted in Fig. 5(a), which admits the U-shaped type phase distribution, as shown in Fig. 5(d2). The black line is the symmetric DVDS after the collision, and the relation between $\gamma_{1}$ and $\gamma_{2}$ is re-calculated as $\gamma_{2}=\gamma_{1}+\frac{(w_{1}-w_{2})\ln(K_{2}^{+})-(w_{1}+w_{2})[\ln(K_{4}^{+})-\ln(K_{3}^{+})]}{4w_{1}w_{2}}$.

By comparing Fig. 4 and Fig. 6, we can see that the existence region of each type of phase distributions before and after the collision is changed obviously in the $(\gamma_{1},\gamma_{2})$ plane. During a collision process, the phase variation for the DVDS includes four kinds: double-step type to U-shaped type, U-shaped type to double-step type, U-shaped type to U-shaped type, and double-step type to double-step type. Particularly, the symmetric DVDS always keeps the double-step type phase distribution. These results reveal the extraordinary phase properties behind the inelastic collision and further deepens our understanding of solitons collision in essence. Actually, the emergence of phase transition for the DVDS attribute to the relation of case II shown in Fig. 1. It should be mentioned that Fig. 4 and Fig. 6 are not the only phase diagrams. When the related parameters are varied, the corresponding phase diagram will also be changed.

Above detailed analysis has revealed the intriguing phase transition of DVDS induced by the inelastic collision. It was reported that the collision of DVDS also can be elastic zhangds . Then, we will investigate the phase properties of DVDS which is undergone the elastic interaction. The elastic collision condition of DVDS have been proved in the Appendix C, which is a sufficient condition Eq. (26). For an example, we show the elastic interaction between a DVDS and an SVDS in Fig. 7(a). The speeds of two solitons are $v_{3}=0.014$ and $v_{1}=-0.009$. The meanings of all marks herein are the same as Fig. 5, we won’t reiterate them here. Based on the expressions $q^{[3]}$, Eq. (7) and Eq. (8) and performing the complex extending method, the topological phase of this elastic collision can also be studied well.

The total phase characteristics of two solitons before and after the collision have been shown in Fig. 7(c1) and (c2). It is seen that the phase distribution of two solitons keep triple-step structure before and after the collision. By analyzing the magnetic fields associated with the vector potential, we find that there are three pairs of magnetic monopoles in the same direction, scattered on three separate lines at $x_{j}$ $(j=1,2,3)$. Considering complexity of vector potential for two solitons solution, we will analyze the phase jump $\Delta_{j}$ on each line through analyzing the topological phase of each of the solitons.

The phase jump induced by the magnetic field for SVDS before and after collision have been shown in Fig. 7(e1) and (d2). A pair of monopoles is initially on the line $x=x_{1}$, and the corresponding singularity near the real axis is $(-10.11,\pm 0.10)$. Finally, it moves to the line $x=x_{4}$ located at $(3.20,\pm 0.10)$. Therefore, topological property of SVDS has remained the same after colliding with DVDS, associated phase jump $\Delta_{4}=\Delta_{1}=-0.94\pi$.

In this case, the DVDS also admits a similar feature, as shown in Fig. 7(e2) and (d1). Before the collision, two pairs of monopoles are scattered on the lines $x=x_{2}$ and $x=x_{3}$, and the monopoles are located at $(-0.40,\pm 0.11)$ and $(2.03,\pm 0.11)$. After the collision, they are located at $(-6.06,\pm 0.11)$ and $(-8.49,\pm 0.11)$ on the line $x=x_{6}$ and $x=x_{5}$, respectively. Obviously, two pairs of monopoles only have undergone a translation along the real axis. Moreover, two pairs of monopoles for the DVDS are symmetrically distributed in the complex panel, which leads to a symmetric phase distribution. This also reveals that the associated two parallel SVDSs have the relation of case II-A mentioned in Fig. 1. We get $\Delta_{2}=\Delta_{3}=\Delta_{6}=\Delta_{5}=-0.93\pi$ numerically, so the total phase jump of DVDS is equal to $-1.86\pi$. Then, we can obtain the total phase jump of two solitons is $\Delta=\Delta^{\prime}=-0.94\pi-1.86\pi=-2.8\pi$.

These results indicate that we can just analyze the topological phase of each soliton to understand and predict the collision properties, by combining the succinct asymptotic expressions and topological vector potential. In general, the inelastic interactions of solitons bring the variations of both the intensity profiles and topological phase, while the elastic collisions just accompany a translation in the real axis. Similar phenomena can also be observed in the collision between two DVDSs, and even multiple collisions involving the DVDSs, based on the $n$-dark soliton solution Eq. (13) and the asymptotic expressions Eq. (19)-Eq. (23) for $n$-dark solitons including $n_{1}$ SVDSs and $n_{2}$ DVDSs ($n=n_{1}+2n_{2}$).