Generation and Control of Shock Waves in Exciton-Polariton Condensates

Jin-Ling Wang(王金玲)$^{1}$, Wen Wen(文文)$^{2}$, Ji Lin(林机)$^{1}$, and Hui-Jun Li(李慧军)$^{1\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/7/070302

⇦Chinese Physics Letters, 2023, Vol. 40, No. 7, Article code 070302⇨ Generation and Control of Shock Waves in Exciton-Polariton Condensates Jin-Ling Wang (王金玲)1, Wen Wen (文文)2, Ji Lin (林机)1, and Hui-Jun Li (李慧军)1* Affiliations 1Institute of Nonlinear Physics and Department of Physics, Zhejiang Normal University, Jinhua 321004, China 2College of Science, Hohai University, Nanjing 210098, China Received 4 April 2023; accepted manuscript online 7 June 2023; published online 10 July 2023 ^*Corresponding author. Email: hjli@zjnu.cn Citation Text: Wang J L, Wen W, Lin J et al. 2023 Chin. Phys. Lett. 40 070302 Abstract We propose a scheme to generate and control supersonic shock waves in a non-resonantly incoherent pumped exciton-polariton condensate, and different types of shock waves can be generated. Under conditions of different initial step waves, the ranges of parameters about various shock waves are determined by the initial incidence function and the cross-interaction between the polariton condensate and the reservoir. In addition, shock waves are successfully found by regulating the incoherent pump. In the case of low condensation rate from polariton to condensate, these results are similar to the classical nonlinear Schrödinger equation, and the effect of saturated nonlinearity resulted from cross interaction is equivalent to the self-interaction between polariton condensates. At high condensation rates, profiles of shock waves become symmetrical due to the saturated nonlinearity. Compared to the previous studies in which the shock wave can only be found in the system with repulsive self-interaction (defocusing nonlinearity), we not only discuss the shock wave in the exciton-polariton condensate system with the repulsive self-interaction, but also find the shock wave in the condensates system with attractive self-interaction. Our proposal may provide a simple way to generate and control shock waves in non-resonantly pumped exciton-polariton systems.

DOI:10.1088/0256-307X/40/7/070302 © 2023 Chinese Physics Society Article Text In semiconductor microcavities, excitons from semiconductor materials and cavity photons form polariton through strong coupling, and Bose–Einstein condensates (BECs) can be generated at room temperature, which are called exciton polariton condensates.[1-12] The polariton BEC is a non-thermodynamic equilibrium system.[13] At the same time, the system has the characteristics[14] of short lifetime, small effective mass,[15,16] strong interaction, coexistence of gain and loss,[17] and so on. Bright solitons,[18,19] gap solitons,[20] multipolar solitons,[21] vortex lattices,[22] spatial patterns,[23] and spinor dark soliton trains[24] were discovered in previous studies by introducing different pumps and external potentials. In addition, by introducing a composite pump field to achieve a balance between gain and loss, stable dark solitons can also be obtained.[25] Shock waves are characterized by the presence of jumps in the wavefront and usually involve two scales, the fast scale (phase) and the slow scale (amplitude, etc.).[26,27] Shock waves have been found in different fields of physics, such as plasmas,[28] optics,[29] water waves,[30] and polariton BECs.[13,31] Under weak dispersion approximation conditions, through the discussion of the Riemann problem[32-36] of the defocusing nonlinear Schrödinger equation (NLSE), the wave structure of the solution is composed of two kinds of rarefaction waves and two kinds of shock waves, and the parameter ranges of various shock waves are shown.[26,37-39] However, there is no report about the effects of saturated nonlinearity and normal dispersion on shock waves. Exciton polariton condensates can provide an ideal platform to study the shock waves controlled by the gain and loss, the dispersion, and the Kerr and saturated nonlinearity. One kind of shock waves has been obtained by theoretical research,[13] in which the dispersion term and the cross interaction are neglected, and an approximation perturbation method is adopted. A type of shock waves can also be acquired when an obstacle potential is introduced, and the cross interaction is also not considered.[40] In addition, one kind of shock waves has been experimentally reported,[31,41] in which resonantly coherent pumping is used. Therefore, there are three goals in our study. Firstly, various shock waves are found in a non-resonantly incoherent pumped exciton polariton condensate by controlling the initial input waves or the profiles of pump. Secondly, the effects of saturation nonlinearity and Kerr nonlinearity on shock waves are discussed, the novel symmetrical profiles of shock waves are found, and we extend the nonlinearity from the classical repulsive interaction to the attractive interaction to support the existence of shock waves. Finally, after determining the existence parameter range of various shock waves, the waveform of shock waves can be controlled by adjusting the relevant parameters. In this Letter, by introducing the initial step wave, various shock waves are found by the numerical evolution method, and the ranges of parameters existing are determined by a lot of numerical evolutions. We not only discuss the shock waves with repulsive interaction in the case of low condensation rate, but also find the novel shock waves with the symmetrical profiles at the high condensation rate. We can also seek out the shock waves by adjusting the profiles of incoherent pump. In addition, we extend the existence range of shock waves from the weak dispersion system to a normal dispersion system, from the repulsive interaction to the attractive interaction, from the Kerr nonlinearity to the saturated nonlinearity. Our proposal can pave a way to generate and control shock waves in a non-resonantly pumped exciton-polariton condensate system. In the following, firstly we introduce the model of study. Secondly, various shock waves and their properties are discussed. Finally, the main results are summarized. Model. Compared with the traditional BEC, the polariton BEC is a dynamically balanced but non-thermodynamic equilibrium system. Considering the mean-field theory, the motion of BEC is described by a coupling equation, which consists of the Gross–Pitaevskii equation of polariton field $\varPsi$ and a rate equation related to the density of the reservoir $n_{\scriptscriptstyle{\rm R}}$ \begin{align} &i\hbar \frac{\partial \varPsi}{\partial t}=\Big[-\!\frac{\hbar^2}{2m^{*}} \frac{\partial^{2}}{\partial x^{2}}\!+\!g_{\scriptscriptstyle{\rm C}}|\varPsi|^2\!+\!g_{\scriptscriptstyle{\rm R}}n_{\scriptscriptstyle{\rm R}}\!+\!i\frac{\hbar}{2}(Rn_{\scriptscriptstyle{\rm R}}\!-\!\gamma_{\scriptscriptstyle{\rm C}})\Big]\varPsi, \tag {1} \\ &\frac{\partial n_{\scriptscriptstyle{\rm R}}}{\partial t}=P_{i}(x)-(\gamma_{\scriptscriptstyle{\rm R}}+R|\varPsi|^{2})n_{\scriptscriptstyle{\rm R}}, \tag {2} \end{align} where $P_{i}(x)$ is the incoherent pump strength, $m^{*}$ is the effective mass of the polariton BEC, $R$ defines the condensation rate from polariton to condensate, and $\gamma_{\scriptscriptstyle{\rm C}}$ and $\gamma_{\scriptscriptstyle{\rm R}}$ are the loss rates of polariton and reservoir, respectively. Both $g_{\scriptscriptstyle{\rm R}}$ and $g_{\scriptscriptstyle{\rm C}}$ are interaction parameters. The former is between polaritons and the reservoir, and the latter is between polaritons. Here, the model corresponds to a (1+1)-dimensional nanowire system.[42] By introducing the characteristic time $\tau_{0}$, the spacial length $R_{x}$, the condensate density $\psi_{0}^{2}$, the reservoir density $n_{\scriptscriptstyle{\rm R}}^{0}$, and the intensity of pump $P_{0}$, we can present the dimensionless forms of Eqs. (1) and (2) as follows: \begin{align} &i\frac{\partial u}{\partial s}=-\frac{1}{2}\frac{\partial^{2} u}{\partial \xi^{2}}-\sigma_{1}|u|^2 u-\sigma_{2}n u +i(\sigma_{3}n-\sigma_{4})u, \tag {3} \\ &\frac{\partial n}{\partial s}=\sigma_{5}P(\xi)-\sigma_{6}(1+\sigma_{7}|u|^{2})n, \tag {4} \end{align} where $s=t/\tau_{0}$, $\xi=x/R_{x}$, $u=\varPsi/\psi_{0}$, $n=n_{\scriptscriptstyle{\rm R}}/n_{\scriptscriptstyle{\rm R}}^{0}$, and $P(\xi)=P_{i}/P_{0}$. The relevant parameters are defined as follows: \begin{eqnarray*} &&\sigma_{1}=-g_{\scriptscriptstyle{\rm C}}\psi_{0}^{2}\tau_{0}/\hbar,~~ \sigma_{2}=-g_{\scriptscriptstyle{\rm R}}n_{\scriptscriptstyle{\rm R}}^{0}\tau_{0}/\hbar, \\ &&\sigma_{3}=Rn_{\scriptscriptstyle{\rm R}}^{0}\tau_{0}/2,~~~~ \sigma_{4}=\gamma_{\scriptscriptstyle{\rm C}}\tau_{0}/2, \\ &&\sigma_{5}=\tau_{0}P_{0} / n_{\scriptscriptstyle{\rm R}}^{0},~~~~ \sigma_{6}=\gamma_{\scriptscriptstyle{\rm R}}\tau_{0}, \\ &&\sigma_{7}=R\psi_{0}^{2}/\gamma_{\scriptscriptstyle{\rm R}},~~~~ \tau_{0}=m^{*}R_{x}^{2}/\hbar. \\ \end{eqnarray*} We assume the constant reservoir density ${\partial n / \partial s=0}$ in Eq. (4). Then, substituting the result $n=P(\xi)/(1+\sigma_{7}|u|^{2})$ into Eq. (3) (here taking $\sigma_{5}=\sigma_{6}=1$), we obtain \begin{align} i\frac{\partial u}{\partial s}=\,&-\frac{1}{2}\frac{\partial^{2} u}{\partial \xi^{2}}- \sigma_{1}|u|^{2}u-\frac{\sigma_{2}P(\xi) u}{1+\sigma_{7}|u|^{2}}\notag\\ &+i\frac{\sigma_{3}P(\xi) u}{1+\sigma_{7}|u|^{2}}-i\sigma_{4} u, \tag {5} \end{align} where $\sigma_{1} < 0$ denotes the intensity of repulsive self-interaction (analogy with defocusing nonlinearity) between polariton condensates, $\sigma_{1}>0$ is the attractive self-interaction (analogy with focusing nonlinearity). We also call the self-interaction the Kerr nonlinearity. Noticeably, $\sigma_{2}$ denotes the intensity of the saturated nonlinearity which results from the cross interaction between polariton condensates and the reservoir; $\sigma_{3}$ and $\sigma_{4}$ denote the nonlinear gain and constant loss, respectively; and $\sigma_{7}$ is the condensation rate. When the value of $\sigma_{7}$ is very small, that is, at the low condensation rate, we can take the approximation $1/(1+\sigma_{7}|u|^{2})\approx 1-\sigma_{7}|u|^{2}$, and consider the uniform pump $P(\xi)=1$, then Eq. (5) becomes \begin{align} i\frac{\partial u}{\partial s}=\,&-\frac{1}{2}\frac{\partial^{2} u}{\partial \xi^{2}}-[\sigma_{2}+(\sigma_{1}-\sigma_{2}\sigma_{7})|u|^{2}]u\notag\\ &+i[(\sigma_{3}-\sigma_{4})-\sigma_{3}\sigma_{7}|u|^{2}]u. \tag {6} \end{align} After taking $\sigma_{2}=\sigma_{4}=0$, Eq. (6) will be reduced to the simple model given in Ref. [13]. Equation (5) permits the following plane wave solution, and its amplitude can be derived from the balance between gain and loss, \begin{align} &u=a e^{i(\phi_{0} \xi-\mu s)},~~~a=\sqrt{\frac{\sigma_{3} P -\sigma_{4}}{\sigma_{4} \sigma_{7}}},\notag\\ &\mu=\frac{1}{2} \phi_{0}^{2}-\sigma_{1} a^{2}-\frac{\sigma_{2} P}{1+\sigma_{7} a^{2}}, \tag {7} \end{align} where $\phi_{0}$ indicates the flow velocity of the condensate, the dispersion relation can be obtained from the analysis of the modulation stability of this plane wave ${u=[a+a_{1} e^{i(k \xi-w s)}+b_{1} e^{-i(k \xi-w s)}] e^{i(\phi_{0} \xi-\mu s)}}$, with $a_{1}, b_{1}\ll 1$. At the same time, the sound velocity of the system can also be acquired, \begin{align} w=\,&\frac{1}{2\,G}\Big[\pm \sqrt{G^{2} k^{4}\!+\!4 a^{2}G(P \sigma_{2} \sigma_{7}\!-\!\sigma_{1}G) k^{2}\!-\!4 P^{2} a^{4} \sigma_{3}^{2} \sigma_{7}^{2}}\notag\\ &+2 i(P\sigma_{3}-\sigma_{4}G)+2 k u_{0} G\Big], \tag {8} \end{align} \begin{align} C_{\scriptscriptstyle{\rm S}}=a \sqrt{-\sigma_{1}+\frac{P \sigma_{2} \sigma_{7}}{G}}, \tag {9} \end{align} where $w$ and $k$ represent the frequency and the wave number, respectively, $C_{\scriptscriptstyle{\rm S}}$ denotes sound velocity, $G=(\sigma_{7} a^{2}+1)^{2}$, and $P(\xi)$ is taken as the uniform distribution $P$. Various Shock Waves and Their Properties. In general, there are two schemes to generate shock waves. One is to choose a step wave as the initial condition, the other is to input a narrow wave superimposed onto a continuous wave background. It is well known that the resolution of breaking singularities in dispersive media occurs through generation of short-wavelength nonlinear oscillations. We discuss the first and second mechanisms in the following. In order to discuss the shock waves of Eq. (5), we select $u=\sqrt{\rho} e^{i \phi \xi}$, and the evolution initial functions are taken as \begin{align} {\rho(\xi, 0)=\left\{\begin{array}{cc}\rho_{0}, & \xi < 0, \\ 1, & \xi>0, \end{array} ~~\phi(\xi, 0)=\left\{\begin{array}{cc} \phi_{0}, & \xi < 0, \\ 0, & \xi>0. \end{array}\right.\right.} \tag {10} \end{align} The shock waves can be obtained by using the pseudospectral method combined with the fourth-order Runge–Kutta method.[43] In the following, we will find various shock waves and discuss their properties at the low and high condensation rates, that is, $\sigma_{7}=0.1$ and $\sigma_{7}=2$. Also, $P(\xi)=1$ in the following. Shock Waves at the Low Condensation Rate. We study the shock waves at the weak condensation rate, that is, $\sigma_{7}=0.1$. In this case, we take $\sigma_{3}=0.12$, $\sigma_{4}=0.1$ to balance the gain and loss. We consider the repulsive interaction $\sigma_{1}=-1$ and the cross interaction coefficient $\sigma_{2}=1$ firstly. In this case, sound velocity $C_{\scriptscriptstyle{\rm S}}=1.5$. In Fig. 1, various wave structures are obtained. The existence intervals of wave structures as functions of $\rho_{0}$ and $\phi_{0}$ are shown in Fig. 1(a). The different color shaded regions marked by I, II, III, IV, V are used to distinguish the types of waves, which means that the waveform can be controlled by the initial values $\rho_{0}$ and $\phi_{0}$. It is worth noting that there are five types when $\rho_{0}>1.1$ but one type disappears when $\rho_{0}\leq 1.1$.

Fig. 1. (a) Existence intervals of various wave structures as functions of $\rho_{0}$ and $\phi_{0}$. Here, $s$ and $v$ represent evolution time and wave velocity, respectively. Different color shaded regions are used to distinguish the types of waves. We use I, II, III, IV, V to denote the types of waves, and the blue pentagram denotes the coordinate point (1.1, 0), which is the demarcation point of four and five kinds of waves. (b)–(f) Profiles $|u|^{2}$ of various waves, with corresponding parameters marked by the black dots in (a). The red dashed lines represent the profiles of the initial incidence waves.

After fixing $\rho_{0}=2$, we plot the profiles of various waves by choosing $\phi_{0}=7,\,4,\, 0,\, -3,\,-7$, respectively. In Fig. 1(b), we show the waves composed of two shock waves, one shock wave is on the left side, and the other is on the right side. With the decreasing $\phi_{0}$, a slightly inclined platform appears between two shock waves as shown in Fig. 1(c). The average value of shock wave on both sides is lower than platform. With further decreasing $\phi_{0}$, as shown in Fig. 1(d), the oscillation of the shock wave becomes weaker, the left side wave becomes a smooth rarefaction wave with the small amplitude, the inclined platform decreases, its value is less than the average value of the left side, and lager than one of the right side, but the right side wave still retains the characteristics of shock wave. When $\phi_{0}=-3$, the oscillations on both sides become rarefaction waves, and the platform is lower than the average value of both sides waves, as shown in Fig. 1(e). When $\phi_{0}=-7$, the platform reaches zero as shown in Fig. 1(f). According to the numerical results, the shock wave and rarefaction wave can be distinguished based on whether the ratio of the maximum amplitude of the oscillating wave train to the average value reaches 0.1. In addition, we also find that the velocities of three kinds of shock waves in Figs. 1(b)–1(d) are supersonic by comparing their velocities $v$ and sound velocity $C_{\scriptscriptstyle{\rm S}}$. Compared to the results of the classical NLSE with Kerr nonlinearity,[26,37] there are the slopes in the platform in Figs. 1(c) and 1(d). By comparing with the initial input wave, the average value of $|u|^{2}$ on the right side increases significantly with the evolution, but it will always maintain the profile as shown in Figs. 1(c)–1(f). The main reasons are the gain and loss. On the left side, the gain equals the loss, but the gain is greater than the loss on the right hand side, so we can find that the right side of platform is higher than the left side of platform, and the average value of right side wave is higher than that of the initial incidence wave. In Fig. 1, the profiles of shock waves can be controlled by the amplitude $\rho_{0}$ and phase $\phi_{0}$ of initial incidence wave, which are similar to the ones of the classical NLSE,[26,37] except that the gain and loss result in the sightly inclined platform. However, there are the saturated nonlinearity terms in Eq. (5), their roles are also worth studying. After taking $\sigma_{1}=-1$ and $\rho_{0}=2$, we obtain the existence intervals of various waves as functions of $\sigma_{2}$ and $\phi_{0}$, as shown in Fig. 2(a). In the intervals of $\sigma_{2}\in [1,\,10]$, there are five kinds of waves including three supersonic shock waves. Choosing the parameters marked by the black points in Fig. 2(a), the profiles of shock waves are shown in Figs. 2(b)–2(d). In comparison of the profiles shown in Fig. 1 and the profiles of the classical NLSE,[26,37] we can find that they are all similar. Therefore, the roles of saturated nonlinearity can be transformed into Kerr nonlinearity in the case of low condensation rate. The conclusion has been proved by the derivation of Eq. (6), and the coefficient of Kerr nonlinearity becomes $\beta=\sigma_{1}-\sigma_{2}\sigma_{7}$. Furthermore, we also prove the equivalence of self-interaction Kerr nonlinearity and cross-interaction saturated nonlinearity by numerical evolution of Eq. (5). In Fig. 2(e), the existence intervals of various waves as functions of $\beta$ and $\phi_{0}$ are obtained. The red solid lines denote the boundary curves of different waves when $\sigma_{2}=1$, the value of $\beta$ will change with $\sigma_{1}$. When we take $\sigma_{1}=-0.1$, the blue dotted lines denote the boundary curves of different waves versus $\beta$ (or $\sigma_{2}$). That is, for the same $\beta$, we choose different $\sigma_{1}$ and $\sigma_{2}$ to satisfy $\beta=\sigma_{1}-\sigma_{2}\sigma_{7}$. We find that the two boundary curves are almost consistent, except for the curves between region I and II. These results prove that the approximation of Eq. (6) is reasonable and effective at the low condensation rate. Taking the parameters marked by the black points in Figs. 2(e), the profiles of shock waves are shown in Figs. 2(f)–2(h). These profiles are similar to those shown in Figs. 2(b)–2(d) except for some details. All these results prove that the saturated nonlinearity can be replaced by the Kerr nonlinearity at low condensation rate.

Fig. 2. (a) Existence intervals of various wave structures as functions of $\sigma_{2}$ and $\phi_{0}$. The different color shaded regions marked by I, II, III, IV, V are used to distinguish the types of waves. Here, $\rho_{0}=2$ and $\sigma_{1}=-1$. (b)–(d) Profiles of shock waves with $\phi_{0}=6,\,2,\,0$, respectively. Here, $\sigma_{1}=-1$, $\sigma_{2}=1$, $C_{\scriptscriptstyle{\rm S}}=1.5$. (e) Existence intervals of various wave structures as functions of $\beta$ and $\phi_{0}$. Here, $\beta=\sigma_{1}-\sigma_{2}\sigma_{7}$ is the coefficient of Kerr nonlinearity in Eq. (6). The red solid lines are the boundary curves of various waves when $\sigma_{2}=1$, the blue dotted lines are for $\sigma_{1}=-0.1$. (f)–(h) Profiles of shock waves with $\phi_{0}=6,\,2,\,0$, respectively. Here, $\sigma_{1}=-0.1$, $\sigma_{2}=10$, $C_{\scriptscriptstyle{\rm S}}=1.3$. The corresponding parameters of these profiles are marked by the black points in (a) and (e). The red dashed lines represent the profiles of initial incidence waves.

Fig. 3. (a) Existence intervals of various wave structures as functions of $\sigma_{2}$ and $\phi_{0}$. The different color shaded regions marked by I–V are used to distinguish the types of waves. Here, $\rho_{0}=2$ and $\sigma_{1}=0.1$. (b)–(d) Profiles of shock waves with $\phi_{0}=6,\,2,\,0$, respectively. Here, $\sigma_{2}=10$, $C_{\scriptscriptstyle{\rm S}}=1$. The corresponding parameters are marked by the black dots in (a). The red dotted lines represent the profiles of initial incidence waves.

In the related studies on shock waves, the KdV equation[44] and the defocusing NLSE[45] are mostly involved, however the shock wave in the focusing NLSE is rarely reported. From the above results, we can find that the Kerr nonlinearity and the saturated nonlinearity are cooperative in relationship. When $\sigma_{2}>0$, the saturated nonlinearity is equivalent to the nonlinearity of repulsive interaction. If the self-interaction is attractive, $\sigma_{1}>0$, then the relationship between Kerr and saturated nonlinearity will appear in competition. With the help of saturated nonlinearity, it is possible to find the shock wave in the attractive interaction system (the focusing nonlinear system). After taking $\sigma_{1}=0.1$ and $\rho_{0}=2$, we show the shock waves in Fig. 3. In this case, $\sigma_{2}$ should be lager than 1. In Fig. 3(a), the existence intervals of various waves as functions of $\sigma_{2}$ and $\phi_{0}$ are obtained. Comparing with the existence regions in Figs. 1 and 2, we can find that the regions marked by II–IV become narrower. The profiles of supersonic shock waves are also plotted in Figs. 3(b)–3(d). Shock Waves at High Condensation Rate. In previous studies, there is no report about the shock waves of the saturated NLSE due to its nonintegrability. However, as a unique nonlinearity, studying its impact on shock waves is also important. In the above text, we have obtained some results of saturated nonlinearity which can be reduced into Kerr nonlinearity. To further study the effect of saturated nonlinearity on the shock wave, we consider the case of high condensation rate. Next, we take $\sigma_{7}=2$, and we choose $\sigma_{3}=0.5$ and $\sigma_{4}=0.1$ to balance the gain and loss. In Fig. 4, we show the shock waves at the high condensation rate $\sigma_{7}=2$. The existence intervals of various waves as functions of $\rho_{0}$, $\phi_{0}$ and $\sigma_{2}$, $\phi_{0}$ are shown in Figs. 4(a) and 4(b), respectively. From these diagrams, we only find three kinds of supersonic shock waves and one kind of subsonic rarefaction wave, while the rarefaction wave with zero platform disappears. After choosing these parameters marked by the black points in Figs. 4(a) and 4(b), the profiles of various waves are obtained as shown in Figs. 4(c)–4(f). We find that the profiles of the first and second shock waves become even symmetric structures as shown in Figs. 4(c) and 4(d), the profile of the third shock wave in Fig. 4(e) becomes odd symmetric form, and the small oscillations on both sides of the rarefaction wave will disappear and its profiles are even symmetric as shown in Fig. 4(f).

Fig. 4. [(a), (b)] Existence intervals of various wave structures as functions of $\rho_{0}$, $\phi_{0}$ and $\sigma_{2}$, $\phi_{0}$, respectively. The different color shaded regions marked by I–IV are used to distinguish the types of waves. Here, $\sigma_{1}=-1$, $\sigma_{2}=1$ in (a) and $\sigma_{1}=-1$, $\rho_{0}=2$ in (b). (c)–(f) Profiles of shock waves with $\phi_{0}=6.5,\,4,\,0,\, -3$, respectively. Here, $\rho_{0}=2$, $\sigma_{2}=1$, $C_{\scriptscriptstyle{\rm S}}=1.5$. The corresponding parameters of these profiles are marked by the black dots in (a) and (b).

Compared to the results in Fig. 1, the shock waves in Fig. 4 have several characteristics: (1) On both the sides of shock waves, there are more waves in the oscillating structures. (2) The profiles have obvious symmetry for every kind of waves. (3) The inclined platform in the middle of two waves disappears as shown in Figs. 4(d) and 4(e). This is the first time to report the shock wave with symmetric structure.

Fig. 5. The projection of the evolution results. (a) Shock wave with $\phi_{0}=3$. (b) Shock wave with $\phi_{0}=1$. (c) Shock wave with $\phi_{0}=0$. Here, $\sigma_{1}=0$, $\sigma_{2}=1$, $\rho_{0}=2$, $C_{\scriptscriptstyle{\rm S}}=0.4$. (d)–(f) Profiles of shock waves with $\phi_{0}=3,\,1,\,0$, for $\sigma_{1}=0$, $\sigma_{2}=0$, $\rho_{0}=2$. In these insets, the profiles of shock waves at a specific time $s$ is shown.

Due to the high condensation rate, the above results are caused by the saturated nonlinearity and Kerr nonlinearity. To explore the contribution of saturated nonlinearity to symmetric shock waves, we show these three kinds of shock waves by taking $\sigma_{1}=0$ or $\sigma_{1}=\sigma_{2}=0$, respectively. At this time, the velocities of these three shock waves are still supersonic. In Fig. 5, the projections of their evolutions are shown. In these insets, the profiles of shock waves are also plotted. From these insets, we find that the symmetry of shock waves in Fig. 5 is better than that in Fig. 4. Thus, it is obvious that the symmetry of shock waves is caused by the saturated nonlinear gain. We also find that the third shock wave composed of one shock wave and one rarefaction wave becomes two shock waves as shown in Figs. 5(c) and 5(f). In addition, we also calculate some results of $\sigma_{1}>0$, and find that the attractive interaction is not conducive to the generation of shock waves. Shock Waves by Superimposing a Narrow Wave on the Uniform Background. So far, the first mechanism of generation shock waves has been discussed. Next, we discuss how to generate shock waves by the second mechanism. We take the initial profiles of condensates $u$ as $2-e^{-(\frac{\xi}{15})^{2}}$, $2-e^{-(4\xi)^{2}}$, and $2-\tanh (\frac{\xi}{3})$, respectively, then three kinds of different shock waves are found as shown in Figs. 6(a)–6(c). From Figs. 6(a) and 6(b), with the decreasing width, the platform narrows and lowers, and the wavelength becomes shorter and the wave number becomes larger, and there are two symmetrical shock waves under these two cases. However, after taking the asymmetrical initial wave, we obtain the shock wave shown in Figs. 6(c), which is similar to the dispersion shock wave of KdV equations.[37] It is well known that the pump is more convenient to adjust in the experiment. We choose the nonuniform pump as $2-e^{-(4\xi)^{2}}$ to generate the shock wave. The result is shown in Fig. 6(d). We discover that the narrower the pump width is, the easier the shock wave generates; and the shock waves are also supersonic.

Fig. 6. (a)–(c) Profiles of shock waves by taking $u(s=0,\xi)=2-e^{-(\frac{\xi}{15})^{2}}, 2-e^{-(4\xi)^{2}}, 2-\tanh (\frac{\xi}{3})$ respectively. (d) Profile of the shock wave by taking $P=2-e^{-(4\xi)^{2}}$ and $u(s=0,\xi)=1$. Here, $\sigma_{1}=-1$, $\sigma_{2}=1$, $\sigma_{3}=0.12$, $\sigma_{4}=0.1$, $\sigma_{7}=0.1$.

Fig. 7. (a)–(c) Profiles of shock waves by taking the same initial values as in Figs. 1(b)–1(d), respectively. (d)–(f) Profiles of shock waves by taking the same initial values as in Figs. 6(b)–6(d), respectively. Here, $s$ and $v$ represent evolution time and wave velocity, respectively. Equations (3) and (4) are chosen for evolution.

Finally, we also calculate all the results according to Eqs. (3) and (4) by taking the same initial values as given above. In Figs. 7(a)–7(c), we obtain the shock waves by taking the same initial values as given in Figs. 1(b)–1(d) and in Figs. 7(d)–7(f), we obtain the shock waves by taking the same initial values as in Figs. 6(b)–6(d). In comparison, we find that these results are similar except for some minor details. Thus, the approximation $\partial n /\partial s=0$ is reasonable. However, the approximation is conducive to our understanding of the reasons for the above results, so we still retain the approximation model and the results in this study. In summary, we have selected the piecewise constant or superimposing a narrow wave on the uniform background as the initial incidence wave to generate the shock wave in the exciton-polariton condensate, and found the parameters region of various waves, which can be used to control shock waves. When the condensation rate is small, we have obtained three kinds of shock waves, which are similar with these results in the classical NLSE. In this case, we have found the inclined platform in the middle of shock waves, and the saturated nonlinearity has been replaced by the Kerr nonlinearity. Furthermore, we have also found the shock waves in the case of attractive interaction. At the high condensation rate, we have obtained the shock waves with the symmetrical structures caused by the saturated nonlinearity. In addition, the narrow waves superimposed on the uniform background are chosen as the initial condensate or incoherent pump, the symmetrical and asymmetrical shock waves are generated. Comparing with previous studies, we not only have obtained various novel shock waves in the incoherent pumped polariton condensates, but also have extended the existence range of shock waves, from the repulsive interaction to attractive interaction, from Kerr nonlinearity to saturated nonlinearity, from weak dispersion systems to normal dispersion systems (the coefficient of $\partial^{2}/\partial \xi^{2}$). Compared to the weak dispersion system, shock waves have a spatial larger scale in a normal dispersion system. The present results may be useful for understanding physical properties of condensates out of equilibrium and for guiding experimental work on condensate shock waves, which may have potential applications in polariton condensates for information storages and processing or quantum simulators. Acknowledgments. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. LZ22A050002), and the National Natural Science Foundation of China (Grant Nos. 11835011 and 12074343). 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