3.1. Poincare–Lightill–Kuo (PLK) perturbation method

To study the collisions between solitary waves, we adopt the PLK perturbation method and introduce the following coordinate transformations:

(3.1)$$\begin{gather} \xi = \epsilon \left( {x + {k_1}y - {\upsilon _{s{\rm{1}}}}t} \right) + {\epsilon ^{2}}{P_0}\left( {\eta ,\tau } \right) + {\epsilon ^{3}}{P_1}\left( {\xi ,\eta ,\tau } \right) + \cdots, \end{gather}$$

(3.2)$$\begin{gather}\eta = \epsilon \left( {x + {k_2}y - {\upsilon _{s{\rm{2}}}}t} \right) + {\epsilon ^{2}}{Q_0}\left( {\xi ,\tau } \right) + {\epsilon ^{3}}{Q_1}\left( {\xi ,\eta ,\tau } \right) + \cdots, \end{gather}$$

(3.3)$$\begin{gather}\tau = {\epsilon ^{3}}t, \end{gather}$$

where $\epsilon$ is a small parameter, $\xi$ and $\eta$ represent the trajectories of two solitary waves, ${v_{s1}}$ and ${v_{s2}}$ are the velocities of solitary waves propagating in different directions, and ${k_1}$ and ${k_2}$ are the wave numbers in the $y$ direction of the first and second solitary waves. Here, ${P_0}( {\eta,\tau } )$ and ${Q_0}( {\xi,\tau } )$ are two quantities which will be determined later, and ${P_1}( {\xi,\eta,\tau } )$ and ${Q_1}( {\xi,\eta,\tau } )$ are another two quantities. We expand the physical quantities as follows:

(3.4)$$\begin{gather} n = 1 + {\epsilon ^{2}}{n_1}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{3}}{n_2}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{4}}{n_3}\left( {\xi ,\eta ,\tau } \right) + \cdots, \end{gather}$$

(3.5)$$\begin{gather}u = {\epsilon ^{2}}{u_1}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{3}}{u_2}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{4}}{u_3}\left( {\xi ,\eta ,\tau } \right) + \cdots, \end{gather}$$

(3.6)$$\begin{gather}v = {\epsilon ^{2}}{v_1}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{3}}{v_2}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{4}}{v_3}\left( {\xi ,\eta ,\tau } \right) + \cdots, \end{gather}$$

(3.7)$$\begin{gather}\phi = {\epsilon ^{2}}{\phi _1}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{3}}{\phi _2}\left( {\xi ,\eta ,\tau } \right) + {\epsilon ^{4}}{\phi _3}\left( {\xi ,\eta ,\tau } \right) + \cdots. \end{gather}$$

3.2. Korteweg de Vries (KdV) equations and their solution

Substituting (3.1)–(3.7) into (2.1)–(2.4), we have the following equations: ${\phi _1} = {\phi _\xi }( {\xi,\tau } ) + {\phi _\eta }( {\eta,\tau } )$, ${n_1} = {n_\xi }( {\xi,\tau } ) + {n_\eta }( {\eta,\tau } )$, ${u_1} = {u_\xi }( {\xi,\tau } ) + {u_\eta }( {\eta,\tau } )$, ${v_1} = {v_\xi }( {\xi,\tau } ) + {v_\eta }( {\eta,\tau } )$, ${v_\xi } = ({{{k_1}{\upsilon _{s{\rm {1}}}}}}/({{{\gamma '}( {1 + k_1^{2}} ) - {\upsilon _{s{\rm {1}}}^{2}}}})){\phi _\xi }$, ${u_\xi } = ({{{\upsilon _{s{\rm {1}}}}}}/({{{\gamma '}( {1 + k_1^{2}} ) - {\upsilon _{s{\rm {1}}}^{2}}}})){\phi _\xi }$, ${n_\xi } = (({{1 + {k_1}^{2}}})/ ({{{\gamma '}( {1 + k_1^{2}} ) - {\upsilon _{s{\rm {1}}}^{2}}}})){\phi _\xi }$, ${v_\eta } = ({{{k_2}{\upsilon _{s{\rm {2}}}}}}/({{{\gamma '}( {1 + k_2^{2}} ) - {\upsilon _{s{\rm {2}}}^{2}}}})/{\phi _\eta }$, ${u_\eta } = ({{{\upsilon _{s{\rm {2}}}}}}/({\gamma '}( {1 + k_2^{2}} ) - \upsilon _{s{\rm {2}}}^{2})){\phi _\eta }$, ${n_\eta } = (({{1 + {k_2}^{2}}})/({{{\gamma '}( {1 + k_2^{2}} ) - {\upsilon _{s{\rm {2}}}^{2}}}})){\phi _\eta }$, ${n_1} = - s( {\nu \beta + \mu } ){\phi _1}$, ${\upsilon _{s{\rm {1}}}^{2}} = ( {1 + k_1^{2}} ) ( {{\gamma '} + {1}/{Q}} )$ and ${\upsilon _{s{\rm {2}}}^{2}} = ( {1 + k_2^{2}} )( {{\gamma '} + {1}/{Q}} )$, where $Q = ( {\nu \beta + \mu } )s$. The unknown functions ${\phi _\xi }( {\xi,\tau } )$ and ${\phi _\eta }( {\eta, \tau } )$ will be given later. We find from the above equations that there are two waves propagating in two different directions of $\xi$ and $\eta$.

From the higher-order approximation, we have

(3.8)$$\begin{gather} \frac{{\partial {\phi _\xi }}}{{\partial \tau }} + b{\phi _\xi }\frac{{\partial {\phi _\xi }}}{{\partial \xi }} + c\frac{{{\partial ^{3}}{\phi _\xi }}}{{\partial {\xi ^{3}}}} = 0, \end{gather}$$

(3.9)$$\begin{gather}\frac{{\partial {\phi _\eta }}}{{\partial \tau }} + b'{\phi _\eta }\frac{{\partial {\phi _\eta }}}{{\partial \eta }} + c'\frac{{{\partial ^{3}}{\phi _\eta }}}{{\partial {\eta ^{3}}}} = 0, \end{gather}$$

where

(3.10)\begin{equation} \left.\begin{gathered} b ={-} \frac{{1 + k_1^{2}}}{{2\upsilon _{s{\rm{1}}}^{\rm{2}}{Q^{2}}}} \left[ {\upsilon _{s1} ( {\nu {\beta ^{\rm{2}}} - \mu } ){s^{2}}}+\frac{{2\upsilon _{s{\rm{1}}}^{\rm{3}}{Q^{3}} + {\upsilon _{s{\rm{1}}}}{Q^{2}}( {1 + k_1^{2}} )}}{{( {1 + k_1^{2}} )}} \right],\\ c = \frac{{1 + k_1^{2}}}{{2{\upsilon _{s1}}{Q^{2}}}},\quad c' = \frac{{1 + k_2^{2}}}{{2{\upsilon _{s2}}{Q^{2}}}},\\ b'={-} \frac{{1 + k_2^{2}}}{{{\rm{2}}\upsilon _{s2}^{2}{Q^{\rm{2}}}}}\left[{\upsilon _{s2} ( {\nu {\beta ^{\rm{2}}} - \mu } ){s^{2}}} + \frac{{2\upsilon _{s2}^{3}{Q^{3}} + {\upsilon _{s2}}{Q^{2}}( {1 + k_2^{2}} )}}{{( {1 + k_2^{2}} )}} \right]. \end{gathered}\right\} \end{equation}

Equations (3.8) and (3.9) are two KdV equations that describe two solitary waves propagating in the $\xi$ and $\eta$ directions. The KdV equations have many solutions. One solitary wave solutions of both (3.8) and (3.9) are as follows:

(3.11)$$\begin{gather} {\phi _\xi } = \frac{{3{u_{0\xi }}}}{b}{{{\rm sech}} ^{2}}\left[ {{{\left( {\frac{{{u_{0\xi }}}}{{4c}}} \right)}^{1/2}}\left( {\xi - {u_{0\xi }}\tau } \right)} \right], \end{gather}$$

(3.12)$$\begin{gather}{\phi _\eta } = \frac{{3{u_{0\eta }}}}{{b'}}{{{\rm sech}}^{2}}\left[ {{{\left( {\frac{{{u_{0\eta }}}}{{4c'}}} \right)}^{1/2}}\left( {\eta - {u_{0\eta }}\tau } \right)} \right]. \end{gather}$$

The amplitudes and the widths of two solitary waves are ${\phi _{m\xi }} = {{\textrm {{3}}{u_{0\xi }}}}/{\textrm {{b}}}$, ${\phi _{m\eta }} = {{\textrm {{3}}{u_{0\eta }}}}/{{\textrm {{b'}}}}$, ${W_\xi } = {( {{{4c}}/{{{u_{0\xi }}}}} )^{1/2}}$, ${W_\eta } = {( {{{4c'}}/{{{u_{0\eta }}}}} )^{1/2}}$, where ${u_{0\xi }}$ and ${u_{0\eta }}$ are two arbitrary constants.

3.3. The solitary wave solution in the experimental coordinate

To compare out results with the experimental ones, we let all the physical quantities be in the experimental coordinate. Then, one solitary wave solutions of the incident wave, reflected wave and transmitted wave in the experimental coordinate are as follows:

(3.13)$$\begin{gather} {u^{I}} = u_m^{I}{{{\rm sech}}^{2}}\frac{{X - {V^{I}}t + \epsilon P\left( {\eta ,\tau } \right)}}{{{W^{I}}}}, \end{gather}$$

(3.14)$$\begin{gather}{u^{R}} = u_m^{R}{{{\rm sech}}^{2}}\frac{{X' - {V^{R}}t + \epsilon Q\left( {\xi ,\tau } \right)}}{{{W^{R}}}}, \end{gather}$$

(3.15)$$\begin{gather}{u^{T}} = u_m^{T}{{{\rm sech}}^{2}}\frac{{X - {V^{T}}t + \epsilon P\left( {\eta ,\tau } \right)}}{{{W^{T}}}}, \end{gather}$$

where $u_m^{\gamma } = u_{m0}^{\gamma } {e^{( { - ({{{\nu _d}}}/{2})t} )}}$ (Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021), where $\gamma =I, R, T$ which represent incident, reflected and transmitted waves, respectively, $u_{m0}^{\gamma }$ is the initial wave amplitude, ${V^{\gamma } } = {C_d}^ \mp + {{{u_m}^{\gamma } }}/{2}$ and ${W^{\gamma } } = ( {1 - {{{u_m}^{\gamma } }}/{{2{C_d}^ \mp }}} )\sqrt {{{4{C_d}^ \mp }}/{{{u_m}^{\gamma } }}}\, {\lambda _{Dd}}^ \mp$. Notice that the amplitude of the solitary wave decays exponentially due to the viscosity of the dusty plasma. For the low viscosity of a dusty plasma, ${\nu _d} = 0$, i.e. the amplitude of the solitary wave remains constant. It seems that the propagation speed of the solitary wave increases with the increase of the amplitude of the solitary wave, while the width of the solitary wave decreases with the increase of the amplitude of the solitary wave. Moreover, we have the following equations in the experimental coordinates: ${{{n^{I}}}}/{{{n_{d0}}^ - }} = {{{u^{I}}}}/{{{C_d}^ - }} = {{{v^{I}}}}/{{{C_d}^ - }} = - {{{\phi ^{I}}}}/{{( {{{{T_{\textrm {eff}}}^ - } / e}} )}}$, ${{{n^{R}}}}/{{{n_{d0}}^ - }} = - {{{u^{R}}}}/{{{C_d}^ - }} = {{{v^{R}}}}/{{{C_d}^ - }} = - {{{\phi ^{R}}}}/{{( {{{{T_{\textrm {eff}}}^ - } / e}} )}}$, ${{{n^{T}}}}/{{{n_{d0}}^ + }} = {{{u^{T}}}}/{{{C_d}^ + }} = {{{v^{T}}}}/{{{C_d}^ + }} = - {{{\phi ^{T}}}}/{{( {{{{T_{\textrm {eff}}}^ + } / e}} )}}$.

4.1. An interface of the dusty plasma

Recently, the propagation of solitary waves in a dusty plasma system composed of different dust particles has been studied theoretically and experimentally. In the experiment, there are two different dusty plasma in two different regions (Sun et al. Reference Sun, Schwabe, Thomas, Lipaev, Molotkov, Fortov, Feng, Lin, Zhang and Guo2018; Du et al. Reference Du, Nosenko, Thomas, Lin, Morfill and Ivlev2019; Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021). The reflection and transmission of a solitary wave at the interface are studied experimentally and theoretically.

Based on these experiments (Sun et al. Reference Sun, Schwabe, Thomas, Lipaev, Molotkov, Fortov, Feng, Lin, Zhang and Guo2018; Du et al. Reference Du, Nosenko, Thomas, Lin, Morfill and Ivlev2019; Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021), we now consider a dusty plasma which is made up of two regions. The dusty plasma with smaller dust particles is in the region $x < 0$, while the dusty plasma with larger dust particles is in the region $x > 0$, see figure 1. Suppose that there is an incident solitary wave in the region $x < 0$ initially. As it travels to the interface $x = 0$, it will be reflected and transmitted at the interface. Therefore, we must consider both reflected and incident waves in the region $x < 0$, whereas we only need to consider transmitted waves in the region $x > 0$. For simplicity, we use superscripts ‘$I$’, ‘$R$’ and ‘$T$’ to represent incident, reflected and transmitted waves, respectively.

4.2. Evolution of solitary waves from an initial condition

For the sake of convenience, we assume that the incident wave is a single solitary wave and try to know the reflected wave and the transmitted wave. For this reason, we use a previous result (Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021). It is well known that the number of solitary waves and their amplitudes can be given from the standard KdV equation and its ‘initial conditions’.

For the standard KdV equation: ${{\partial \varphi }}/{{\partial t}} + 6\varphi ({{\partial \varphi }}/{{\partial \xi }}) + {{{\partial ^{3}}\varphi }}/{{\partial {\xi ^{3}}}} = 0$ and its ‘initial conditions’: $\varphi |_{t = 0} = - ({A}/{{{L_0}^{2}}}){{\textrm {sech}}^{2}}({\xi }/{{{L_0}}})$, where ${L_0}$ is the characteristic width of the initial pulse, ${{{A_0}}}/{{{L_0}^{2}}}$ is the characteristic amplitude of the initial pulse. The number $N$ of generated solitary waves and their wave amplitudes for each solitary wave can be given by the following equations: $\sqrt {{A_0} + \frac {1}{4}} + \frac {1}{2} - N > 0$, $2{\left( {\sqrt {{A_0} + \frac {1}{4}} + \frac {1}{2} - j} \right)^{2}}{L_0}^{-2}$, where $j=1,2,\ldots,N$. The number $N$ is the maximum integer.

The three KdV equations for the incident wave, reflected wave and transmitted wave can be rewritten as follows:

(4.1)$$\begin{gather} \frac{{\partial \phi _\xi ^{I}}}{{\partial \tau }} + {b^ - }\phi _\xi ^{I}\frac{{\partial \phi _\xi ^{I}}}{{\partial \xi }} + {c^ - }\frac{{{\partial ^{3}}\phi _\xi ^{I}}}{{\partial {\xi ^{3}}}} = 0, \end{gather}$$

(4.2)$$\begin{gather}\frac{{\partial \phi _\eta ^{R}}}{{\partial \tau }} + {\left( {b'} \right)^{ }- }\phi _\eta ^{R}\frac{{\partial \phi _\eta ^{R}}}{{\partial \eta }} + {\left( {c'} \right)^ - }\frac{{{\partial ^{3}}\phi _\eta ^{R}}}{{\partial {\eta ^{3}}}} = 0, \end{gather}$$

(4.3)$$\begin{gather}\frac{{\partial \phi _\xi ^{T}}}{{\partial \tau }} + {b^ + }\phi _\xi ^{T}\frac{{\partial \phi _\xi ^{T}}}{{\partial \xi }} + {c^ + }\frac{{{\partial ^{3}}\phi _\xi ^{T}}}{{\partial {\xi ^{3}}}} = 0, \end{gather}$$

where superscript ‘$-$’ represents the values in the region $x < 0$, while superscript ‘$+$’ stands for the values in the region$x > 0$, where ${\phi ^{\gamma } } = {\epsilon ^{2}}{\phi _1^{\gamma }}$, $\gamma = I,R,T$, ${\phi ^{\gamma } }$ represents the electrostatic potential of the wave $\gamma$ in the experimental coordinate, and the coefficients of the KdV equation are

(4.4)\begin{equation} \left.\begin{gathered} b ={-} \frac{{1 + k_1^{2}}}{{2\upsilon _{s{\rm{1}}}^{\rm{2}}{Q^{2}}}} \left[ {\upsilon _{s1} ( {\nu {\beta ^{\rm{2}}} - \mu } ){s^{2}}}+\frac{{2\upsilon _{s{\rm{1}}}^{\rm{3}}{Q^{3}} + {\upsilon _{s{\rm{1}}}}{Q^{2}}( {1 + k_1^{2}} )}}{{( {1 + k_1^{2}} )}} \right],\\ c = \frac{{1 + k_1^{2}}}{{2{\upsilon _{s1}}{Q^{2}}}},\quad c' = \frac{{1 + k_2^{2}}}{{2{\upsilon _{s2}}{Q^{2}}}},\\ b'={-} \frac{{1 + k_2^{2}}}{{{\rm{2}}\upsilon _{s2}^{2}{Q^{\rm{2}}}}}\left[{\upsilon _{s2} ( {\nu {\beta ^{\rm{2}}} - \mu } ){s^{2}}} + \frac{{2\upsilon _{s2}^{3}{Q^{3}} + {\upsilon _{s2}}{Q^{2}}( {1 + k_2^{2}} )}}{{( {1 + k_2^{2}} )}} \right]. \end{gathered}\right\} \end{equation}

5.1. Continuity conditions at the interface

Neglecting higher-order quantities, we give the continuity conditions at the interface $x = 0$. The continuous conditions at the interface are electrostatic-potential and momentum:

(5.1)$$\begin{gather} {\left. {\left[ {{\phi ^{I}} + {\phi ^{R}}} \right]} \right|_{x = 0}} = {\left. {{\phi ^{T}}} \right|_{x = 0}}, \end{gather}$$

(5.2)$$\begin{gather}{{{m}}_d}^ - {n_{d0}}^ - \left.\left[ {{u^{I}} + {u^{R}}} \right]\right| _{x = 0}= \left. {{{m}}_d}^ + {n_{d0}}^ + {u^{T}}\right| _{x = 0}, \end{gather}$$

(5.3)$$\begin{gather}{{{m}}_d}^ - {n_{d0}}^ - \left.\left[ {{v^{I}} + {v^{R}}} \right]\right| _{x = 0} = \left.{{{m}}_d}^ + {n_{d0}}^ + {v^{T}}\right| _{x = 0} , \end{gather}$$

where momentum is a vector; therefore, there are two components of momentum in the $x$ and $y$ directions. Equations (5.1), (5.2) and (5.3) are in the experimental coordinate. We have the ‘initial conditions’ of the reflected and the transmitted waves from (5.1), (5.2) and (5.3):

(5.4)$$\begin{gather} \left.{\phi ^{T}}\right| _{x = 0} = \frac{2}{{1 + \chi }}\left.{\phi ^{I}}\right|_{x = 0}, \end{gather}$$

(5.5)$$\begin{gather}\left.{\phi ^{R}}\right| _{x = 0} = \frac{{1 - \chi }}{{1 + \chi }}\left.{\phi^{I}}\right| _{x = 0}, \end{gather}$$

where $\chi = {{{{{m}}_d}^ + {n_{d0}}^ + {C_D}^ + {T_{\textrm {eff}}}^ - }}/{{{{{m}}_d}^ - {n_{d0}}^ - {C_D}^ - {T_{\textrm {eff}}}^ + }}$. We assume that the incident wave is a single solitary wave given by the standard KdV equation and the following initial conditions:

(5.6)$$\begin{gather} \frac{{\partial {\varphi ^{{{I}}}}}}{{\partial t}} + 6{\varphi ^{{{I}}}}\frac{{\partial {\varphi ^{{{I}}}}}}{{\partial \xi }} + \frac{{{\partial ^{3}}{\varphi ^{{{I}}}}}}{{\partial {\xi ^{3}}}} = 0, \end{gather}$$

(5.7)$$\begin{gather}{\varphi ^{\left( I \right)}}_{\left( {{\xi}, \tau } \right)} ={-} \frac{2}{{{W^{2}}}}{{{\rm sech}}^{2}}\left( {\frac{\xi }{W} - \frac{{4\tau }}{{{W^{2}}}}} \right), \end{gather}$$

where $W$ is the wave width. When the incident solitary wave propagates from the region $x < 0$ to the interface $x = 0$, it will be reflected and transmitted. The equivalent ‘initial conditions’ of reflected wave and transmitted wave can be given by the boundary conditions of (5.4) and (5.5):

(5.8)$$\begin{gather} {\varphi ^{\left( T \right)}}_{\left( {{t_T},0} \right)} = \frac{{{A_T}}}{{{{\left( {{L_T}} \right)}^{2}}}}{{{\rm sech}} ^{2}}\left( {\frac{{{t_T}}}{{{L_T}}},0} \right), \end{gather}$$

(5.9)$$\begin{gather}{\varphi ^{\left( R \right)}}_{\left( {{t_R},0} \right)} = \frac{{{A_R}}}{{{{\left( {{L_R}} \right)}^{2}}}}{{{\rm sech}} ^{2}}\left( {\frac{{{t_R}}}{{{L_R}}},0} \right), \end{gather}$$

where ${A_T} = 2{A_0}({1}/{{(1 + \chi )}})({{{T_{\textrm {eff}}}^ - }}/{{{T_{\textrm {eff}}}^ + }})$, ${L_T} = {L_0}$, ${A_R} = {A_0}({{(1 - \chi )}}/{{(1 + \chi )}})$, ${L_R} = L_0$, ${A_0} = 2$. The ‘initial conditions’ of the reflected and transmitted waves can be used to determine the number of reflected and transmitted solitary waves and the amplitudes of each reflected solitary wave and each transmitted solitary wave generated by the incident solitary wave after a period of evolution.

It seems that the number of reflected and transmitted solitary waves is in agreement in the special case that the propagation direction of the incident wave is parallel to the normal direction of the interface (Hong et al. Reference Hong, Sun, Schwabe, Du and Duan2021).

5.2. Dependence of the transmitted wave angle $\alpha$ on the incident wave angle $\theta$

In this section, we will discuss how the quantity $\alpha$ depends on the quantity $\theta$ when the incident wave hits the interface arbitrarily. We assume that the angle of the incident solitary wave is $\theta$ and the angle of the reflected solitary wave is equal to that of the incident solitary wave. However, the angle of the transmitted solitary wave is usually different from that of the incident solitary wave. Therefore, we assume that the angle of the transmitted solitary wave is $\alpha$. Notice from figure 1 that ${v^{I}} = {u^{I}}ctg\theta$, ${v^{R}} = - {u^{R}}ctg\theta$ and ${v^{T}} = {u^{T}}ctg\alpha$. Then we have

(5.10)\begin{equation} \frac{{\tan \alpha }}{{\tan \theta }} = \chi, \end{equation}

where $\chi = {{{{{m}}_d}^ + {n_{d0}}^ + {C_d}^ + {T_{\textrm {eff}}}^ - }}/{{{{{m}}_d}^ - {n_{d0}}^ - {C_d}^ - {T_{\textrm {eff}}}^ + }}$, and (5.10) is the relation between $\theta$ and $\alpha$. To know the dependence of $\alpha$ on $\theta$, we use the result (Wang et al. Reference Wang, Schwan, Hsu, Grün and Horányi2016): ${Q_d} = {k_q}^{\prime } \cdot {m_d}^{{2}/{3}}$, where ${k_q}^{\prime }$, is a constant, and it is estimated that ${k_q}^{\prime } = - 3.204 \times {10^{ - 6}}$. It is easy to be verified as follows. The charge of the dust particles is generally proportional to the square of their radius (Wang et al. Reference Wang, Schwan, Hsu, Grün and Horányi2016), while the mass of the dust particles is proportional to the cubic power of their radius.

Notice that the dependence of $\alpha$ on $\theta$ is equivalent to the dependence of the parameter $\chi$ on the parameters of the dusty plasma, such as the number density, the mass, the temperature, the charge of dust particles, as well as the number density, and the temperatures of both electrons and ions.

Figure 2 shows the dependence of $\chi$ on the mass ${m^ + }$ and ${m^ - }$ of dust particles in the region $x > 0$ and $x < 0$, respectively, for the three-dimensional case, where ${n_{d0}}^ + = {n_{d0}}^ - = 10 \times {10^{9}}\ {\textrm {m}^{ - 3}}$, ${n_{e0}} = 1.0 \times {10^{14}}\ {\textrm {m}^{ - 3}}$, ${T_e} = 5\ \textrm {eV}$, ${T_i} = 0.1\ \textrm {eV}$ and ${T_d} = 298\ \textrm {K}$ (Du et al. Reference Du, Sütterlin, Jiang, Räth, Ivlev, Khrapak, Schwabe, Thomas, Fortov and Lipaev2012). Notice from figure 2 that $\chi =1$ when ${m^ + }={m^ - }$, i.e. if the masses of the dust particles in two different regions are same, $\alpha =\theta$. It is noted from figure 2 that $\alpha$, or $\chi$, increases as the mass of the dust particles in the region $x>0$ increases, while it decreases as the mass of the dust particles in the region $x<0$ increases.

Figure 2. In the three-dimensional case, the influence of dust particle mass ${m^ + }$ and ${m^ - }$ on parameter $\chi$, where the orange line is $\chi = \frac {1}{4}$, $\tan \alpha =\frac {1}{4} \tan \theta$, the green line is $\chi = \frac {1}{2}$, $\tan \alpha =\frac {1}{2} \tan \theta$, the black line is $\chi = 1$, $\tan \alpha = \tan \theta$, the purple line is $\chi = \frac {3}{2}$, $\tan \alpha = \frac {3}{2} \tan \theta$, and the other system parameters are ${n_{d0}}^ + = {n_{d0}}^ - = 10 \times {10^{9}}\ {\textrm {m}^{ - 3}}$, ${n_{e0}} = 1.0 \times {10^{14}}\ {\textrm {m}^{ - 3}}$, ${T_e} = 5\ \textrm {eV}$, ${T_i} = 0.1\ \textrm {eV}$ and ${T_d} = 298\ \textrm {K}$.

Figure 3 shows the dependence of $\chi$ on the number density ${n_{d0}}^ +$ and ${n_{d0}}^ -$ of the dust particles in the region $x > 0$ and $x < 0$, respectively, for the three dimensional case, where ${m^ + } = {m^ - } = 5.0 \times {10^{ - 15}}\ \textrm {kg}$, ${n_{e0}} = 1.0 \times {10^{14}}\ {\textrm {m}^{ - 3}}$, ${T_e} = 5\ \textrm {eV}$, ${T_i} = 0.1\ \textrm {eV}$ and ${T_d} = 298\ \textrm {K}$ (Du et al. Reference Du, Sütterlin, Jiang, Räth, Ivlev, Khrapak, Schwabe, Thomas, Fortov and Lipaev2012). Notice from figure 3 that $\chi =1$ when ${n_{d0}}^ + ={n_{d0}}^ -$, i.e. if the number density of dust particles in two different regions are the same, $\alpha =\theta$. It is also noted from figure 3 that $\alpha$, or $\chi$, increases as the number density of the dust particles in the region $x>0$ increases, while it decreases as the number density of the dust particles in the region $x<0$ increases.

Figure 3. In the three-dimensional case, the influence of dust particle number density ${n_{d0}}^ +$ and ${n_{d0}}^ -$ on the system parameter $\chi$, where the green line is $\chi = \frac {3}{5}$, $\tan \alpha = \frac {3}{5} \tan \theta$, the black line is $\chi = 1$, $\tan \alpha = \tan \theta$, the purple line is $\chi = \frac {3}{2}$, $\tan \alpha = \frac {3}{2} \tan \theta$, and the other system parameters are ${m^ + } = {m^ - } = 5.0 \times {10^{ - 15}}\ \textrm {kg}$, ${n_{e0}} = 1.0 \times {10^{14}}\ {\textrm {m}^{ - 3}}$, ${T_e} = 5\ \textrm {eV}$, ${T_i} = 0.1\ \textrm {eV}$ and ${T_d} = 298\ \textrm {K}$.

The dependences of $\chi$ on the ion number density ${n_{e0}}$, the dust particle temperature ${T_d}$, the electron temperature ${T_e}$ and the ion temperature ${T_i}$ in both regions of $x > 0$ and $x < 0$ for the given system parameters have also been studied. It seems that all have no effect on $\chi$. Therefore, $\alpha$ is independent of ion number density, and the temperatures of electrons, ions and dust particles.

The dependence of $\alpha$ on $\theta$ has potential applications. For example, though the dependence of the electric charge on the dust size is usually in the form (Wang et al. Reference Wang, Schwan, Hsu, Grün and Horányi2016): ${Q_d} \propto r_d^{p}$, where $r_d$ is the size of a dust particle, $1< p\leqslant 2$ (Wang et al. Reference Wang, Schwan, Hsu, Grün and Horányi2016). The electric charge of a dust particle may be positive in certain conditions. To know the electric charge of a dust particle, we may try to devise an experiment to detect the electric charge of a dust particle by the following process. The parameter $\chi$ can be rewritten as follows:

(5.11)\begin{equation} \chi=\frac{\sqrt{m_d^{+}}n_{d0}^{+}\sqrt{Z_{d0}^{+}}\sqrt{T_{{\rm eff}}^{-}} }{\sqrt{m_d^{-}}n_{d0}^{-}\sqrt{Z_{d0}^{-}}\sqrt{T_{{\rm eff}}^{+}}}. \end{equation}

It is easy to give the mass, the number density and the effective temperature of the dust particles in two different regions. After given these parameters, let an incident solitary wave propagate in an incident wave angle $\theta$, measure the transmitted wave angle $\alpha$, and then we can obtain the ratio of parameter $Z_{d0}^{+}$ to $Z_{d0}^{-}$. If one of them is given, then the other is obtained. This result can be used to measure the electric charge of a dust particle in a dusty plasma.