2. Basic equations and their solutions

The description of nonlinear waves of surface charge is given in Gradov (Reference Gradov2020, Reference Gradov2021) based on using the theory of the potential (Jeffreys & Swirles Reference Jeffreys and Swirles1999) in a model of cold plasma with a sharp boundary occupying the region x > 0 along the 0X axis, and stationary ions with density N and charge e. The plane boundary placed at the point x = 0 on the 0X axis can be distorted for electrons. Then their surface is described by the equation $x = {x_0}(t,y,z)$. For the simple case of complete homogeneity along the 0Y axis, the dependence of the parameters on the y coordinate can be excluded. The environment surrounding the plasma can be characterized by a dielectric having permittivity ε_d. Under these conditions, it is possible to consider the peculiarities of the occurrence of various nonlinear phenomena on the plasma surface for the selected initial and boundary conditions specified at the origin. The motion of electrons with density n, charge −e and mass m is described by the equation for velocity v as a function of time t and coordinate z, which is represented in dimensionless form in Gradov (Reference Gradov2020, Reference Gradov2021, Reference Gradov2022) as the following relation:

(2.1)\begin{gather}\frac{{\partial {u_0}}}{{\partial \tau }} + \frac{\partial }{{\partial \eta }}\left( {\frac{{\partial {u_0}}}{{\partial \eta }} \cdot u} \right) + u = 0,\end{gather}

(2.2)\begin{gather}\left. {\begin{array}{*{20}{c@{}}} {{u_0} = \dfrac{{\partial u}}{{\partial \tau }} + {a^2}\dfrac{{\partial {u^2}}}{{\partial \eta }} + {a_2}\dfrac{\partial }{{\partial \eta }}\left\{ {{u^2}{{\left( {\dfrac{{\partial \varLambda }}{{\partial \eta }}} \right)}^2}} \right\};\quad {a_2} = {a_1}(1 + {a_1});}\\ {{a_1} = \dfrac{{\ln (\sqrt 2 + 1)}}{{\sqrt 2 {\rm \pi}}} \approx 0.2;\quad {a^2} = \dfrac{{1 + a_1^2}}{2}.} \end{array}} \right\}\end{gather}

Here the dimensionless coordinates and variables $\varLambda = {k_1}{x_0}$; ${k_1} = {\omega _S}/s$; ${\omega _S} = {\omega _L}/{(1 + {\varepsilon _d})^{ - 1/2}}$; ${\omega _L} = {(N{e^2}/m{\varepsilon _0})^{1/2}}$; $\eta = z{k_1}$; $s = \textrm{const}\textrm{.}$; and $\tau = {\omega _S}t,\;u = v/s$ were introduced with using ε ₀ as the notation of the vacuum permittivity. As shown in Gradov (Reference Gradov2020), the constant a ₁ arises when the value of the electrostatic potential near the boundary is presented like an expansion on a small parameter (x − x ₀) as a result of the integration in the corresponding term of this expansion determined by the potential theory of Jeffreys & Swirles (Reference Jeffreys and Swirles1999). The use of the dimensional constant s is associated with the conversion of all equations into a dimensionless form, and its value with the velocity dimension is determined by the velocity at which the points of the profile of the nonlinear wave under consideration move, as shown by the solutions obtained below.

During derivation of (1) in Gradov (Reference Gradov2021), the velocity of electrons was presented in the form $\boldsymbol{v} = \boldsymbol{\nabla }\varPsi$ for the case when their transportation is driven by the potential Ψ, which permits one to transform the equation of the motion and leads to the relation

(2.3)\begin{gather}\frac{{\partial {\varPsi _0}}}{{\partial t}} + {a^2}{\frac{{\partial {\varPsi _0}}}{{\partial z}}^2} = \frac{e}{m}\varPhi (z,t);\end{gather}

(2.4)\begin{gather}{\varPsi _\textrm{0}}(z\textrm{,}t) = \varPsi (x = {x_0}(z\textrm{,}t) + 0,z\textrm{,}t\textrm{)}.\end{gather}

Equation (2.3) was derived with the help of the proposition that inside plasma $x > {x_0}(z,t) + 0$ quasi-neutrality takes place, N = n, which provides the validity of the relation ΔΨ = 0 there in accordance with the equation of the continuity (see Gradov Reference Gradov2021). Equation (2.1) can be used to find the velocity of nonlinear motion as well as the associated density of the surface charge for a given form Λ = Λ(τ,η) of the distortion of the boundary. Examples of the solution obtained in this case for some special cases of physically justified types of the function Λ(τ,η) are given in Gradov (Reference Gradov2020, Reference Gradov2021).

On the other hand, the nonlinear motion of electrons can also lead to distortion of their equilibrium surface at the plasma boundary. In order to find an exact self-consistent solution, it is necessary to obtain another new equation describing the relationship of the nonlinear surface charge with the curvature of the plasma surface. Such an equation can be obtained on the basis of a generalization of the nonlinear theory of ‘shallow water’ in a liquid (see Stoker Reference Stoker1957) to a plasma. Using this approach, it is possible to derive this equation by integrating the continuity equation

(2.5)\begin{equation}{\frac{{\partial n}}{{\partial t}} + \boldsymbol{\nabla }\boldsymbol{\cdot }(n\boldsymbol{v}) = 0}\end{equation}

over the region of homogeneous electron density of plasma in the range ${x_0}(t,z) < x < \infty$. It should be borne in mind that the change in the density of n(t,x,z) near the boundary is associated with its movement along the axis 0X, when a jump in density from 0 to N can occur at a fixed point, both over time and with a change of z in the surface of the plasma. Therefore, the function ∂n/∂t,z behaves like a δ-function of the argument $x - {x_0}\textrm{(}t,z\textrm{)}(n = N\delta [{k_1}(x - {x_0})])$ and can be written

(2.6)\begin{equation}\frac{{\partial n(t,z)}}{{\partial t,z}} = \mathop {\lim }\limits_{x \to {x_0}} \left\{ {\frac{{\partial {x_0}(t,z)}}{{\partial t,z}}\frac{{\partial {n_e}(t,x,z)}}{{\partial x}}} \right\}.\end{equation}

Given that for a nonlinear surface charge there is a relation of Δψ = 0 in the region ${x_0}(t,z) < x < \infty$ (see Gradov Reference Gradov2020, Reference Gradov2021, Reference Gradov2022), the second term in (2.5) can be represented as $\boldsymbol{\nabla }\boldsymbol{\cdot }({n_e}\boldsymbol{v}) = \boldsymbol{\nabla }{n_e}\boldsymbol{\cdot }\boldsymbol{\nabla }\psi$. Using (2.6), after integrating the values in (2.5) over this region of the z coordinate, the following equation can be obtained:

(2.7)\begin{equation}{\frac{{\partial {x_0}}}{{\partial t}} + \left( {{a_1} + \frac{{\partial {x_0}}}{{\partial z}}} \right)\frac{{\partial {\psi _0}}}{{\partial z}} = 0}.\end{equation}

The following formula is also used here:

(2.8)\begin{equation}{\left. {\frac{{\partial \psi }}{{\partial x}}} \right|_{x = {x_0} + 0}} = {a_1}\frac{{\partial {\psi _0}}}{{\partial z}},\end{equation}

As obtained and used in works (Gradov Reference Gradov2020, Reference Gradov2021, Reference Gradov2022) concerning nonlinear surface charge. The relation (2.8) is a consequence of the potential theory (Jeffreys & Swirles Reference Jeffreys and Swirles1999) used to describe the nonlinear surface charge.

As a result of performing the necessary calculations, the following differential equation can be derived, which describes the connection between the distortion value of the boundary Λ(τ,η) and the velocity u(τ,η) of electron motion:

(2.9)\begin{equation}{\frac{{\partial \varLambda }}{{\partial \tau }} + \left( {{a_1} + \frac{{\partial \varLambda }}{{\partial \eta }}} \right)u = 0}.\end{equation}

Equation (2.9) makes it possible to study the behaviour of the curvature of the surface of electrons at the plasma boundary depending on the speed of their motion along this boundary. At the same time, the system (2.1), (2.9) gives the possibility of searching for such solutions that describe pulses of a certain shape, both boundary distortion and nonlinear surface charge parameters, capable of propagating over long distances without changing their properties. It also follows from (2.9) that the curvature of the surface Λ(τ,η) does not change in time if there is no movement of electrons along the boundary.

The obtained dependences on time and spatial coordinates for the velocities of electrons and the shape of curvature of the boundary make it possible to determine the spatially temporal structure of other characteristics of the process, such as, for example, the electrostatic potential φ_e(ξ) and the surface charge density n_S(ξ), which is determined by the following formula (Gradov Reference Gradov2020, Reference Gradov2021):

(2.10)\begin{equation}{n_S}(z,t) = \mathop {\lim }\limits_{\delta \to 0} \int_{{x_0} - \delta }^{{x_0} + \delta } {\textrm{d}xn(x,z,t)} .\end{equation}

The connection between this density and magnitudes of the electrostatic potential at the plasma boundary $\varPhi (z,t) = {\varphi _e}(x = {x_0}(z,t),z)$ is described by the following equation (Gradov Reference Gradov2020) derived with help of the definition (2.10):

(2.11)\begin{equation}{{n_S}(z,t) = \frac{{{a_1}(1 + {\varepsilon _d}){\varepsilon _0}}}{e}\frac{{\textrm{d}\varPhi }}{{\textrm{d}z}}}.\end{equation}

As follows from the theory of the potential (Jeffreys & Swirles Reference Jeffreys and Swirles1999), the space distribution of the electrostatic potential φ_e of the surface charge is defined in whole space by its magnitude at the boundary Φ(z,t) for the case when the half-space plasma has a boundary in the form of the infinite surface x = x ₀(z) consistent with Gradov (Reference Gradov2020):

(2.12)\begin{gather}{\varphi _e}(r,t) ={\pm} \frac{1}{\rm \pi}\int_{ - \infty }^\infty {\textrm{d}z^{\prime}} \frac{{{{[x - {x_0}(z^{\prime},t)]}_{}}\theta (z^{\prime},t)\varPhi (z^{\prime},t)}}{{{{(z - z^{\prime})}^2} + {{[x - {x_0}(z^{\prime},t)]}^2}}},\end{gather}

(2.13)\begin{gather}\theta (z,t) = 1\textrm{ } + {\left( {\frac{{\partial {x_0}}}{{\partial z}}} \right)^2}.\end{gather}

Here the assumption was used that inside plasma x > x ₀(z,t) + 0 the quasi-neutrality (N = n) is supported by an equilibrium of forces confining the plasma which transforms the Poisson equation to a Laplace one. It gives the possibility of using the theory of the potential for describing a space structure of a considered function as was done above for the electrostatic potential. In (2.12) the sign ‘+’ should be taken for any point with coordinates {x,z} inside the plasma (x > x ₀(z,t)) and the sign ‘−’ is used for points of observations outside plasma (x < x ₀(z,t)). Therefore, the spatial distribution of the velocity potential Ψ(z,t) is described by the formula of the potential theory in the same way as was done using formula (2.12) for the electrostatic potential, i.e. through its own value at the plasma boundary. The corresponding expression can be easily written by analogy with (2.12).

The description of the function Ψ ₀(z,t) obtained as a result of the solution of the initial system (2.1), (2.3), (2.9) makes it possible to establish the spatial–temporal behaviour of the electrostatic potential at the plasma boundary Φ(z,t) with using (2.12). Formula (2.11) provides the same possibility of obtaining similar information for the surface charge density n_S(z,t). Finally, with the help of expressions (2.11), (2.12), it is possible to determine all the main characteristics of the process under consideration, if the functions Ψ ₀(z,t) and x ₀(z,t) are known, the description of which must be obtained as a result of solving the original system of (2.1), (2.3), (2.9). The approximate expressions obtained in this paper for these functions can be useful in estimating all the main parameters of the process. It should be noted that the adopted simplifying assumptions significantly narrow the range of conditions under which the phenomena under consideration may occur. For example, the approach of a cold plasma imposes certain restrictions on the temperature of electrons and the parameters of the wave process under study. This means, first of all, a requirement for a relationship between the Debye radius of the plasma and the characteristic size of the nonlinear wave. The smaller the ratio of these quantities, the weaker the effect of the thermal motion of electrons on the phenomena under consideration.

The study of the nonlinear wave process is based primarily on the assumption that the desired values u(τ,η) and Λ(τ,η) depend on the variables τ and η in the form of a combination of $\xi = \eta - \varOmega \tau \;(\varOmega = \textrm{const}\textrm{.})$, that is, the solution has the form of a wave. As can be seen from the definition of ξ, the velocity at which the profile points of the nonlinear wave in question move is equal to the value Ωs. In this case, the original system of (2.1), (2.3), (2.9) can be written in the following form:

(2.14)\begin{gather}\varOmega \frac{{\textrm{d}{u_0}}}{{\textrm{d}\xi }} - \frac{\textrm{d}}{{\textrm{d}\xi }}\left( {\frac{{\textrm{d}{u_0}}}{{\textrm{d}\xi }} \cdot u} \right) + u = 0;\end{gather}

(2.15a,b)\begin{gather}{u_0} = \frac{\textrm{d}}{{\textrm{d}\xi }}\left\{ {\varOmega u - {a^2}{u^2} - {a_2}{u^2}{{\left( {\frac{{\textrm{d}\varLambda }}{{\textrm{d}\xi }}} \right)}^2}} \right\};\quad {\frac{{\textrm{d}\varLambda }}{{\textrm{d}\xi }} = \frac{{a_1^2u(\xi )}}{{\varOmega - u(\xi )}}}.\end{gather}

The numerical solution (2.15) for some selected characteristics of the problem is shown in figure 1. The impulse form of the desired quantities is provided by the selection of the values of the parameters of the initial conditions by their mutual agreement. In order to have an approximate idea of the form of the solution in an analytical form and to obtain comparative estimates, the following functions can be used, which are not an exact solution of the system of (2.15) but sufficiently accurately convey the shape of solitary exact curves:

(2.16a,b)\begin{equation}{u_1}(\xi ) = {b_1}Sech ({b_2}{\xi ^2});\quad {\varLambda _1}(\xi ) = \frac{{{b_3}}}{{1 + {b_4}{\xi ^2}}}.\end{equation}

Functions (2.16) are represented in figure 1 by dotted curves.

Figure 1. The spatial dependence of electron velocity and the boundary distortion caused by the surface charge for some chosen set of parameters of nonlinear perturbations: b ₁ = 1, b ₂ = 0.5, b ₃ = 0.5, b ₄ = 0.5; Ω = 2.2, u(ξ = 0) = 2; Λ(ξ = 0) = 2; $u^{\prime}(\xi = 0) = 0$; $u^{\prime\prime}(\xi = 0) = 0$.

From the nonlinear surface charge theory (Gradov Reference Gradov2020, Reference Gradov2021), it is known that the velocity of electrons ${u_x}(\tau ,\eta )$ along the 0X axis at the boundary $x = {x_0}(\tau ,\eta )$ is associated with a similar parameter of motion along the u(τ,η) boundary by the ratio ${u_x}(\tau ,\eta ) = {a_1}u(\tau ,\eta )$. Therefore, if at some selected point η = η ₀ the electrons are given the velocity ${u_x}(\tau ,\eta = {\eta _0}) = {a_1}u({\eta _0}\unicode{x2013}\varOmega \tau )$, then a nonlinear wave of curvature of the electron boundary will begin to propagate along the boundary having the form Λ(ξ) satisfying, like the function Λ(ξ), the system of (2.15). This wave is a moving electric charge and may be of interest for solving some applied problems. It should be noted that the function ${u_1}({\eta _0}\unicode{x2013}\varOmega \tau )$ described by formula (2.16) can be used approximately as a boundary condition, which greatly simplifies the problem.

Thus, by creating, for example, with the help of a modulated or impulse laser beam the deviation of the surface of electrons from the original position, it is possible, under certain conditions, to initiate solitary pulses of an electrical signal that are able to propagate along the surface together with the curvature of the boundary. This allows one not only to redirect the force effect to places inaccessible to direct impact, but also to send control signals and information there. Such a scheme is especially relevant for plasma objects of an extended shape such as an ellipsoid, in which large areas of the surface fully meet the conditions of the problem presented in this paper.

The reverse process is also possible, when the curvature of the boundary is generated by electrical impulses along the boundary. Here, pulsating electrostatic fields having, according to (2.12), a large localization area near the boundary, which can not only accelerate charged particles, but also create electromagnetic radiation due to their nonlinear properties, may be of possible interest.

3. Summary and conclusion

Within the framework of a simple model of cold plasma with a sharp boundary, a system of two equations was obtained describing the self-consistent propagation of the distortion of the electron surface at the plasma boundary together with electrical impulses along the boundary. This self-consistent process is possible if certain conditions arising from the solution of the system (2.5) are met when propagation along the boundary is carried out over a long distance without distorting the waveform. Taking into account electromagnetic effects, performed similarly to the calculations presented in Yu & Zhelyazkov (Reference Yu and Zhelyazkov1978) and Brodin & Stenflo (Reference Brodin and Stenflo2014), can help to identify new aspects of this phenomenon, for example, the description of radiation along the normal to the boundary in the form of outflowing waves (Tamir & Oliner Reference Tamir and Oliner1963). But this requires more research. A closed system of equations derived in the present work includes a description of the shape of the surface of electrons, which may help to study a wide range of phenomena associated with the occurrence of a nonlinear surface charge. There is every reason to believe that its widespread use will make it possible to discover and study many interesting and important effects connected with plasma boundaries. For example, special features of the surface charge could help in understanding many interesting nonlinear phenomena in the case when this theory would be generalized in the future to use also phase-mixing (Karmakar et al. Reference Karmakar, Maity, Chakrabarti and Sengupta2018; Pramanik & Maity Reference Pramanik and Maity2018), prospective numerical schemes (Verma Reference Verma2018) as well as quantum effects (Shahmansouri, Aboltaman & Misra Reference Shahmansouri, Aboltaman and Misra2018). The described approach can be also useful for the design and production of new functional materials and processes (Ye et al. Reference Ye, Zhang, Zhang, Hu, Ji, Ding and Jiang2013). Also, influences of transversal electromagnetic phenomena on nonlinear surface waves have to be taken into account (Yu & Zhelyazkov Reference Yu and Zhelyazkov1978; Vladimirov, Yu & Tsytovich Reference Vladimirov, Yu and Tsytovich1994; Shukla Reference Shukla1999) in some cases of an essential practical interest. Moreover, as was shown in Lee, Jung & Jung (Reference Lee, Jung and Jung2018) and Chandler-Wilde & Zhang (Reference Chandler-Wilde and Zhang1998), they can play sometimes a noticeable role in processes across the interface between various physical and biological substances.