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A centrally symmetric exact solution of the Maxwell's equations in logarithmically nonlinear media
Accepted Manuscript Title: A centrally symmetric exact solution of the Maxwell’s equations in logarithmically nonlinear media Author: Guozhu Yu Dazhi Zhao PII: DOI: Reference:
S0030-4026(16)31543-1 http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.204 IJLEO 58620
To appear in: Received date: Accepted date:
10-10-2016 30-11-2016
Please cite this article as: Guozhu Yu, Dazhi Zhao, A centrally symmetric exact solution of the Maxwell’s equations in logarithmically nonlinear media, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.204 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
A centrally symmetric exact solution of the Maxwell’s equations in logarithmically nonlinear media Guozhu Yu
1†
, Dazhi Zhao
2∗
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School of Mathematics, Southwest Jiaotong University Chengdu – 610031, China 2 School of Mathematics, Sichuan University Chengdu – 610065, China † Email: [email protected] ∗ Corresponding author. Email: [email protected]
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Abstract
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In this paper, we present a new method to get explicit exact solutions of the Maxwell’s equations for the spherical wave in logarithmically nonlinear media. Two kinds of logarithmically nonlinear media are discussed and solutions of both cases are derived. We also prove that the solution seeking routine is effective for a large class of nonlinear wave equations and power or logarithmic nonlinearity is necessary for multiplicative separable solutions. Physical interpretation of the multiplicative separable solution is given, which is regarded as a modulation of magnitude on the basis of an electrostatic field with the same multiple at all spatial points.
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Keywords: Nonlinear Maxwell’s Equations, Exact Solution, Logarithmic Nonlinearity, Spherical Wave PACS: 03.50.De, 41.20.Jb, 42.65.-k
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1. Introduction
Electromagnetic wave propagation in a nonlinear medium is a fundamental topic in physics [1] which always attracts much attention [2]. However, it is difficult to handle nonlinear problems in a unified framework and hard to find exact solutions of nonlinear equations due to the superposition principle no longer holds in nonlinear systems. Thus computational electromagnetics (CEM) becomes the main way to deal with electromagnetic problems in applications [3]. In CEM, in order to compare the numerical results with Preprint submitted to Elsevier
October 3, 2016
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the true ones, exact solutions are expected to be benchmarks. For example, Martin Pototschnig et al. [4] presented an iterative exact solution to test the effectiveness of numerical methods in a Kerr medium. So even exact solutions for simpler problems, for example, one-dimensional electromagnetic wave propagation in typical nonlinear media, are welcome. The fundamental theory of electromagnetic fields is based on the Maxwell’s equations. Meanwhile, for some models in the fields of physics and engineering, especially in nonlinear optics, the nonlinear Schr¨ odinger equation (NLSE) plays key roles [5, 6, 7]. Hence in the literature of electromagnetic wave propagation, the Maxwell’s equations and NLSE are the main topic of interest. Recently, E.Yu. Petrov et al. [8] discussed the exact axisymmetric solutions of the Maxwell’s equations in a nonlinear nondispersive medium where the relationship between the electric displacement D and the electric field E was an exponential function. Since then, some researchers have followed to find exact solutions in different nonlinear effects [9] and to discuss potential applications [10, 11]. The solutions seeking of Ref. [8] and Ref. [9] are carried out via some complicated transformations to make nonlinear PDEs transform to linear PDEs, however, those transformations are difficult to construct. Meanwhile, the obtained solutions by that method are expressed in implicit functions, which need numerical calculations for further applications. We have presented another way to find explicit exact solutions for cylindrical waves in exponentially nonlinear media and discussed their physical interpretation [12]. Compared with the cylindrical wave, the spherical wave is another important electromagnetic form with central symmetry which is always used to describe the wave is from a point source. It was studied intensively in various applications, such as electromagnetic cloaking in spherical structure [13], nonlinear plasma waves [14], diffraction of electromagnetic spherical waves [15], and so on. Logarithmically nonlinear media is a different medium behaves oppositely with the exponentially nonlinear medium. The current works studying electromagnetic wave propagation in a logarithmic medium mainly focus on the field of optical solitons [16, 17, 18, 19, 20, 21] and self-trapping spatially incoherent radiation [22]. For example, Kr´olikowski et al. discussed partially coherent solitons in logarithmically nonlinear media [16] where the refractive index change is a logarithmic function of the beam intensity. In what follows, we present a different method from Ref. [8] to find two
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{
and D2 =
ϵ0 cos(ηt)( Rr )β ln(γE α ), if E > E0 ϵ0 E, if E ≤ E0 ,
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new explicit exact solutions of the spherical wave in logarithmically nonlinear media where the constitutive relations are { ϵ0 ( Rr )β ln(γE α ), if E > E0 D1 = (1) ϵ0 E, if E ≤ E0 ,
(2)
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2. Model and Solution
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0 < E0 ≪ 1 is a very small positive constant, ϵ0 is the permittivity of free space, R > 0 is a constant, α (̸= 0), β, γ > 0 and η (̸= 0) are arbitrary real numbers. Compared with eq. (1), eq. (2) has a time-variant term cos(ηt) which suggests the medium is changing periodically. It is noted that the expression Di = ϵ0 E, i = 1, 2 is reasonable when E ≤ E0 . For example, nonlinearity often occurs only if the intensity is strong enough in nonlinear optics. A wide variety of situations can be attributed to the nonlinearity we present here, according to different parameters of α, β, γ and η respectively.
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Assume the medium holds a central symmetry and there exists a field source at the central point. Suppose the central point is the original point, at any points of the sphere whose radius is r, the directions of electric field E and magnetic filed H are both in the tangent space of the sphere at a given point. At any direction ⃗r we are interested in, take (O, e⃗r , e⃗θ , e⃗ϕ ) as the spherical coordinate system, where O is the original point, e⃗θ and e⃗ϕ are the directions of E and H respectively, thus only the Eθ and Hϕ components are nonzero. Because we have supposed that the medium holds a central symmetry and the fields are independent of θ and ϕ, both E and H can be denoted as E(r, t) and H(r, t) for simplicity. This model (the Maxwell’s equations) can be written in the form
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−
H ∂E ∂E E ∂H ∂H − = ε(E) , + = −µ0 , ∂r r ∂t ∂r r ∂t
(3)
where ε(E) = dD . As examples, ε(E) = ϵ0 ε1 exp(αE) in Ref. [8], ε(E) = dE r β ϵ0 ε1 ( R ) exp(αE) in Ref. [9] where R is a constant. Obviously, the former nonlinearity is a special case of the latter only if the parameter β = 0.
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Eq. (3) can be reduced to ∂ 2 E 2 ∂E ∂E ∂ + = µ0 (ε(E) ), 2 ∂r r ∂r ∂t ∂t
(4)
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which is our main study object in the following context. In this paper, we will discuss the electromagnetic wave propagation in logarithmically nonlinear media as the constitutive relations are expressed in eq. (1) and eq. (2). When E ≤ E0 , eq. (4) is a linear equation, so we only consider the case for E > E0 . Apparently, we get ε(E) = ϵ0 ( Rr )β Eα for eq. (1) and ε(E) = ϵ0 cos(ηt)( Rr )β Eα for eq. (2) in this situation. For simplicity, we only discuss the solving routine for the case ε(E) = ϵ0 ( Rr )β Eα and give the solution for the case ε(E) = ϵ0 cos(ηt)( Rr )β Eα direct. Rewrite eq. (4) as
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∂ 2 E 2 ∂E ∂ r α ∂E + = (µ0 ϵ0 ( )β ). 2 ∂r r ∂r ∂t R E ∂t
(5)
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Suppose that eq. (5) has a multiplicative separable solution E(r, t) = u(r)v(t). Putting it into eq. (5) leads to
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2 ′ r ∂ 1 ′ ′′ v(t)(u (r) + u (r)) = µ0 ϵ0 α( )β ( v (t)) r R ∂t v(t) r 1 ′ ′ = µ0 ϵ0 α( )β ( v (t)) . R v(t)
(6)
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It is noted that variables can be divided in eq. (6). Moving all terms containing r to the left side and all terms containing t to the right side results in 1 2 ′ 1 1 ′ ′ ′′ µ0 ϵ0 ( v (t)) . (7) r β (u (r) + u (r)) = α( R ) r v(t) v(t)
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Since variables r and t are independent, then both the left side and the right side of eq. (7) are constant numbers. Let C0 be an arbitrary number, eq. (7) is equivalent to the following ordinary differential equations r 2 ′ ′′ u (r) + u (r) = C0 α( )β r R
(8)
1 ′ ′ v (t)) = C0 v(t). v(t)
(9)
and µ0 ϵ0 (
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First, we try to find the solution of eq. (8). Let K1 = is transformed to ′′ uη (η) = K1 η −(4+β) ,
C0 α , Rβ
η = 1r , eq. (8) (10)
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where η indicates the second derivative to η. Eq. (10) is a simple ODE, we can get the solution by integrating it twice, if β = −3 K1 η(ln(η) − 1) + C1 η + C2 −K ln(η) + C η + C if β = −2 (11) u(η) = 1 1 2 K1 −(2+β) η + C η + C if β = ̸ −3, β = ̸ −2. 1 2 (β+2)(β+3)
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Replacing η = 1r yields to ln(r) if β = −3 −K1 r + C1 1r + C2 1 K1 ln(r) + C1 r + C2 if β = −2 u(r) = K1 1 (2+β) r + C + C ̸ −3, β ̸= −2. 1r 2 if β = (β+2)(β+3)
′′
C0 , µ0 ϵ0
(12)
w(t) = ln(v(t)),
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Next, we try to find the solution of eq. (9). Let K2 = then eq. (9) can be written as wtt = K2 exp(w).
(13)
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Eq. (13) has an exact solution [23] expressed as 2 K2 if K2 > 0 − ln( 2C32 sinh (C3 t + C4 )) C3 t + C4 if K2 = 0 w(t) = − ln(− K22 cosh2 (C3 t + C4 )) if K2 < 0. 2C
(14)
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Considering v(t) = exp(w(t)), 2C32 K2 sinh2 (C3 t+C4 ) C4 exp(C3 t) v(t) = exp(w(t)) = −2C32
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K2 cosh2 (C3 t+C4 )
if K2 > 0 if K2 = 0 if K2 < 0.
To sum up, the explicit exact solution of eq. (4) when β ̸= −3, β 2C32 K1 (2+β) + C1 1r + C2 ) K2 sinh2 (C3 t+C4 ) ( (β+2)(β+3) r K1 C4 exp(C3 t)( (β+2)(β+3) r(2+β) + C1 1r + C2 ) E(r, t) = u(r)v(t) = −2C32 K1 ( r(2+β) + C1 1 + C2 ) 2 K2 cosh (C3 t+C4 ) (β+2)(β+3)
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(15) ̸= −2 is if K2 > 0 if K2 = 0 if K2 < 0. (16)
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Cases for β = −3 and β = −2 are similar. For the case ε(E) = ϵ0 cos(ηt)( Rr )β Eα , eq. (9) is changed to 1 ′ ′ v (t)) = C0 v(t). v(t)
(17)
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µ0 ϵ0 (cos(ηt)
v1 (t) = −
η2 cos(ηt). K2
cr
One can verify that the following v1 (t) is a special solution of eq. (17),
(18)
When E(r, t) is obtained, one can easily get the result of H(r, t) according = µ0 ∂H in eq. (3). That is ∂t ∫ u′ (r) v(t)dt. (20) H(r, t) = µ0
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Thus the explicit exact solution of eq. (4) when ε(E) = ϵ0 cos(ηt)( Rr )β Eα can be got accordingly, E1 (r, t) = u(r)v1 (t). (19)
For the case ε(E) = ϵ0 cos(ηt)( Rr )β Eα , eq. (20) has a simple expression
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H1 (r, t) = −u′ (r) {
where ′
−K1 ln(r) + C1 if β = −3 K1 (3+β) − β+4 r + C1 if β ̸= −3.
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u (r) =
η sin(ηt), K2 µ0
(21)
(22)
3. Physical interpretation of the solution
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We have got the general solution. There are some arbitrary numbers C0 , C1 , C2 , C3 and C4 . When the initial fields distributions E(r, 0), H(r, 0), and their derivative functions, boundary fields distributions E(R, t), H(R, t), where R is a fixed distance, are determined, we can calculate results of those arbitrary numbers. ∫ Typical graphs of u(r), v(t), u′ (r) and v(t)dt for ε(E) = ϵ0 cos(ηt)( Rr )β Eα are as follows (See Figures 1 - 6), where C0 = 1, C1 = 1, C2 = 1, K2 = 1, R = 1, α = 2, η = 2. From Figures 1 and 4, we note that the parameter β plays a vital role in the spatial distributions of E(r, t) and H(r, t). Figures 2 6
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spatial distribution of E
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spatial distribution of E
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Figure 1: Spatial distribution of E – u(r).
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time modulation of E
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Figure 2: Time modulation of E – v(t)
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electric field E
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Figure 3: E(r, t) when β = −3.5
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spatial distribution of H
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Figure 4: Spatial distribution of H – u′ (r).
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time modulation of E
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magnetic field H
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Figure 5: Time modulation of H –
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Figure 6: H(r, t) when β = −3.5
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and 5 show the plots of time modulations. Finally, Figures 3 and 6 display the characteristics of E(r, t) and H(r, t) respectively. Suppose an electrostatic field Es (r) has the spatial distribution u(r), then at any given time t, E(r, t) = u(r)v(t) is a modulation of magnitude based on Es (r) with the same multiple at all spatial points. Note that H(r, t) is also in the form of multiplicative separable solutions according to eq. (20). Then H(r, t) has a similar physical interpretation to E(r, t). 4. Discussions
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4.1. This method is effective for a large class of nonlinear wave equations Along the routine we do in the solving process, it is an easy task to declare the following general nonlinear wave equation has a multiplicative separable solution, ∂ 2E ∂E ∂ ∂E f1 (r) 2 + f2 (r) = h1 (t) (g(r)h2 (t)E α ). (23) ∂r ∂r ∂t ∂t Actually, suppose that eq. (23) has a multiplicative separable solution E(r, t) = u(r)v(t). Putting it into eq. (23) leads to ′′
′
(24)
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f1 (r)u (r) + f2 (r)u (r) h1 (t) ′ ′ = (h2 (t)v α (t)v (t)) , α+1 g(r)u (r) v(t)
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which can be divided into two second order ODEs. In other words, a large class of nonlinear wave equations in the form of eq. (23) can be transformed to nonlinear ODEs, and we can obtain solutions by solving ODEs. 4.2. Power or logarithmic nonlinearity is necessary for multiplicative separable solutions Suppose that eq. (4) has a multiplicative separable solution E(r, t) = u(r)v(t). Putting it into eq. (4) leads to
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2 ′ ∂ ′′ ′ v(t)(u (r) + u (r)) = µ0 (ε(E)u(r)v (t)) r ∂t ′′ ′ ′ = µ0 u(r)(ε(E)v (t) + ε (E)u(r)v 2 (t)).
(25)
Since eq. (4) has a multiplicative separable solution, then ε(E) satisfies ′
ε (E)E = C0 , ε(E)
(26)
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where C0 is an arbitrary number, and we have replaced u(r) by E because E(r, t) = u(r)v(t) and to ignore any function of t on the right side of eq. (4) does not affect the process of variables separation. It is easy to solve eq. (26) and conclude that ε(E) is in the form of power function. In other words, the constitutive relation between D and E must be a power function or a logarithmic function.
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Acknowledgements
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This work was supported partially by the Fundamental Research Funds for the Central Universities of China (under Grant No.2682015CX053). References
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[23] Valentin F Zaitsev and Andrei D Polyanin. Handbook of exact solutions for ordinary differential equations. CRC Press, 2002.
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S0030-4026(16)31543-1 http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.204 IJLEO 58620
To appear in: Received date: Accepted date:
10-10-2016 30-11-2016
Please cite this article as: Guozhu Yu, Dazhi Zhao, A centrally symmetric exact solution of the Maxwell’s equations in logarithmically nonlinear media, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.204 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
A centrally symmetric exact solution of the Maxwell’s equations in logarithmically nonlinear media Guozhu Yu
1†
, Dazhi Zhao
2∗
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School of Mathematics, Southwest Jiaotong University Chengdu – 610031, China 2 School of Mathematics, Sichuan University Chengdu – 610065, China † Email: [email protected] ∗ Corresponding author. Email: [email protected]
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Abstract
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In this paper, we present a new method to get explicit exact solutions of the Maxwell’s equations for the spherical wave in logarithmically nonlinear media. Two kinds of logarithmically nonlinear media are discussed and solutions of both cases are derived. We also prove that the solution seeking routine is effective for a large class of nonlinear wave equations and power or logarithmic nonlinearity is necessary for multiplicative separable solutions. Physical interpretation of the multiplicative separable solution is given, which is regarded as a modulation of magnitude on the basis of an electrostatic field with the same multiple at all spatial points.
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Keywords: Nonlinear Maxwell’s Equations, Exact Solution, Logarithmic Nonlinearity, Spherical Wave PACS: 03.50.De, 41.20.Jb, 42.65.-k
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1. Introduction
Electromagnetic wave propagation in a nonlinear medium is a fundamental topic in physics [1] which always attracts much attention [2]. However, it is difficult to handle nonlinear problems in a unified framework and hard to find exact solutions of nonlinear equations due to the superposition principle no longer holds in nonlinear systems. Thus computational electromagnetics (CEM) becomes the main way to deal with electromagnetic problems in applications [3]. In CEM, in order to compare the numerical results with Preprint submitted to Elsevier
October 3, 2016
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the true ones, exact solutions are expected to be benchmarks. For example, Martin Pototschnig et al. [4] presented an iterative exact solution to test the effectiveness of numerical methods in a Kerr medium. So even exact solutions for simpler problems, for example, one-dimensional electromagnetic wave propagation in typical nonlinear media, are welcome. The fundamental theory of electromagnetic fields is based on the Maxwell’s equations. Meanwhile, for some models in the fields of physics and engineering, especially in nonlinear optics, the nonlinear Schr¨ odinger equation (NLSE) plays key roles [5, 6, 7]. Hence in the literature of electromagnetic wave propagation, the Maxwell’s equations and NLSE are the main topic of interest. Recently, E.Yu. Petrov et al. [8] discussed the exact axisymmetric solutions of the Maxwell’s equations in a nonlinear nondispersive medium where the relationship between the electric displacement D and the electric field E was an exponential function. Since then, some researchers have followed to find exact solutions in different nonlinear effects [9] and to discuss potential applications [10, 11]. The solutions seeking of Ref. [8] and Ref. [9] are carried out via some complicated transformations to make nonlinear PDEs transform to linear PDEs, however, those transformations are difficult to construct. Meanwhile, the obtained solutions by that method are expressed in implicit functions, which need numerical calculations for further applications. We have presented another way to find explicit exact solutions for cylindrical waves in exponentially nonlinear media and discussed their physical interpretation [12]. Compared with the cylindrical wave, the spherical wave is another important electromagnetic form with central symmetry which is always used to describe the wave is from a point source. It was studied intensively in various applications, such as electromagnetic cloaking in spherical structure [13], nonlinear plasma waves [14], diffraction of electromagnetic spherical waves [15], and so on. Logarithmically nonlinear media is a different medium behaves oppositely with the exponentially nonlinear medium. The current works studying electromagnetic wave propagation in a logarithmic medium mainly focus on the field of optical solitons [16, 17, 18, 19, 20, 21] and self-trapping spatially incoherent radiation [22]. For example, Kr´olikowski et al. discussed partially coherent solitons in logarithmically nonlinear media [16] where the refractive index change is a logarithmic function of the beam intensity. In what follows, we present a different method from Ref. [8] to find two
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{
and D2 =
ϵ0 cos(ηt)( Rr )β ln(γE α ), if E > E0 ϵ0 E, if E ≤ E0 ,
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new explicit exact solutions of the spherical wave in logarithmically nonlinear media where the constitutive relations are { ϵ0 ( Rr )β ln(γE α ), if E > E0 D1 = (1) ϵ0 E, if E ≤ E0 ,
(2)
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2. Model and Solution
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0 < E0 ≪ 1 is a very small positive constant, ϵ0 is the permittivity of free space, R > 0 is a constant, α (̸= 0), β, γ > 0 and η (̸= 0) are arbitrary real numbers. Compared with eq. (1), eq. (2) has a time-variant term cos(ηt) which suggests the medium is changing periodically. It is noted that the expression Di = ϵ0 E, i = 1, 2 is reasonable when E ≤ E0 . For example, nonlinearity often occurs only if the intensity is strong enough in nonlinear optics. A wide variety of situations can be attributed to the nonlinearity we present here, according to different parameters of α, β, γ and η respectively.
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Assume the medium holds a central symmetry and there exists a field source at the central point. Suppose the central point is the original point, at any points of the sphere whose radius is r, the directions of electric field E and magnetic filed H are both in the tangent space of the sphere at a given point. At any direction ⃗r we are interested in, take (O, e⃗r , e⃗θ , e⃗ϕ ) as the spherical coordinate system, where O is the original point, e⃗θ and e⃗ϕ are the directions of E and H respectively, thus only the Eθ and Hϕ components are nonzero. Because we have supposed that the medium holds a central symmetry and the fields are independent of θ and ϕ, both E and H can be denoted as E(r, t) and H(r, t) for simplicity. This model (the Maxwell’s equations) can be written in the form
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−
H ∂E ∂E E ∂H ∂H − = ε(E) , + = −µ0 , ∂r r ∂t ∂r r ∂t
(3)
where ε(E) = dD . As examples, ε(E) = ϵ0 ε1 exp(αE) in Ref. [8], ε(E) = dE r β ϵ0 ε1 ( R ) exp(αE) in Ref. [9] where R is a constant. Obviously, the former nonlinearity is a special case of the latter only if the parameter β = 0.
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Eq. (3) can be reduced to ∂ 2 E 2 ∂E ∂E ∂ + = µ0 (ε(E) ), 2 ∂r r ∂r ∂t ∂t
(4)
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which is our main study object in the following context. In this paper, we will discuss the electromagnetic wave propagation in logarithmically nonlinear media as the constitutive relations are expressed in eq. (1) and eq. (2). When E ≤ E0 , eq. (4) is a linear equation, so we only consider the case for E > E0 . Apparently, we get ε(E) = ϵ0 ( Rr )β Eα for eq. (1) and ε(E) = ϵ0 cos(ηt)( Rr )β Eα for eq. (2) in this situation. For simplicity, we only discuss the solving routine for the case ε(E) = ϵ0 ( Rr )β Eα and give the solution for the case ε(E) = ϵ0 cos(ηt)( Rr )β Eα direct. Rewrite eq. (4) as
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∂ 2 E 2 ∂E ∂ r α ∂E + = (µ0 ϵ0 ( )β ). 2 ∂r r ∂r ∂t R E ∂t
(5)
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Suppose that eq. (5) has a multiplicative separable solution E(r, t) = u(r)v(t). Putting it into eq. (5) leads to
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2 ′ r ∂ 1 ′ ′′ v(t)(u (r) + u (r)) = µ0 ϵ0 α( )β ( v (t)) r R ∂t v(t) r 1 ′ ′ = µ0 ϵ0 α( )β ( v (t)) . R v(t)
(6)
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It is noted that variables can be divided in eq. (6). Moving all terms containing r to the left side and all terms containing t to the right side results in 1 2 ′ 1 1 ′ ′ ′′ µ0 ϵ0 ( v (t)) . (7) r β (u (r) + u (r)) = α( R ) r v(t) v(t)
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Since variables r and t are independent, then both the left side and the right side of eq. (7) are constant numbers. Let C0 be an arbitrary number, eq. (7) is equivalent to the following ordinary differential equations r 2 ′ ′′ u (r) + u (r) = C0 α( )β r R
(8)
1 ′ ′ v (t)) = C0 v(t). v(t)
(9)
and µ0 ϵ0 (
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First, we try to find the solution of eq. (8). Let K1 = is transformed to ′′ uη (η) = K1 η −(4+β) ,
C0 α , Rβ
η = 1r , eq. (8) (10)
′′
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where η indicates the second derivative to η. Eq. (10) is a simple ODE, we can get the solution by integrating it twice, if β = −3 K1 η(ln(η) − 1) + C1 η + C2 −K ln(η) + C η + C if β = −2 (11) u(η) = 1 1 2 K1 −(2+β) η + C η + C if β = ̸ −3, β = ̸ −2. 1 2 (β+2)(β+3)
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Replacing η = 1r yields to ln(r) if β = −3 −K1 r + C1 1r + C2 1 K1 ln(r) + C1 r + C2 if β = −2 u(r) = K1 1 (2+β) r + C + C ̸ −3, β ̸= −2. 1r 2 if β = (β+2)(β+3)
′′
C0 , µ0 ϵ0
(12)
w(t) = ln(v(t)),
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Next, we try to find the solution of eq. (9). Let K2 = then eq. (9) can be written as wtt = K2 exp(w).
(13)
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Eq. (13) has an exact solution [23] expressed as 2 K2 if K2 > 0 − ln( 2C32 sinh (C3 t + C4 )) C3 t + C4 if K2 = 0 w(t) = − ln(− K22 cosh2 (C3 t + C4 )) if K2 < 0. 2C
(14)
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Considering v(t) = exp(w(t)), 2C32 K2 sinh2 (C3 t+C4 ) C4 exp(C3 t) v(t) = exp(w(t)) = −2C32
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K2 cosh2 (C3 t+C4 )
if K2 > 0 if K2 = 0 if K2 < 0.
To sum up, the explicit exact solution of eq. (4) when β ̸= −3, β 2C32 K1 (2+β) + C1 1r + C2 ) K2 sinh2 (C3 t+C4 ) ( (β+2)(β+3) r K1 C4 exp(C3 t)( (β+2)(β+3) r(2+β) + C1 1r + C2 ) E(r, t) = u(r)v(t) = −2C32 K1 ( r(2+β) + C1 1 + C2 ) 2 K2 cosh (C3 t+C4 ) (β+2)(β+3)
r
(15) ̸= −2 is if K2 > 0 if K2 = 0 if K2 < 0. (16)
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Cases for β = −3 and β = −2 are similar. For the case ε(E) = ϵ0 cos(ηt)( Rr )β Eα , eq. (9) is changed to 1 ′ ′ v (t)) = C0 v(t). v(t)
(17)
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µ0 ϵ0 (cos(ηt)
v1 (t) = −
η2 cos(ηt). K2
cr
One can verify that the following v1 (t) is a special solution of eq. (17),
(18)
When E(r, t) is obtained, one can easily get the result of H(r, t) according = µ0 ∂H in eq. (3). That is ∂t ∫ u′ (r) v(t)dt. (20) H(r, t) = µ0
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Thus the explicit exact solution of eq. (4) when ε(E) = ϵ0 cos(ηt)( Rr )β Eα can be got accordingly, E1 (r, t) = u(r)v1 (t). (19)
For the case ε(E) = ϵ0 cos(ηt)( Rr )β Eα , eq. (20) has a simple expression
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H1 (r, t) = −u′ (r) {
where ′
−K1 ln(r) + C1 if β = −3 K1 (3+β) − β+4 r + C1 if β ̸= −3.
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u (r) =
η sin(ηt), K2 µ0
(21)
(22)
3. Physical interpretation of the solution
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We have got the general solution. There are some arbitrary numbers C0 , C1 , C2 , C3 and C4 . When the initial fields distributions E(r, 0), H(r, 0), and their derivative functions, boundary fields distributions E(R, t), H(R, t), where R is a fixed distance, are determined, we can calculate results of those arbitrary numbers. ∫ Typical graphs of u(r), v(t), u′ (r) and v(t)dt for ε(E) = ϵ0 cos(ηt)( Rr )β Eα are as follows (See Figures 1 - 6), where C0 = 1, C1 = 1, C2 = 1, K2 = 1, R = 1, α = 2, η = 2. From Figures 1 and 4, we note that the parameter β plays a vital role in the spatial distributions of E(r, t) and H(r, t). Figures 2 6
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spatial distribution of E
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spatial distribution of E
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Figure 1: Spatial distribution of E – u(r).
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time modulation of E
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Figure 2: Time modulation of E – v(t)
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electric field E
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Figure 3: E(r, t) when β = −3.5
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spatial distribution of H
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spatial distribution of H
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Figure 4: Spatial distribution of H – u′ (r).
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time modulation of E
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magnetic field H
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Figure 5: Time modulation of H –
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Figure 6: H(r, t) when β = −3.5
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and 5 show the plots of time modulations. Finally, Figures 3 and 6 display the characteristics of E(r, t) and H(r, t) respectively. Suppose an electrostatic field Es (r) has the spatial distribution u(r), then at any given time t, E(r, t) = u(r)v(t) is a modulation of magnitude based on Es (r) with the same multiple at all spatial points. Note that H(r, t) is also in the form of multiplicative separable solutions according to eq. (20). Then H(r, t) has a similar physical interpretation to E(r, t). 4. Discussions
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4.1. This method is effective for a large class of nonlinear wave equations Along the routine we do in the solving process, it is an easy task to declare the following general nonlinear wave equation has a multiplicative separable solution, ∂ 2E ∂E ∂ ∂E f1 (r) 2 + f2 (r) = h1 (t) (g(r)h2 (t)E α ). (23) ∂r ∂r ∂t ∂t Actually, suppose that eq. (23) has a multiplicative separable solution E(r, t) = u(r)v(t). Putting it into eq. (23) leads to ′′
′
(24)
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f1 (r)u (r) + f2 (r)u (r) h1 (t) ′ ′ = (h2 (t)v α (t)v (t)) , α+1 g(r)u (r) v(t)
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which can be divided into two second order ODEs. In other words, a large class of nonlinear wave equations in the form of eq. (23) can be transformed to nonlinear ODEs, and we can obtain solutions by solving ODEs. 4.2. Power or logarithmic nonlinearity is necessary for multiplicative separable solutions Suppose that eq. (4) has a multiplicative separable solution E(r, t) = u(r)v(t). Putting it into eq. (4) leads to
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2 ′ ∂ ′′ ′ v(t)(u (r) + u (r)) = µ0 (ε(E)u(r)v (t)) r ∂t ′′ ′ ′ = µ0 u(r)(ε(E)v (t) + ε (E)u(r)v 2 (t)).
(25)
Since eq. (4) has a multiplicative separable solution, then ε(E) satisfies ′
ε (E)E = C0 , ε(E)
(26)
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where C0 is an arbitrary number, and we have replaced u(r) by E because E(r, t) = u(r)v(t) and to ignore any function of t on the right side of eq. (4) does not affect the process of variables separation. It is easy to solve eq. (26) and conclude that ε(E) is in the form of power function. In other words, the constitutive relation between D and E must be a power function or a logarithmic function.
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Acknowledgements
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This work was supported partially by the Fundamental Research Funds for the Central Universities of China (under Grant No.2682015CX053). References
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