Solitary and periodic waves in two-fluid magnetohydrodynamics

Accepted Manuscript

Solitary and periodic waves in two-fluid magnetohydrodynamics M.B. Gavrikov, N.A. Kudryashov, B.A. Petrov, V.V. Savelyev, D.I. Sinelshchikov PII: DOI: Reference:

S1007-5704(16)30029-6 10.1016/j.cnsns.2016.02.010 CNSNS 3768

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

10 December 2015 5 February 2016 8 February 2016

Please cite this article as: M.B. Gavrikov, N.A. Kudryashov, B.A. Petrov, V.V. Savelyev, D.I. Sinelshchikov, Solitary and periodic waves in two-fluid magnetohydrodynamics, Communications in Nonlinear Science and Numerical Simulation (2016), doi: 10.1016/j.cnsns.2016.02.010

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ACCEPTED MANUSCRIPT Highlights

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• A system of equations of two-fluid magnetohydrodynamics is studied. • The Painlev´e analysis of this equation is carried out. • The general solution of the equation is constructed in terms of the Weierstrass elliptic function. • Solitary and periodic wave solutions for the components of magnetic field are found.

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Solitary and periodic waves in two-fluid magnetohydrodynamics M. B. Gavrikov2 , N. A. Kudryashov1 , B. A. Petrov1 , V. V. Savelyev2 , D. I. Sinelshchikov1 1 Department 2 Keldysh

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of Applied Mathematics, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation

Institute of Applied Mathematics, Miusskaya sq., 4, Moscow, 125047, Russia

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Abstract

A system of equations of two-fluid magnetohydrodynamics is studied. An ordinary differential equation describing traveling waves in an ideal cold quasi-neutral plasma is obtained in the case of quasi-stationary electromagnetic field. The Painlev´e analysis of this equation is carried out and the general solution of the equation is constructed in terms of the Weierstrass elliptic function. Solitary and periodic wave solutions for the components of magnetic field are found and analyzed.

Introduction

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Keywords: two-fluid magnetohydrodynamics, plasma, Painlev´e test, periodic waves, solitary waves, exact solutions.

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The electrons and ions in the two-fluid magnetohydrodynamics (MHD) are considered as interpenetrating conducting current liquids. It is also supposed that the main hydrodynamic parameters (a mass density, speed and pressure) depend on spatial coordinates and time. A current density and magnetic field are used as electromagnetic parameters. Taking into account the Maxwell equations, heat transfer equations and state equations for the electrons and ions, one can obtain a closed system of equations (see, e.g. [1–3]). This mathematical model is widely used in astrophysics for description of solar corona [4–7], plasma influenced by the gravitational field of black holes [8] and in others applications [9–19]. However, these works was mainly devoted either to the linear or long wave approximations or numerical investigation of mathematical models of the two-fluid MHD. Recently, an attempt to study Preprint submitted to Elsevier

15 February 2016

ACCEPTED MANUSCRIPT this model analytically was made in [20] and solitary traveling wave solutions for the magnetic field were found. In this work we generalize these results and find the general traveling wave solution of this mathematical model of the two-fluid MHD.

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The rest of this work is organized as follows. In the next section we present a closed system of equation for the description of an ideal cold plasma in the two fluid approximation. Then we introduce traveling wave variables and transform this system of equations into a system of two ordinary differential equations. Section 3 is devoted to constructing and analyzing the general solution of this system of equations. In the last section we briefly summarize our results.

Equations of two-fluid electromagnetic hydrodynamics (EMHD) of plasma

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Let us consider the motion of an ideal cold plasma, so the heat transfer and state equations are not taken into account. In addition, we assume that the plasma is quasi-neutral and the electromagnetic field is quasi-stationarity. In this case to describe motion of the plasma the following system of equations can be used [1, 20, 21]:

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∂ne + div(ne Ve ) = 0, ∂t ∂ni + div(ni Vi ) = 0, ∂t !

(2.1)

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  ∂Ve 1 me ne + (Ve , ∇) Ve = −ene E + [Ve , B] , ∂t c !   ∂Vi 1 mi ni + (Vi , ∇) Vi = eni E + [Vi , B] , ∂t c 1 ∂B + rotE = 0, divB = 0, c ∂t 4π rotB = j, j = e (ni Vi − ne Ve ) , c ne = ni = n.

Here ne , ni are concentrations of electrons and ions and Ve , Vi are their velocities, E is an electric field, B is an magnetic field, j is a current density, t is time, ∇ is the Nabla operator. The first two equations in (2.1) are continuity equations of electrons and ions, the third and fourth equations are equations of motion for electrons and ions, the next four equations are the quasi-stationary Maxwell equations and the last equation expresses the assumption of the quasi-neutrality of plasma. 3

ACCEPTED MANUSCRIPT System of equations (2.1) is a complex system of nonlinear partial differential equations. However, this system of equations can be considerably simplified if we use the following variables [22, 23]: ρ = me ne + mi ni , me Ve + mi Vi , U= me + mi

(2.2) (2.3)

As a result, system (2.1) takes the form:

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where U is a mass hydrodynamic velocity of the plasma and ρ is a mass density of the plasma. We also consider one–dimensional motion of the ideal cold plasma assuming that ∂/∂y = ∂/∂z = 0.

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∂ρ ∂ρUx + = 0, ∂t ∂x ! ∂ρUx ∂ 1 By2 + Bz2 2 + ρUx + = 0, ∂t ∂x 4π 2  ∂ρUy ∂  1 ρUx Uy − + Bx By = 0, ∂t ∂x 4π  ∂ρUz ∂  1 + Bx Bz = 0, ρUx Uz − ∂t ∂x 4π ! ∂Bz c2 me mi ∂ 2 Ey 1 cme mi ∂ mi − me ∂By Ey − = (B U − B U ) − U + Bx , z x x z x 4πe2 ρ ∂x2 c 4πe2 ρ ∂x ∂x 4πeρ ∂x ! ∂By c2 me mi ∂ 2 Ez 1 cme mi ∂ mi − me ∂Bz Ez − = (Bx Uy − By Ux ) + Ux + Bx , 2 2 2 4πe ρ ∂x c 4πe ρ ∂x ∂x 4πeρ ∂x Bx = const, ! mi − me ∂ By2 + Bz2 1 , Ex = (By Uz − Bz Uy ) − c 4πeρ ∂x 2 1 ∂By ∂Ez − = 0, c ∂t ∂x 1 ∂Bz ∂Ey + = 0. c ∂t ∂x (2.4) Thus, we get closed system of equations (2.4) with respect to the variables ρ, Ux , Uy , Uz , Ey , Ez , By , Bz . Let us consider traveling waves governed by system of equations (2.4). Firstly, we introduce dimensionless variables in equations (2.4) ˜ ρ = ρ0 ρ˜, x = L0 x˜, B = B0 B, L0 B0 ˜ E0 = B0 V0 , (2.5) , t = t0 t˜, t0 = , E = E0 E, U = V0 U˜ , V0 = √ 4πρ0 V0 c

Then we use the traveling wave variable χ = x − C0 t. As a result, we get the 4

ACCEPTED MANUSCRIPT following system of equations (waves are omitted) −C0 ρ0 + (ρUx )0 = 0, −C0 By0 − Ez0 = 0, −C0 Bz0 + Ey0 = 0,

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B 2 + Bz2 −C0 (ρUx ) + + y = 0, 2 −C0 (ρUy )0 + (ρUx Uy − Bx By )0 = 0, −C0 (ρUz )0 + (ρUx Uz − Bx Bz )0 = 0, (1 − µ) Bx 0 ξ 2 ξ2 0 Ey − Ey00 = (Bz Ux − Bx Uz ) + ξ √ B − (Ux Bz0 ) , ρ µ ρ y ρ 0 (1 − µ) Bx 0 ξ 2  ξ 2 00 Ez − Ez = (Bx Uy − By Ux ) + ξ √ Bz + Ux By0 , ρ µ ρ ρ 0

(2.6)

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ρUx2

ξ2 =

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where primes denotes differentiation with respect to χ and parameters ξ and µ are given by formulas: c2 me mi , 4πe2 ρ0 L20

µ=

me . mi

(2.7)

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Integrating the first six equations from (2.6) with respect to χ, we have:

(2.8)

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ρVx = Π, Ez = Ez 0 − C0 By , Ey = Ey 0 + C0 Bz , ! By2 + Bz2 1 Vx = Nx − − C0 , Π 2 1 Uy = (Ny + Bx By ), Π 1 Uz = (Nz + Bx Bz ), Π

where Π, Ez 0 . Ey 0 , Nx , Ny , Nz are constants of integration and Vx = Ux − C0 .

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Substituting expressions (2.8) into the last two equations (2.6), we get the system of equations for the components of the magnetic field !

(1 − µ) Bx ξ2 Bx2 Bx Nz 0 Vx (Vx Bz0 )) − ξ √ Vx By0 + Bz − Vx + Ey 0 + = 0, Π µ Π Π Π ! 2 0 (1 − µ) B ξ2  B Bx Ny x x − Vx Vx By0 ) − ξ √ Vx Bz0 − By − Vx + Ez 0 − = 0. Π µ Π Π Π (2.9) 5

ACCEPTED MANUSCRIPT Without loss of generality one can choose B0 so that Bx = 1. Taking this fact into account and assuming that Vx dτ = dχ we obtain ξ 2 By00 − (ΠVx − 1)By + αBz0 = ΠEz 0 − Ny , ξ 2 Bz00 − (ΠVx − 1)Bz − αBy0 = −ΠEy 0 − Nz ,

(2.10)

√ where primes denotes differentiation with respect to τ and α = ξ(1 − µ)/ µ. Let us consider the case where ΠEz 0 − Ny = −ΠEy 0 − Nz = 0 and convert the system of equations (2.10) to a polar form. Using the following transformations Bz = B sin ϕ,

we obtain

B=

q

 2 0 0  ξ (2B ϕ

By2 + Bz2 ,

+ Bϕ00 ) − αB 0 = 0, B2 − B2 − 2  ξ 2 (B 00 − B(ϕ0 )2 ) + αBϕ0 − m B = 0, 2

(2.12)

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2 where Bm = 2Nx − 2C0 Π.

(2.11)

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By = B cos ϕ,

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Multipling the first equation of system (2.12) by B and integrating the result, we get Ω α ϕ0 = 2 2 + 2 , (2.13) ξ B 2ξ where Ω is an integration constant. Substituting (2.13) into the second equation of system (2.12), we obtain an equation for the magnetic field:

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2 Bm 1 6 Ω2 α2 4 − + 1 B + B − 4 = 0. 4ξ 2 2 2ξ 2 ξ

1 B B + 2 ξ 3

(2.14)

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Using a new independent variable τ = ξθ, from (2.14) we get

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1 00 B 3 B + δB 4 + B 6 − β 2 = 0, 2

where

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(2.16)

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2 α2 Bm Ω2 2 δ= − + 1 , β = . 4ξ 2 2 ξ2 The rest of this work is devoted to investigation of (2.15).

(2.15)

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General solution of Eq. (2.15).

In [20] it was found that in the two-fluid MHD can exist solitary waves which are similar to soliton solutions of the Korteweg–de Vries equation. The exis6

ACCEPTED MANUSCRIPT tence of solitary waves point out that the two-fluid MHD can be integrable mathematical model. To investigate integrability of considered mathematical model let us analyze equation (2.15) on the Painlev´e property. To this aim we use the algorithm of the Painlev´e test in the interpretation by Ablowitz, Ramani and Segur [24–26]. According to this algorithm we seek for a solution of (2.15) in the form ∞ X i=0

ai (θ − θ0 )j−p ,

where θ0 is an arbitrary constant.

(3.1)

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B=

Substituting B = a0 /θp into leading terms of (2.15) we find that p = 1 and (1,2) a0 = ±2i. In order to find the Fuchs indices we substitute (1,2)

a0 (θ) = θ

± ar θr−1 .

(3.2)

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B

(1,2)

(1,2)

(1,2)

= = −1, r2 into leading terms of (2.15). As a results, we find that r1 4. Note that the first Fuchs index corresponds to arbitrariness of θ0 , while the second Fuchs index indicates that coefficient a4 has to be arbitrary in expansion (3.1).

2i iδ (1,2) ± (θ − θ0 ) + a4 (θ − θ0 )3 + . . . , θ − θ0 3

(3.3)

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B (1,2) (θ) = ±

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In the third step of the Painlev´e test we substitute (3.1) into (2.15). As a result we find the following expansion

(1,2)

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where a4 are arbitrary constants. The presence of two arbitrary constants (1,2) (θ0 and a4 ) in the expansion (3.3) suggests that the equation (2.15) can have the Painlev´e property. However, this equation does not belong to the list of Painlev´e–type second order differential equations (see, e.g. [27]).

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Indeed, it can be seen that in addition to expansion (3.3) Eq. (2.15) also admits the following expansion

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B = b0 (θ − θ0 )1/2 + b1 (θ − θ0 )3/2 −

2δb20 + 3b21 (θ − θ0 )5/2 + . . . , 6b0

(3.4)

where b0 is a solution of the equation b40 + 4β 2 = 0 and b1 is an arbitrary constant. Since expansion (3.4) is an expansion in a neighborhood of a movable branch point and contains two arbitrary constants (θ0 and b1 ), Eq. (2.15) does not possess the Painlev´e property. As a consequence, taking into account the Ablowitz, Ramani and Segur hypothesis [26, 28, 29], it can be conjectured that the system of the equations of two-fluid magnetohydrodynamics does 7

ACCEPTED MANUSCRIPT not belong to the class of integrable mathematical models and does not have soliton solutions. Equation (2.15) belong to a class of equation which was recently considered in [30–32] and can be integrated in terms of the elliptic functions. Equation (2.15) admits the first integral 1 0 (B )2 + δB 2 + B 4 + β 2 B −2 = E, 4

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where E is an integration constant.

(3.5)

Fig. 1. Exact solution (3.8) of (2.15) and components By , Bz of magnetic field for electron–proton plasma (µ = 1/1836, ξ = 0.1, Ω = 1, Bm = 33, E = 3).

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Let us show that equation (3.5) can be reduced to an equation for the Weierstrass elliptic function. Substituting w = B 2 into (3.5) we obtain (w )2 + 4δw2 + w3 + 4β 2 − 4Ew = 0

(3.6)

(

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The general solution of equation (3.6) has the form )

θ − θ0 4δ w = −℘ , g2 , g3 − , 2 3

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4δ 2 g2 = 16 E + , 3





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4 32 g3 = 16 β + Eδ + δ 3 , 3 27 (3.7) where ℘ is the Weierstrass elliptic function, g2 , g3 are invariants and θ0 is an arbitrary constant. 2

Fig. 2. Exact solution (3.8) of (2.15) and components By , Bz of magnetic field for electron–proton plasma (µ = 1/1836, ξ = 0.1, Ω = 1, Bm = 15, E = 3291949/7344).

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ACCEPTED MANUSCRIPT Thus, the general solution of equations (3.5) and (2.15) can be written as "

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(

4δ θ − θ0 , g2 , g3 − B = −℘ 2 3

#1/2

,

(3.8)

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where invariants g2 , g3 are given in (3.7). Note that solution (3.8) is real. Examples of graphs of solution (3.8) and components By , Bz of magnetic field are demonstrated in Fig.1,2.

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Fig. 3. Dependence of period (3.9) of the solution (3.8) on parameter Ω with E = 0, ξ = 0.1, µ = 1/1836 and Bm = 31 (curve 1), Bm = 32 (curve 2), Bm = 35 (curve 3).

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Let us analyze obtained solution. The behavior of solution (3.8) determined by the sign of the discriminant ∆ = g23 − 27g32 . If ∆ = 0 solution (3.8) becomes degenerate and it can be expressed in terms of trigonometric or hyperbolic functions. In the case of ∆ 6= 0 solution (3.8) can not be expressed in terms of elementary functions and it is a periodic function.

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In the case of ∆ > 0 the real period of solution (3.8) is given by the formula

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4 K(k), T =√ e1 − e3

k=

s

e2 − e3 , e1 − e3

(3.9)

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where ei , i = 1, 2, 3 are the roots of the equation 4p3 − g2 p − g3 which are decreasingly ordered, K(k) is the complete elliptic integral of the first kind. Formula (3.9) allows us to study the dependence of the period of solution (3.8) on physical parameters. For example, in the case of electron-proton plasma the dependence of the period on parameter Ω is represented in Fig.3. Note that while Ω → 0 the period tents to infinity, which corresponds to degeneration of a periodic solution to a solitary wave. In the case of ∆ = 0 solution (3.9) degenerates, and we have the following relation between the parameters: β2 =

i 4 h 2 (4δ + 3E)3/2 − 9Eδ − 8δ 3 27

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(3.10)

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Fig. 4. Exact solution (3.12) of (2.15) and components By , Bz of the magnetic field for electron–proton plasma (µ = 1/1836, ξ = 0.1, Ω = 0, Bm = 31, E = 0).

The general solution of (3.5) in this case is given by the formula:

√ ( )#1/2 2 1/4 √ (3E + 4δ ) 2 3E + 4δ 2 4δ −2 √ B= − − 2 3E + 4δ 2 cos (θ − θ0 ) 3 3 2 (3.11) It can be seen that solution (3.11) has no poles on the real axis provided that E = 0. Then from (3.10) it follows that in this case either β 2 = 0 or β 2 = −64/27δ 3 . Note that the case of β 2 = −64/27δ 3 has no physical meaning, because the solution is real only if δ > 0. In the case of β = 0 and δ < 0 (δ = −|δ|) there is a bounded on the real axis solution in the form of a solitary wave: q

2 |δ|

q

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B=

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"

cosh{ |δ|(θ − θ0 )}

.

(3.12)

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Plots of the solution (3.12) and components By , Bz of the magnetic field are presented in Fig.4.

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Conditions for existence of solution (3.12) can be written as follows

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Ω = 0,

2 Bm <

(1 − µ)2 α2 + 2 = + 2. 2ξ 2 2µ

(3.13)

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Condition (3.13) can be simplified taking into account that the solitary wave solution tends to zero with θ → ±∞. In this case Ny = Nz = Ez0 = Ey0 = 0, Vx − = C0 , Bm = 2C02 and inequality (3.13) takes the following form: 2C02 <

(1 − µ)2 +2 2µ

(3.14)

Note that inequality (3.14) define the range of values of the solitary waves’ velocity. 10

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Conclusion

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In this work we have studied the system of equations of two-fluid magnetohydrodynamics with the assumptions of quasi-neutrality of plasma and quasistationary of electromagnetic field. In the case of an ideal cold plasma from this system of equations the ordinary differential equation describing traveling waves is obtained. Integrability of this equation is studied using the Painlev´e test. It is showed that the resulting equation does not belong to the class of integrable equations. The general solution of the studied equation is found and analyzed. Special cases leading to solitary waves are considered.

Acknowledgments

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This research was supported by Russian Science Foundation Grant No. 14– 11–00258.

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[3] N.A. Krall, A.W. Trivelpiece, Principles of plasma physics. New York. McGrawHill, 1978.

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[26] N.A. Kudryashov, Methods of nonlinear mathematical physics. Dolgoprudny. Intellect, 2010. [27] E.L. Ince, Ordinary Differential Equations. New York. Dover Publications, 1956.

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[28] M.J. Ablowitz, A. Ramani, H. Segur, Journal of Mathematical Physics 21 (1980) 715.

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[29] M.J. Ablowitz, A. Ramani, H. Segur, Lettere Al Nuovo Cimento 23 (1978) 333. [30] N.A. Kudryashov, D.I. Sinelshchikov, J. Phys. A Math. Theor. 47 (2014) 405202.

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[31] N.A. Kudryashov, D.I. Sinelshchikov, Phys. Lett. A. 379 (2015) 798.

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