Research ArticleThe Solitary Wave Solution for Quantum Plasma NonlinearDynamic Model

Yihu Feng 1,2 and Lei Hou1

1Department of Mathematics, Shanghai University, Shanghai 200444, China2Department of Electronics and Information Engineering, Bozhou University, Bozhou, Anhui 236800, China

Correspondence should be addressed to Yihu Feng; fengyihubzxy@163.com

Received 2 October 2019; Accepted 19 November 2019; Published 6 January 2020

Academic Editor: P. Areias

Copyright © 2020 Yihu Feng and Lei Hou. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In this paper, we discussed the quantum plasma system. A nonlinear dynamic disturbed model is studied. We used theundetermined coefficients method, dimensionless transformation and traveling wave transformation for the hyperbolicfunctions, and perturbation theory and method; then, the solitary wave solution for the quantum plasma nonlinear dynamicmodel is solved. Finally, the characteristics of the corresponding physical quantity are described.

1. Introduction

In recent years, there have been many discussions on thestudy of various kinds of plasma, such as laser plasma, densecelestial plasma, and dust magnetic plasma. Jung [1] studiedthe quantum-mechanical effects on electron-electron scatter-ing cross-sections which are investigated in dense high-temperature plasmas; an effective pseudopotential modeltaking into account both quantum-mechanical effects andplasma screening effects is applied to describe electron-electron interactions in dense high-temperature plasmas.Kremp et al. [2] discussed a kinetic theory for quantummany-particle systems in time-dependent electromagneticfields which is developed based on a gauge-invariant formu-lation. The resulting kinetic equation generalizes previousresults to quantum systems and includes many-body effects.It is, in particular, applicable to the interaction of strong laserfields with dense correlated plasmas. Shukla et al. [3] studiedthe decay of a magnetic-field-aligned Alfven wave into iner-tial and kinetic Alfven waves in plasmas and so on [4–9].Some quantum plasma acoustic wave and quantum-relatedproperties have also been discussed in depth [10, 11].

Nowadays, many scholars [12] have made a lot of discus-sions on the solutions of nonlinear problems. Ramos [13]

studied the existence, multiplicity, and shape of positivesolutions of the system −ε2Δu + VðxÞu = KðxÞgðvÞ,−ε2Δv +VðxÞv =HðxÞf ðuÞ in RN , as ε→ 0. The functions f and gare power-like nonlinearities with superlinear and subcriticalgrowth at infinity, and V ,H, andK are positive and locallyHölder continuous.

Teresa and Angela [14] studied the existence of some newpositive interior spike solutions to a semilinear Neumannproblem.

Faye et al. [15] built models for short-term, mean-term,and long-term dynamics of dune and megariple morphody-namics. They are models that degenerated parabolic equa-tions which are, moreover, singularly perturbed. Then, theygive an existence and uniqueness result for the short-termand mean-term models. This result is based on a time-space periodic solution existence result for degenerated para-bolic equation that they set out. Finally, the short-termmodelis homogenized.

Sirendaoerji and Tangetusang [16] constructed some newexact solitary solutions to the generalized mKdV equationand generalized Zakharov-Kuznetsov equation based onhyperbolic tanh-function method and homogeneous balancemethod, and auxiliary equation method, by the method ofauxiliary equation with function transformation with aid of

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 5602373, 6 pageshttps://doi.org/10.1155/2020/5602373

symbolic computation system Mathematica. The method isof important significance in seeking new exact solutions tothe evolution equation with arbitrary nonlinear term.

Mao et al. [17] discussed nonlinear waves in an inhomo-geneous quantum plasma; for an inhomogeneous quantummagnetoplasma system with density and temperature gradi-ents, a two-dimensional nonlinear fluid dynamic equationis derived in the case where the collision frequency betweenions and neutrals is minor. They also discussed how theshock, explosion, and vortex solutions of the potential for thissystem are obtained as well as the changes on the potential inthe dense astrophysical environment.

Mo and others [18–26] have also discussed a class ofnonlinear physical problems by means of some asymptoticanalysis methods.

In this paper, a quantum plasma system with tempera-ture gradient and density is discussed in the case of electronsand ions. The solitary wave solutions of its nonlinear systemare discussed by using the relevant mathematical physicalmethods and theories.

The rest of this paper is organized as follows. In Section 2,we used dimensionless transformation and traveling wavetransformation to discuss a nonlinear partial differentialmathematical physical equation. In Section 3, we obtainedzero solitary wave of plasma electrostatic potential. InSection 4, we obtained each solitary wave of the electrostaticpotential of plasma. In Section 5, we obtained the force func-tion of plasma solitary wave. Finally, a brief conclusion isgiven in Section 6, in which we used the approximate methodto solve it; the hyperbolic function method of undeterminedcoefficients and perturbation theory is an effective way. Themethod of solution is parse operation. So it continues tostudy solitary wave solutions related to physical quantitiesof other physical state.

2. Quantum Plasma Dynamics Model

According to the theory of quantum plasma dynamics, whenthe collision frequency of quantum is relatively low, the non-linear dynamic model of a class of unhomogeneous quantumplasma is as follows [17].

32∂2u∂t2

+ a∂2u2

∂t2+ H2 − λ2 − ρ2� � ∂4u

∂t2∂y2− ωρ2

∂3u∂t∂y2

+ 3u2

∂2u∂t∂y

−D0∂2u2

∂t∂y− c2

∂2u∂z2

= f ε, uð Þ:ð1Þ

In the above expression, u is the electrostatic potential,a = −3e/4lT , l is the Boltzmann constant, e is the electricquantity of electrons, T is the Fermi temperature of electrons,λ is the Fermi wavelength of the electron, ρ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilT/mΩ2p

is theFermi radius of the electron, H is the quantum parameter, ωis the ion cyclotron frequency, m is the mass of ions, and v isthe drift velocity of quantum ions. D0 = 3ðln − ltÞ/4B0, whereln is the density coefficient, lt is the temperature gradient, B0is the measure of uniform external magnetic field, Ω is theion cyclotron frequency, c is the sound velocity of quantumions, and f is the perturbation term caused by the related

factors of quantum plasma. Let it be a sufficiently smoothfunction and f ð0, uÞ = 0.

Equation (1) is a nonlinear partial differential mathemat-ical physical equation. We use a special method to discuss theperturbed solitary wave solutions of Eq. (1).

Dimensionless transformation for Eq. (1): v = eu/lT ,y1 = y/ρ, z1 = z/ρ, t1 =Ωt, form normalization and Eq.(1) becomes

∂2v∂t21

− b1∂2v2

∂t21− b2

∂4v∂t21∂y21

− b3∂3v

∂t1∂y21− b4

∂2v∂z21

− b5∂2v

∂t1∂y1

− b6∂2v2

∂t1∂y1= g ε, vð Þ,

ð2Þ

where gðε, vÞ = f ðε, lT/evÞ and the expressions of dimen-sionless coefficient biði = 1, 2,⋯,6Þ are omitted.

Solitary wave solutions for the nonlinear dimensionlessquantum plasma (Eq. (2)) are obtained by introducing trav-eling wave transformation.

ζ =�l1y1 +�l2z1 −�l3t1, ð3Þ

where �l1 and �l2 are wave numbers and �l3 is the wave fre-quency, substituting Eq. (3) into Eq. (2).

d4v

dζ4+ α

d3v

dζ3+ β

d2v

dζ2+ γv

d2v

dζ2+ κ

d2v2

dζ2= G ε, vð Þ, ð4Þ

where

α = −b3b2�l3

,

β = −�l23 − b4�l

22 + b5�l1�l3

b2�l21�l23

,

γ = −l23 + b4�l22 − b5�l1�l3

b2�l21�l23

,

κ = b1�l3 − b6�l1b2�l

21�l3

,

G ε, vð Þ = −g ε, vð Þb2�l

21�l23:

ð5Þ

3. Zeroth Solitary Wave of the PlasmaElectrostatic Potential

Let the perturbation solution for the nonlinear dimensionlesstraveling wave (Eq. ð4Þ) be [12]

v ζ, εð Þ = 〠∞

i=0vi ζð Þεi: ð6Þ

2 Advances in Mathematical Physics

Substituting Eq. (6) into the traveling wave (Eq. (4)), thenonlinear term is expanded according to perturbationparameter ε, combine the same power terms of εiði = 0, 1,⋯Þ,and let their coefficients equal to zero.

From the coefficients of ε0 equal zero, the nonlinear equa-tion is obtained.

d4v0dζ4

+ αd3v0dζ3

+ βd2v0dζ2

+ γv0d2v0dζ2

+ κd2v20dζ2

= 0, ð7Þ

where α, β, γ, and κ are represented by (5).The solution of Eq. (7) is obtained by the undetermined

coefficient method of hyperbolic function [12]. Let Eq. (7)have a solitary wave solution.

v0 ζð Þ = P0 + P1 tanh ζð Þ + P2 tanh2 ζð Þ + Q1tanh ζð Þ + Q2

tanh2 ζð Þ,

ð8Þ

where Pi,Qjði = 0, 1, 2, j = 1, 2Þ are undetermined constants.By substituting Eq. (8) into the nonlinear Eq. (7), combiningthe positive and negative powers coefficients of tanh ðζÞ, andsetting them to zero, we can determine the coefficients of Piand Qj. Then, we substituted Pi and Qj into Eq. (8), whichhave the following two solitary wave solutions.

v10 ζð Þ = T0 + T12 tanh ζð Þ + 1ð Þ

tanh2 ζð Þ , ð9Þ

v20 ζð Þ = T0 + T1 2 − tanh ζð Þð Þ tanh ζð Þ, ð10Þwhere T0, T1 are constants,

T0 = −12l21b2b23 + 10l1b1b3b5 − 100l22b22b4 + b23

2b3 b1b3 − 10l1b2b6ð Þ ,

T1 =6l21b2b3

b1b3 − 10l1b2b6:

ð11Þ

Let T0 = 0, T1 = 10. The solitary wave solutions (9) aredescribed in the normal solitary wave solution of the singularsoliton v10ðζÞ, which is shown in Figure 1.

Take T0 = 0, T1 = 1; at this time, the solitary wave solu-tions (9) is described in the solitary wave solution �v10ðζÞ,which is shown in Figure 2.

Comparing Figures 1 and 2, it can be seen that the twocorresponding solitary waves (v10 and �v10) are quite differentin the strength (dimensionless scale) of the correspondingphysical quantities by choosing different physical parameters.

4. Each Solitary Wave of the PlasmaElectrostatic Potential

Substituting Eq. (7) into travelling wave Eq. (4), expandnonlinear terms according to perturbation parameter ε andcombine the terms of the same power of εiði = 0, 1,⋯Þ. From

the coefficient of εi which is equal to zero, the linear equationis obtained.

d4v1dζ4

+ αd3v1dζ3

+ β + 2κ + γð Þv0ð Þ d2v1dζ2

+ 4κ dv0dζ

dv1dζ

+ 2κ + γv0ð Þ d2v0dζ2

v1 =G 0, v0ð Þ,ð12Þ

where v0 is a known function determined by Eqs. (9) and(10). The solutions v11ðζÞ and v21ðζÞ under zero initial valuecan be obtained from linear equation (12).

From the perturbation theory ([12] and Eq. (7)), wecan obtain v1ðζ, εÞ, v2ðζ, εÞ of the first approximate solitarywave solutions which are two nonuniform nonlinear

0 1 2 3 4 50

100

200

300

400

500

600

700

800

900

Figure 1: Illustration for the zeroth solitary wave v10ðζÞ curve inquantum plasma electrostatic potential ðT0 = 0, T1 = 10Þ (ζ for theabscissa, v10ðζÞ for the ordinate).

0 1 2 3 4 50

10

20

30

40

50

60

70

80

90

Figure 2: Illustration for the zeroth solitary wave �v10ðζÞ curve inquantum plasma electrostatic potential ðT0 = 0, T1 = 1Þ (ζ for theabscissa, �v10ðζÞ for the ordinate).

3Advances in Mathematical Physics

dimensionless quantum plasmas for the electrostatic poten-tial (Eq. (4)).

v1 ζ, εð Þ = T0 + T12 tanh ζð Þ + 1ð Þ

tanh2 ζð Þ+ εv11 ζð Þ +O ε2

� �,

 0 < ε≪ 1,ð13Þ

v2 ζ, εð Þ = T0 + T1 2 − tanh ζð Þð Þ tanh ζð Þ + εv21 ζð Þ +O ε2� �

, 0 < ε≪ 1:

ð14Þ

Take T0 = 0, T1 = 10; at this time, Eq. (13) of the solitarywave solution describes the first perturbation function v11ðζÞfor the electrostatic potential of solitary wave, which is shownin Figure 3.

By the same way, the sequence of functions fv1ng andfv2ng can be obtained successively from the structure ofthe quantum plasma nonlinear dynamics (Eq. (4)).

According to the perturbation theory [12], the series canbe determined by Eq. (7).

�v1 ζ, εð Þ = 〠∞

i=0v1i ζð Þεi, �v2 ζ, εð Þ = 〠

∞

i=0v2i ζð Þεi, ð15Þ

where is uniformly valid for the finite interval of ζ, and

�v1 ζ, εð Þ = 〠n

i=0v1i ζð Þεi +O εn+1

� �, 0 < ε≪ 1,

�v2 ζ, εð Þ = 〠n

i=0v2i ζð Þεi +O εn+1

� �, 0 < ε≪ 1:

ð16Þ

Therefore, v1nðζ, εÞ and v2nðζ, εÞ are the nth asymptoticapproximate solutions for the electrostatic potential solitarywaves of the two quantum plasmas in Eq. (4).

v1n ζ, εð Þ = T0 + T12 tanh ζð Þ + 1ð Þ

tanh2 ζð Þ

+ 〠n

i=1v1i ζð Þεi +O εn+1

� �, 0 < ε≪ 1,

v2n ζ, εð Þ = T0 + T1 2 − tanh ζð Þð Þ tanh ζð Þ

+ 〠n

i=1v2i ζð Þεi +O εn+1

� �, 0 < ε≪ 1:

ð17Þ

Considering the transformation (3), unðl1y1 + l2z1 − l3t1Þis the nth traveling wave solution for the two electrostaticpotential solitary waves of the quantum plasma nonlineardynamics dimensionless (Eq. (2)).

u1n x1, y1, z1, εð Þ = T0 + T12 tanh �l1y1 +�l2z1 −�l3t

� �+ 1

� �tanh2 �l1y1 +�l2z1 −�l3t

� �+ 〠

n

i=1v1i �l1y1 +�l2z1 −�l3t� �

εi +O εn+1� �

, 0 < ε≪ 1,

u2n x1, y1, z1, εð Þ = T0 + T1 2 − tanh �l1y1 +�l2z1 −�l3t� �� �

tanh

� �l1y1 +�l2z1 −�l3t� �

+ 〠n

i=1v2i �l1y1 +�l2z1 −�l3t� �

εi

+O εn+1� �

, 0 < ε≪ 1:ð18Þ

5. Force Function of the Plasma Solitary Wave

By using the electrostatic potential solitary wave perturbationsolution for the quantum plasma nonlinear dynamics dimen-sionless (Eq. (4)) [12], all kinds of relevant physical quantitiescan be obtained. For example, from Eq. (4) and relationalEqs. (13) and (14), it is not difficult to obtain the subsolitarywave functions F1nðζ, εÞ and F2nðζ, εÞ of the two dimension-less force functions for the quantum plasma nonlineardynamics.

F1n ζ, εð Þ = 2T1 1 + 1tanh ζð Þ −

1tanh2 ζð Þ

−1

tanh3 ζð Þ

!

+ 〠n

i=1

dv1idζ

ζð Þεi +O εn+1� �

, n = 1, 2,⋯, 0 < ε≪ 1,

F2n ζ, εð Þ = 2T1 1 − tanh ζð Þ − tanh2 ζð Þ + tanh3 ζð Þ� �+ 〠

n

1=1

dv2idζ

ζð Þεi +O εn+1� �

, n = 1, 2,⋯, 0 < ε≪ 1:

ð19Þ

Take T1 = 10; the curve of the first solitary wave functionF11ðζ, εÞ for the dimensionless force function to the quantumplasma nonlinear dynamics model is shown Figure 4.

6. Conclusion

From nonuniform quantum dynamics equations of theplasma system, we can discuss the impact of the potentialsolutions of the system, explosion, and vortex solutions and

0 0.5 1 1.5 2 2.5 30

1000

2000

3000

4000

5000

6000

Figure 3: Illustration for the first solitary wave v11ðζÞ curve inquantum plasma electrostatic potential ðT0 = 0, T1 = 10Þ (ζ for theabscissas, v11ðζÞ for the ordinate).

4 Advances in Mathematical Physics

analyze the range of the electric potential of the shock waveand the blast wave width and density for the relationshipbetween the changes to the drift velocity, to understandthe stability of the static potential changing at any timein space.

Unhomogeneous quantum plasma disturbance systemcomes from a complex natural phenomena. In order to studythe nonlinear solitary of more complex models, sometimes,we need to use the approximate method to solve it. In thispaper, the hyperbolic function of undetermined coefficientsmethod and perturbation theory is an effective way. Themethod of solution is parse operation. So it can continue tostudy the solitary wave solutions related to physical quanti-ties of other physical state.

Data Availability

The authors declare that the data in the article is available,shareable, and referenced. Readers can check to each articlehere: http://apps.webofknowledge.com/.

Conflicts of Interest

No potential conflict of interest was reported by the authors.

Acknowledgments

This work has been supported by the National NaturalScience Foundation of China (11271247); the Project ofSupport Program for Excellent Youth Talent in Collegesand Universities of Anhui Province (gxyqZD2016520); theKey Project for Teaching Research in Anhui Province(2018jyxm0594, 2016jyxm0677); the Key Project of TeachingResearch of Bozhou University (2017zdjy02); and the KeyProject of the Natural Science Research of Bozhou University(BYZ2017B02).

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0 0.1 0.2 0.3 0.4 0.5−2.5

−2

−1.5

−1

−0.5

0×107

Figure 4: Illustration for the first solitary wave F11ðζÞ curve inquantum plasma force function ðT1 = 10Þ (ζ for the abscissa,F11ðζÞ for the ordinate).

5Advances in Mathematical Physics

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