# The Solitary Wave Solution for Quantum Plasma Nonlinear Dynamic Model

Authors: Yihu Feng; Lei Hou
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

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


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## Body

## 1. Introduction

In recent years, there have been many discussions on the study of various kinds of plasma, such as laser plasma, dense celestial plasma, and dust magnetic plasma. Jung [1] studied the quantum-mechanical effects on electron-electron scattering cross-sections which are investigated in dense high-temperature plasmas; an effective pseudopotential model taking into account both quantum-mechanical effects and plasma screening effects is applied to describe electron-electron interactions in dense high-temperature plasmas. Kremp et al. [2] discussed a kinetic theory for quantum many-particle systems in time-dependent electromagnetic fields which is developed based on a gauge-invariant formulation. The resulting kinetic equation generalizes previous results to quantum systems and includes many-body effects. It is, in particular, applicable to the interaction of strong laser fields with dense correlated plasmas. Shukla et al. [3] studied the decay of a magnetic-field-aligned Alfven wave into inertial and kinetic Alfven waves in plasmas and so on [4–9]. Some quantum plasma acoustic wave and quantum-related properties have also been discussed in depth [10, 11].Nowadays, many scholars [12] have made a lot of discussions on the solutions of nonlinear problems. Ramos [13] studied the existence, multiplicity, and shape of positive solutions of the system −ε2Δu+Vxu=Kxgv,−ε2Δv+Vxv=Hxfu in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V,H,andK are positive and locally Hölder continuous.Teresa and Angela [14] studied the existence of some new positive interior spike solutions to a semilinear Neumann problem.Faye et al. [15] built models for short-term, mean-term, and long-term dynamics of dune and megariple morphodynamics. They are models that degenerated parabolic equations which are, moreover, singularly perturbed. Then, they give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that they set out. Finally, the short-term model is homogenized.Sirendaoerji and Tangetusang [16] constructed some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuznetsov equation based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.Mao et al. [17] discussed nonlinear waves in an inhomogeneous quantum plasma; for an inhomogeneous quantum magnetoplasma system with density and temperature gradients, a two-dimensional nonlinear fluid dynamic equation is derived in the case where the collision frequency between ions and neutrals is minor. They also discussed how the shock, explosion, and vortex solutions of the potential for this system are obtained as well as the changes on the potential in the dense astrophysical environment.Mo and others [18–26] have also discussed a class of nonlinear physical problems by means of some asymptotic analysis methods.In this paper, a quantum plasma system with temperature gradient and density is discussed in the case of electrons and ions. The solitary wave solutions of its nonlinear system are discussed by using the relevant mathematical physical methods and theories.The rest of this paper is organized as follows. In Section2, we used dimensionless transformation and traveling wave transformation to discuss a nonlinear partial differential mathematical physical equation. In Section 3, we obtained zero solitary wave of plasma electrostatic potential. In Section 4, we obtained each solitary wave of the electrostatic potential of plasma. In Section 5, we obtained the force function of plasma solitary wave. Finally, a brief conclusion is given in Section 6, in which we used the approximate method to solve it; the hyperbolic function method of undetermined coefficients and perturbation theory is an effective way. The method of solution is parse operation. So it continues to study solitary wave solutions related to physical quantities of other physical state.


## 2. Quantum Plasma Dynamics Model

According to the theory of quantum plasma dynamics, when the collision frequency of quantum is relatively low, the nonlinear dynamic model of a class of unhomogeneous quantum plasma is as follows [17].
(1)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=fε,u.In the above expression,u is the electrostatic potential, a=−3e/4lT, l is the Boltzmann constant, e is the electric quantity of electrons, T is the Fermi temperature of electrons, λ is the Fermi wavelength of the electron, ρ=lT/mΩ2 is the Fermi radius of the electron, H is the quantum parameter, ω is the ion cyclotron frequency, m is the mass of ions, and v is the drift velocity of quantum ions. D0=3ln−lt/4B0, where ln is the density coefficient, lt is the temperature gradient, B0 is the measure of uniform external magnetic field, Ω is the ion cyclotron frequency, c is the sound velocity of quantum ions, and f is the perturbation term caused by the related factors of quantum plasma. Let it be a sufficiently smooth function and f0,u=0.Equation (1) is a nonlinear partial differential mathematical physical equation. We use a special method to discuss the perturbed 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
(2)∂2v∂t12−b1∂2v2∂t12−b2∂4v∂t12∂y12−b3∂3v∂t1∂y12−b4∂2v∂z12−b5∂2v∂t1∂y1−b6∂2v2∂t1∂y1=gε,v,where gε,v=fε,lT/ev and the expressions of dimensionless coefficient bii=1,2,⋯,6 are omitted.Solitary wave solutions for the nonlinear dimensionless quantum plasma (Eq. (2)) are obtained by introducing traveling wave transformation.
(3)ζ=l¯1y1+l¯2z1−l¯3t1,where l¯1 and l¯2 are wave numbers and l¯3 is the wave frequency, substituting Eq. (3) into Eq. (2).
(4)d4vdζ4+αd3vdζ3+βd2vdζ2+γvd2vdζ2+κd2v2dζ2=Gε,v,where
(5)α=−b3b2l¯3,β=−l¯32−b4l¯22+b5l¯1l¯3b2l¯12l¯32,γ=−l32+b4l¯22−b5l¯1l¯3b2l¯12l¯32,κ=b1l¯3−b6l¯1b2l¯12l¯3,Gε,v=−gε,vb2l¯12l¯32.


## 3. Zeroth Solitary Wave of the Plasma Electrostatic Potential

Let the perturbation solution for the nonlinear dimensionless traveling wave (Eq.4) be [12]
(6)vζ,ε=∑i=0∞viζεi.Substituting Eq. (6) into the traveling wave (Eq. (4)), the nonlinear term is expanded according to perturbation parameter ε, combine the same power terms of εii=0,1,⋯, and let their coefficients equal to zero.From the coefficients ofε0 equal zero, the nonlinear equation is obtained.
(7)d4v0dζ4+αd3v0dζ3+βd2v0dζ2+γv0d2v0dζ2+κd2v02dζ2=0,where α,β,γ,andκ 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.
(8)v0ζ=P0+P1tanhζ+P2tanh2ζ+Q1tanhζ+Q2tanh2ζ,where Pi,Qji=0,1,2,j=1,2 are undetermined constants. By substituting Eq. (8) into the nonlinear Eq. (7), combining the positive and negative powers coefficients of tanhζ, and setting them to zero, we can determine the coefficients of Pi and Qj. Then, we substituted Pi and Qj into Eq. (8), which have the following two solitary wave solutions.
(9)v10ζ=T0+T12tanhζ+1tanh2ζ,(10)v20ζ=T0+T12−tanhζtanhζ,where T0,T1 are constants,
(11)T0=−12l12b2b32+10l1b1b3b5−100l22b22b4+b322b3b1b3−10l1b2b6,T1=6l12b2b3b1b3−10l1b2b6.LetT0=0,T1=10. The solitary wave solutions (9) are described in the normal solitary wave solution of the singular soliton v10ζ, which is shown in Figure 1.Figure 1
Illustration for the zeroth solitary wavev10ζ curve in quantum plasma electrostatic potential T0=0,T1=10 (ζ for the abscissa, v10ζ for the ordinate).TakeT0=0,T1=1; at this time, the solitary wave solutions (9) is described in the solitary wave solution v¯10ζ, which is shown in Figure 2.Figure 2
Illustration for the zeroth solitary wavev¯10ζ curve in quantum plasma electrostatic potential T0=0,T1=1 (ζ for the abscissa, v¯10ζ for the ordinate).Comparing Figures1 and 2, it can be seen that the two corresponding solitary waves (v10 and v¯10) are quite different in the strength (dimensionless scale) of the corresponding physical quantities by choosing different physical parameters.


## 4. Each Solitary Wave of the Plasma Electrostatic Potential

Substituting Eq. (7) into travelling wave Eq. (4), expand nonlinear terms according to perturbation parameter ε and combine the terms of the same power of εii=0,1,⋯. From the coefficient of εi which is equal to zero, the linear equation is obtained.
(12)d4v1dζ4+αd3v1dζ3+β+2κ+γv0d2v1dζ2+4κdv0dζdv1dζ+2κ+γv0d2v0dζ2v1=G0,v0,where v0 is a known function determined by Eqs. (9) and (10). The solutions v11ζ and v21ζ under zero initial value can be obtained from linear equation (12).From the perturbation theory ([12] and Eq. (7)), we can obtain v1ζ,ε,v2ζ,ε of the first approximate solitary wave solutions which are two nonuniform nonlinear dimensionless quantum plasmas for the electrostatic potential (Eq. (4)).
(13)v1ζ,ε=T0+T12tanhζ+1tanh2ζ+εv11ζ+Oε2,0<ε≪1,(14)v2ζ,ε=T0+T12−tanhζtanhζ+εv21ζ+Oε2,0<ε≪1.TakeT0=0,T1=10; at this time, Eq. (13) of the solitary wave solution describes the first perturbation function v11ζ for the electrostatic potential of solitary wave, which is shown in Figure 3.Figure 3
Illustration for the first solitary wavev11ζ curve in quantum plasma electrostatic potential T0=0,T1=10 (ζ for the abscissas, v11ζ for the ordinate).By the same way, the sequence of functionsv1n and v2n can be obtained successively from the structure of the quantum plasma nonlinear dynamics (Eq. (4)).According to the perturbation theory [12], the series can be determined by Eq. (7).
(15)v¯1ζ,ε=∑i=0∞v1iζεi,v¯2ζ,ε=∑i=0∞v2iζεi,where is uniformly valid for the finite interval of ζ, and
(16)v¯1ζ,ε=∑i=0nv1iζεi+Oεn+1,0<ε≪1,v¯2ζ,ε=∑i=0nv2iζεi+Oεn+1,0<ε≪1.Therefore,v1nζ,ε and v2nζ,ε are the nth asymptotic approximate solutions for the electrostatic potential solitary waves of the two quantum plasmas in Eq. (4).
(17)v1nζ,ε=T0+T12tanhζ+1tanh2ζ+∑i=1nv1iζεi+Oεn+1,0<ε≪1,v2nζ,ε=T0+T12−tanhζtanhζ+∑i=1nv2iζεi+Oεn+1,0<ε≪1.Considering the transformation (3), unl1y1+l2z1−l3t1 is the nth traveling wave solution for the two electrostatic potential solitary waves of the quantum plasma nonlinear dynamics dimensionless (Eq. (2)).
(18)u1nx1,y1,z1,ε=T0+T12tanhl¯1y1+l¯2z1−l¯3t+1tanh2l¯1y1+l¯2z1−l¯3t+∑i=1nv1il¯1y1+l¯2z1−l¯3tεi+Oεn+1,0<ε≪1,u2nx1,y1,z1,ε=T0+T12−tanhl¯1y1+l¯2z1−l¯3ttanhl¯1y1+l¯2z1−l¯3t+∑i=1nv2il¯1y1+l¯2z1−l¯3tεi+Oεn+1,0<ε≪1.


## 5. Force Function of the Plasma Solitary Wave

By using the electrostatic potential solitary wave perturbation solution for the quantum plasma nonlinear dynamics dimensionless (Eq. (4)) [12], all kinds of relevant physical quantities can be obtained. For example, from Eq. (4) and relational Eqs. (13) and (14), it is not difficult to obtain the subsolitary wave functions F1nζ,ε and F2nζ,ε of the two dimensionless force functions for the quantum plasma nonlinear dynamics.
(19)F1nζ,ε=2T11+1tanhζ−1tanh2ζ−1tanh3ζ+∑i=1ndv1idζζεi+Oεn+1,n=1,2,⋯,0<ε≪1,F2nζ,ε=2T11−tanhζ−tanh2ζ+tanh3ζ+∑1=1ndv2idζζεi+Oεn+1,n=1,2,⋯,0<ε≪1.TakeT1=10; the curve of the first solitary wave function F11ζ,ε for the dimensionless force function to the quantum plasma nonlinear dynamics model is shown Figure 4.Figure 4
Illustration for the first solitary waveF11ζ curve in quantum plasma force function T1=10 (ζ for the abscissa, F11ζ for the ordinate).


## 6. Conclusion

From nonuniform quantum dynamics equations of the plasma system, we can discuss the impact of the potential solutions of the system, explosion, and vortex solutions and analyze the range of the electric potential of the shock wave and the blast wave width and density for the relationship between the changes to the drift velocity, to understand the stability of the static potential changing at any time in space.Unhomogeneous quantum plasma disturbance system comes from a complex natural phenomena. In order to study the nonlinear solitary of more complex models, sometimes, we need to use the approximate method to solve it. In this paper, the hyperbolic function of undetermined coefficients method and perturbation theory is an effective way. The method of solution is parse operation. So it can continue to study the solitary wave solutions related to physical quantities of other physical state.



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