Exact Static Plane Symmetric Soliton-Like Solutions to the Nonlinear Interacting Electromagnetic and Scalar Field Equations in General Relativity Electromagnetic, Gravitational Fields Interaction’s, Description of Elementary Particles

This research work is related to soliton solutions considered as models that can describe the complex configuration of elementary particles from the study of the interactions of their fields. It is interested in the interaction of fields between two different elementary particles by expressing their physical properties. For that, we have obtained, exact static plane symmetric soli-ton-like solutions to the nonlinear equations of interacting electromagnetic and scalar fields taking into account the own gravitational field of elementary particles using the calibrated invariance function ( ) P I . It has been proved that all solutions of the Einstein, nonlinear electromagnetic and scalar field equations are regular with the localized energy density. Moreover, the total charge of particles is finite and the total energy of the interaction fields is bounded. It have been emphasized the importance to the own gravitational field of elementary particles and the role of the nonlinearity of fields in the determination of these solutions. In flat space-time, soliton-like solutions ex-ist but the total energy of the interaction fields is equal to zero. We have also shown that in the linear case, soliton-like solutions are absent.


Introduction
The exact solutions of nonlinear differential equations to describe physical phenomena have become a scientific priority after the development of general relativity (GR) and quantum field theory (QFT). New concepts have been created for this purpose. For example, the strange attractor is linked to the notion of chaos and applied to systems with a low degree of freedom but also the notion of soliton. The soliton, considered as an exact, regular, localized energy density, finite total energy and stable solution of nonlinear differential equations is widely used in pure science [1]. In particle physics, the soliton-like solution is used to describe the complex internal configuration of elementary particles experimentally proven [2] [3] [4]. Several works related to soliton-like solutions have been done. Shikin [5] [6] has elaborated the theory of solitons in general relativity.
Bronnikov et al. [7]  To answer this aim, Section 2 addresses the model and fundamental equations.
In Section 3, the solutions of Einstein's equation and of interacting nonlinear electromagnetic and scalar fields are established by taking into account the own gravitational field of the elementary particles. Section 4, gives an account of the influence of the own gravitational field of the elementary particles and the role of the nonlinearity of the fields on the obtained solutions. A conclusion follows in Section 5.

Model and Fundamental Equations
In general relativity, the Einstein equation has the form: From (1), (2) and (3), we find the components of Einstein's tensor equation [16]: Let us choose the Lagrangian [17]: = + is some arbitrary function characterizing the interaction between the nonlinear electromagnetic and scalar fields; is the 4-vector potential λ represents the parameter of the nonlinearity.
From the Lagrangian (9), we establish the nonlinear scalar and the electromagnetic field equations [8]: In the absence of a current source and a magnetic monopole, the chronometric invariant I is written: The scalar field Equation (10) is reduced to: which has the solution: On the other hand, the nonlinear electromagnetic field Equation (11) develops into four related equations: 0, e e 0, e 0.
Focusing exclusively on the electric part of the electromagnetic field, the differential equation system (14) reduces to: .
The energy-momentum metric tensor of interacting nonlinear electromagnetic and scalar fields is: Its non-zero components verify the equalities: The sum of Einstein's tensors which has the solution: Summation of Einstein's tensors Substituting (21) The first integration of (22) gives: For 0 K = , the solution of (23) is: The Equation (15) The energy density per unit invariant volume, ( ) 0 3 The total energy f E of interacting nonlinear electromagnetic and scalar fields verifies: In generally from (11) one gets [9]: where j ν is the 4-vector current density tensor. Its non-zero components are:

Exact Static Plane Symmetric Solutions of the Einstein Equation, the Nonlinear Interacting Electromagnetic and Scalar Field Equations
From (12), (24), (25) and (32), we obtain the expression of the electric scalar potential: where ( )  In Figure 1(a) and Figure 1(b):  The electric scalar potential ( ) A x is a regular function, tending respectively to: From (33), (34), (35); the relations (12), (17), (26), (29) and (30) become:    In Figure 2(a), the energy density is a regular, asymptotic and localized function. The increase of the nonlinearity parameter, decreases its localization width but increases its depth. It varies as follows: Figure 2(b), the charge density is a regular, asymptotic, localized function with high depth and low localization width as the nonlinear parameter increases. It takes the values: The charge density per unit volume being an odd function, the charge total of elementary particles (31) is: The total energy f E of the interacting nonlinear electromagnetic and scalar fields in the presence of the own gravitational field of the elementary particles verifies:  cosh d cosh In this expression, ( )

Solutions in Flat Space-Time
In the absence of the own gravitational field of elementary particles, the metric Journal of High Energy Physics, Gravitation and Cosmology (2) is written: The nonlinear electromagnetic field equation takes the form: which has the solution: With the calibrated invariance function ( ) P I , the relation (45) leads to: The energy density verify: Figure 3 makes a comparative numerical study between the electric scalar potentials (33) and (46) as well as between the energy densities (37) and (47) obtained respectively in the presence and absence of the own gravitational field of elementary particles.
In Figure 3(a), without the own gravitational field of the elementary particles, the electric scalar potential is a regular function of almost equal amplitude.
In Figure 3(b), the energy density is a regular, asymptotic, localized function. Its depth is more flared and its localization width is almost equal to that obtained in the presence of the own gravitational field of the elementary particles.
The total energy of the interacting nonlinear electromagnetic and scalar fields; the charge density per unit invariant volume and the total charge of elementary particles are given by the expressions: The results obtained in Section 4.1, prove that in the absence of the own gravitational field of the elementary particles, we obtained soliton-like solutions. This soliton-like solution confuses elementary particles with material points. It is opposed to the experimental results obtained in high energy physics. It does not allow to explain the evidence of fermions (leptons and quarks) as well as the gauge bosons and the Higgs boson of the standard model. The gravitational field of the elementary particles plays a role of interaction of the fields and should not be neglected. Let's look at the role of the nonlinearity of the fields.

Solutions in Linear Case
In the linear case, the parameter of the nonlinear λ is equal to zero. The coupling is minimal and we obtain: The energy-momentum metric tensor (16) gives in this case the non-zero components: .

Conclusion
The exact static plane symmetric soliton-like solutions for the Einstein equation The energy density is localized. Its width and depth depend on the constant integration and the parameter of the nonlinearity. The total energy of the interacting fields is bounded and the total charge of elementary particles is finite.