New impressive analytical optical soliton solutions to the Schrödinger–Poisson dynamical system against its numerical solutions

In our current study, we will derive new diverse enormous impressive analytical optical soliton solutions for the Schrödinger–Poisson dynamical system. The proposed model is applied in gravity field with the corresponding quantum state that produces coupling between different gravity states. Moreover, this model has a significant role in the field of many quantum phenomena. Hereby, we will construct diverse forms of the soliton behaviors that arising from this dynamical system via the solitary wave ansatze method. This technique is one of the ansatze methods that doesn’t surrenders to the homogeneous balance and continuously achieves good results. Moreover, we will construct the numerical solutions that are identical for all achieved exact solutions by using two-dimensional differential transform method (TDDTM). The extracted soliton solutions are new compared with that realized before by other authors who used various techniques. The achieved solutions will give new distinct configurations to soliton behaviors arising from this model and show the fact of charges regular distributions on conductors’ materials surface.


Introduction
Schrödinger (Schrödinger 1926) in successive published articles derived the wave equation for time-independent systems that it gave the correct energy eigenvalues for a hydrogen-like atom, solved the quantum harmonic oscillator, rigid rotor, and diatomic molecule problems, proofed consistent to Heisenberg approach and gave the treatment of the Stark effect. Furthermore, he treated problems of time system changes, as in scattering problems. The Schrödinger-Poisson dynamical system (SPDS) is a quantum-model that surrenders to outside quantum operator, gives lose information contained system compared to its environment. From the fact that any quantum system cannot separated from its surroundings, it is important to improve a theoretical formalism treating these interactions to get correct interpretation of quantum systems. The SPDS is essentially the Schrödinger equation as well as the gravitational potential that has application of gravity's role in quantum state in Nakatsuji (2012). Moreover, it can also be seen as the connection of Einstein-Dirac system and Einstein-Klein Gordon in Giulini and Großardt (2012). In addition, it can approximate the coupling between quantum mechanics with gravitation in Penrose (1996). Many models which are derived from Schrödinger equation in various fields of science have been discussed via in Bekir et al. (2020); Bekir and Zahran xxxx;Bekir and Zahran 2021a;Shehata et al. 2019;Bekir and Zahran 2021b;Zahran et al. 2021a;Zahran et al. 2021b;Bekir and Zahran 2020;Bekir and Zahran 2021c;Bekir and Zahran 2021d;Bekir and Zahran 2021e;Bekir and Zahran 2021f;Bekir and Zahran 2021g;Hosseini et al. 2020;Mirzazadeh et al. 2018;Younis et al. 2020;Rehman et al. 2020;Mao et al. 2019;Wang and Wei 2020;Liu et al. 2019;Chen et al. 2021). In related subject, we will use the two-dimensional differential transform method (Zhou 2010;Chen and Ho 1999;Ziyaee and Tari 2015;Ayaz 2003;Patel and Dhodiya 2022) to obtain the numerical solutions corresponding to the achieved analytical solutions. The Schrödinger -Poisson dynamical system (Younis et al. 2021) can be written as; When this system is surrenders to the definition of Δ it will be converted to where Ψ describe the expected shape of the wave pulse, x, t denote to the independent spatial and temporal variables respectively, Φ which has elliptic behaviors is a solution of Schrödinger-Poisson system, Φ(x) represent the charge of the wave function itself.
Our article is prepared as follow, in the 1 st -secession the introduction is present, in the 2 nd -secession the solitary wave ansatze method (SWAM) Zahran 2020, 2021d;Zahran and Bekir 2022) and its application to detect the bright and dark optical soliton solutions of the suggested model are present. In the 3th-secession the twodimensional differential transform method and their application to obtain the numerical solutions of the achieved analytical solutions is present. In the 4th-secession we admit the conclusion of our current work (Figs. 1,2,3,4,5,6,7,8,9,10).

The SWAM schema
The solution perception of the SWAM will be firstly introduced by suppose this transformation where q(x, t) represent the portion amplitude,R(x, t) is the portion phase of the soliton, k, Ω are constants that represent the coefficients of the spatial and temporal x, t of the portion phase respectively, by calculus operations for Eq. (3) we obtain By inserting the relations (3)-(9) into Eq. (1) leads to Equation (10) implies these real and imaginary parts; (11) q xxxx + 2kq xt − (Ω + Φ + 6k 2 )q xx + 2Φ � q x + (k 4 − Φk 2 )q + q 3 = 0.

(2-I) The bright solitary wave solution
where B 1 , w 1 that appears in the bright solution are constants that represent the coefficients of the spatial and temporal x, t of the portion amplitude respectively and A 1 is the coefficient of the amplitude.
By introducing the relations (13)-(20) into the real and imaginary parts Eqs. (11), (12) we get The equivalence between the higher orders of sech i in Eq. (21) implies P = 2, by substituting about this value into Eq. (21) and equating the coefficients of various powers of sech i to zero we obtain the following relations: (14) q t = A 1 Bw 1 Psech P t 1 tanh t 1 .
By put P = 2 in the imaginary part Eq. (22) we get the following relations It is clear that the two systems given by Eqs. (23) and (24) imply that Φ � = 0 which leads to Φ = C which physically denotes to the regular distribution of the charge that must occur in the conducting materials. Moreover, when the two systems in Eqs. (23) and (24) are merged with each other and solving the resultant system the following results are detected. While, from the first part we can extract We will construct the solutions that are identical only for the first and the third results after putting B = 1.

(1) For the first result which is
This result can be simplified to be Thus the solution is (2) For the third result which is: This result can be simplified to be: Thus the solution is: By the same method we can plot solutions that are corresponding to the other two results.

The dark solitary wave solution
where B 2 , w 2 that appears in the dark solution are constants that represent the coefficients of the spatial and temporal x, t of the portion amplitude respectively and A 2 is the coefficient of the amplitude By introducing the relations (37-44) into the real and imaginary parts Eqs. (11), (12) we get By equating the coefficients of various powers of sech i in Eq. (45) to zero, it implies P = 2, by substituting this value into Eq. (45) again, this leads to the following relations: By put P = 2 in the imaginary part Eq. (46) we get The resultant relations Eqs. (47) and (48) imply Φ � = 0 and by substituting from Eq. (48) into Eq. (47) about 4k + w 2 = 0, Φ � = 0 we obtain the following system of equations When this system is solved by any computer program the following results will be detected: −4kw 2 B 2 − 72B 4 − 2B 2 (Ω + Φ + 6k 2 ) = 0.
For the above achieved results, we will extract the solutions that are identical for the first and the third one.
(1) For the first result which is: This can be simplified to be: The dark solution is: (2) For the third result which is: This can be simplified to be: The dark solution is: (60) Ω = 91, B = −0.7, A 2 = 5 i, k = 1, w 2 = −4.

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The other two solutions can be implemented by the same technique.

Numerical treatment for the Schrödinger-Poisson dynamical system using two dimensional differential transform method (TDDTM)
The idea of differential transform was first introduced by Zhou (Patel and Dhodiya 2022), who used it to solve linear and nonlinear initial value problems in electric circuit analysis. Chen and Ho (Younis et al. 2021) extended this method for partial differential equations (PDEs) and got on a closed form series solution for linear and nonlinear initial value problems. The method applied to PDEs, is called two-dimensional differential transform method (TDDTM) Zahran 2020, 2021d;Zahran and Bekir 2022), this method is different from the high-order Taylor series method, which consists of computing the coefficients of the Taylor series of the solution using the initial data and the partial differential equation. But the Taylor series method needs more computational work for large orders. It is stated that ''the differential transform is an iterative procedure for obtaining Taylor series solutions of differential equations''. This method decreases the size of computational domain and suitable to many problems easily.

The two-dimensional differential transform method (TDDTM)
TDDTM is defined as where where q(x, t) is the original function and Q(k, h) is the transformed function. Using Eq. (66) in Eq. (65) we obtain The following relations (rules) are important for solution, .
By the same technique we can implement the numerical solutions of the other achieved dark solutions.

Conclusion
The Schrödinger-Poisson dynamical system which applied in gravity field with the corresponding quantum state that produce coupling between different gravity states is one of very effective models in physics and applied mathematics. In fact, it responsible the charges distribution on material that signify if it conductor or insulated. Hereby, the SWAM has been used for the first time to extract the bright and dark optical soliton solutions which weren't realized before. The suggested method admits new forms of the achieved soliton solutions as M-shaped soliton and W-shaped as well as bright and dark soliton solutions. The amplitude of the bright solutions is real while amplitude of the dark solutions is complex but this contradiction is acceptable because the intensity of the wave is directly proportion with the square of the amplitude whatever the q(x, t) = 4 ∑ k=0 4 ∑ h=0 Q(k, h)x k t h = 8.5 − 4341.46t 2 + 1.478296 × 10 6 t 4 + −384.2t + 261645.3226t 3 x + −8.5 + 17365t 2 − 1.2565516 × 10 7 t 4 x 2 + 513t − 738423t 3 x 3 + 5.554 − 24655.8t 2 + 4.74545 × 10 7 t 4 x 4 (88) Ψ(x, t) = q(x, t) × (5.7x − 14.5t) = (8.5 − 4341.46t 2 + 1.478296 × 10 6 t 4 + −384.2t + 261645.3226t 3 x + −8.5 + 17365t 2 − 1.2565516 × 10 7 t 4 x 2 + 513t − 738423t 3 x 3 values of choose parameters. The obtained solutions show the regular charge distribution for all conducting materials as well as its show significant the quantum state that produces coupling between different gravity states. The novelties of our obtained solutions appear when we compared it's with that realized before by Younis et al. (2021) who use (G'/G)-expansion method and the extended direct algebraic methods. Furthermore, the numerical solutions corresponding to the achieved analytical solutions have been constructed by using the two-dimensional differential transform method. The compatible between the analytical and the numerical solutions clears because there exists agreement between the McLaurin expansion to the achieved exact solutions, the obtained numerical solutions and we demonstrated this in the above table. These new types of solutions will play many fundamental roles in the quite understanding of qualitative features of many phenomenon arising in various areas of natural science and will add future studies for all related phenomenon in plasma physics, applied mathematics and engineering.