Novel analytic solutions of strain wave model in micro-structured solids

: In this article, the modi ﬁ ed extended direct algebraic method is implemented to investigate the strain wave model that governs the wave propagation in micro-struc-tured solids. The proposed method provides many new exact traveling wave solutions with certain free parameters. Exact solutions are extremely important in interpreting the inner structures of the natural phenomena. Solitary and other wave solutions are provided for this model, such as bright solitary solutions, dark solitary solutions, singular solitary solutions, singular-dark combo solitary solutions. Also, periodic solutions and Jacobi elliptic function solutions are presented. To show the physical characteristics of the raised solutions, the graphical illustration of some solutions is presented.


Introduction
In science, many important phenomena can be described by nonlinear partial differential equations (NLPDEs).Seeking the exact solutions for these equations plays an important role in the study on the dynamics of those phenomena that appear in various scientific and engineering fields, such as solid-state physics, fluid mechanics, chemical kinetics, plasma physics, population models, and nonlinear optics.Many authors have been investigated the exact solutions of these models.Topsakal and Tascan [1] obtained the exact traveling wave solutions for space-time fractional Klein-Gordon equation and (2 + 1)-dimensional time-fractional Zoomeron equation via the auxiliary equation method.Jafari et al. [2] established the exact solutions of two NLPDEs using the first integral method.El-Horbaty and Ahmed [3] studied the solitary traveling wave solutions of some NLPDEs using the modified extended tanh function method with Riccati equation.Seadawy et al. [4] studied the weakly nonlinear wave propagation theory for the Kelvin-Helmholtz instability in magnetohydrodynamics flows.Arshad et al. [5] discussed the elliptic function solutions, modulation instability, and optical soliton analysis of the paraxial wave dynamical model with Kerr media.Arshad et al. [6] obtained the dispersive solitary wave solutions of strain wave dynamical model and its stability.Ali et al. [7] investigated the new solitary wave solutions of some nonlinear models and their applications.Lu et al. [8] discussed the structure of traveling wave solutions for some nonlinear models via the modified mathematical method.Mohyud-Din and Irshad [9] derived the solitary wave solutions of some nonlinear PDEs arising in electronics.Seadawy et al. [10] constructed the solitary wave solutions of some nonlinear dynamical systems arising in nonlinear water wave models.Samir et al. [11] introduced the exact wave solutions of the fourth-order nonlinear partial differential equation of optical fiber pulses by using different methods.El-Sheikh et al. [12] discussed the dispersive and propagation of shallow water waves as a higher-order nonlinear Boussinesq-like dynamical wave equations.Donfack et al. [13] studied the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method.Park et al. [14] introduced novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg-de Vries equations.Nisar et al. [15] established novel multiple soliton solutions for some nonlinear PDEs via multiple Exp-function method.Siddique et al. [16] obtained the exact traveling wave solutions for two prolific conformable M-fractional differential equations via three diverse approaches.Djennadi et al. [17] introduced the Tikhonov regularization method for the inverse source problem of time-fractional heat equation in the view of ABCfractional technique.Malik et al. [18] studied a (2 + 1)-dimensional Kadomtsev-Petviashvili equation with competing dispersion effect: Painlev analysis, dynamical behavior, and invariant solutions.Liu and Osman [19] discussed the nonlinear dynamics for different nonautonomous wave structure solutions of a 3D variable-coefficient generalized shallow water wave equation.Yao et al. [20] studied the analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches.
Micro-structured materials such as ceramics, alloys, crystallites, and functionally graded materials, which are used as "super resistant" materials in fuselage coatings and propulsion systems in spacecraft in order to improve thermal resistivity and reduce generated thermal stress, have gained wide applications.The strain wave model that governs the wave propagation in micro-structured solids has been studied in some works: Silambarasan et al. [21] discussed the longitudinal strain waves propagating in an infinitely long cylindrical rod composed of generally incompressible materials and its Jacobi elliptic function solutions.Arshad et al. [22] studied the brightdark solitons of strain wave equation in micro-structured solids and its applications.Porubov and Pastrone [23] constructed nonlinear bell-shaped and kink-shaped strain waves in micro-structured solids.Kumar et al. [24] obtained new exact solitary wave solutions of the strain wave equation in micro-structured solids via the generalized exponential rational function method.Taher et al. [25] established the new solitary wave solutions to the strain wave model in micro-structured solids.Arshad et al. [6] established dispersive solitary wave solutions of strain wave dynamical model and its stability.
Recently, many new approaches to obtain the exact solutions of nonlinear differential equations have been proposed, such as Kudryashov's method, Generalized Jacobi's elliptic function expansion, extended trial equation method, and P 6 model approach [26][27][28].The modified extended direct algebraic method has been suggested to obtain the exact complex solutions of NLPDEs [29][30][31][32].
In this work, the modified extended direct algebraic method is applied to obtain solitary wave solutions and other wave solutions for the strain wave model.By comparing our results with the outcomes of previous studies [6,[21][22][23][24][25], we obtain various types of solutions such as, bright solitary solutions, dark solitary solutions, singular solitary solutions, singular-dark combo solitary solutions, periodic solutions, Jacobi elliptic function solutions, and other solutions.In the end of this article, we present some 3D and 2D graphs to illustrate the obtained results.
We assume an NLPDE as follows: D( ) = P P P P P P , , , , , , … 0, where D denotes a polynomial function in ( ) and also its partial derivatives.
In order to obtain the solution of Eq. ( 1) using this method, the following steps are followed: Step 1: Suppose that Eq. ( 1) has a traveling wave solu- tion as follows: where μ is a constant that is not equal to zero.Now, the ordinary differential equation is obtained by substituting Eq. ( 2) into Eq.( 1) as follows: Step 2: The equation of ancillary for this methodology is as follows: where (5) Step 3: The positive integer m is determined by balancing deri- vatives with the highest order and nonlinear terms in Eq. (3).
Step 4: By replacing Eq. ( 4) with Eq. ( 5) in Eq. ( 3) and sum- ming all the coefficients of the same forces and setting them equal to zero, we obtain a system of algebraic equations, which can be solved by Mathematica packages, and finding the values of the unknown parameters.
Step 5: From the different possible values of ( ) = h i 0, 1, …,6 i , Eq. ( 5) has many fundamental solutions and some of them can be formulated as follows: Case 1: , the following solutions are raised: Case 2: 4 , the following solutions are raised: ( ) Case 3:

4
, and , the following solutions are raised: Case 4: , the following solutions are raised:

Governing system
The system of stress waves in micro-structured solids is represented as [6,24] ( ) where u represents the strain wave in micro-structured solids, while ε and δ represent the elastic strains and the ratio between wavelength and micro-structure size, respectively and the influence of dissipation is represented by β, while i are nonzero real numbers.Solutions of stress waves in solids with micro-structure were studied by Porubov and Pastrone [23].The nondispersed state arose when = β 0, where β is the influence of dissipation and is shown by the following double dispersed equation [33]: The proposed model (7) was studied in the study by Arshad et al. [22] under the constraint = ε δ, but in our work, we study (7) without this constraint.Compared to results in the study by Arshad et al. [22], various types of solutions are obtained.

Dynamical behavior for solutions of strain wave model
To find the solitary wave solutions of Eq. ( 7), we use the following transformation: where λ is the real constant, and r represents the velocity of the wave.By substituting from Eq. ( 8) into Eq.( 7), we obtain the following: By integrating Eq. ( 9) twice, we obtain the following: and the second constant of integration is represented by c with the first assumed to be zero.By performing step 2 in Section 2 with the application of the equilibrium principle, we obtain the solution of Eq. (10)  .By compensating Eq. (11) with Eq. ( 5) into Eq.( 10), we can obtain the solutions to Eq. ( 7) as follows: Through this case, we have two types, namely: , εα 1 and λ≠0 then we have the combo singular and bright solitary solutions for Eq. ( 7) as follows: where = ±1.
(3.1) Through Result (3.1), we are able to find the traveling wave solutions for Eq. ( 7) under the restrictions εα 1 and λ≠0 as follows: , and = h m 4 2 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.
Special case, if = m 1 or = m 0, then Eq. ( 7) has a sin- gular solitary solution or the periodic wave solution as follows: , and = h 1 4 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: or where < ≤ m 0 1.Special case, if = m 1, then Eq. ( 7) has a dark solitary solution as follows: , and = − h 1
, and we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ < m 0 1.
, then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where < ≤ m 0 1.Special case, if = m 1, then Eq. ( 7) has a bright solitary solution as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ < m 0 1.Special case, if = m 0, then Eq. ( 7) has a periodic wave solution as follows: 2 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.

Special case, if
= m 1 or = m 0, then Eq. ( 7) has a bright solitary solution or a periodic solution as follows: , and , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.

Special case, if
= m 1 or = m 0, then Eq. ( 7) has a singular solitary solution or a periodic solution as follows: , and , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where < < m 0 1.
, then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.Special case, if = m 1 or = m 0, then Eq. ( 7) has the bright and the singular solitary solutions or periodic solution as follows:  , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ < m 0 1.Special case, if = m 0, then Eq. ( 7) has a periodic solu- tion as follows: , and = h 4 1 4 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where < ≤ m 0 1.
, then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ ≤ m 0 1.

Special case. If
= m 1 or = m 0, then Eq. ( 7) has a singular solitary solution or periodic solution as follows: , and , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where R ( ) , and ≤ < m 0 1 .
Special case, if = m 0, then Eq. ( 7) has a periodic solution as follows: , and 4 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where R ( ) , and ≤ < m 0 1 .
, and = h m 4 4 , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: , and , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where R ( ) , and < ≤ m 0 1 .
Strain wave equation in micro-structured solids  7 and , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where T ( ) 2 , and < ≤ m 0 1 .
Special case, if = m 0, then Eq. ( 7) has a periodic solution as follows: , and , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where T ( ) 2 , and < ≤ m 0 1 .Special case, if = m 1, then Eq. ( 7) has the combo bright- singular solitary solutions as follows: , and 2 , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where < ≤ m 0 1.
Special case.If = m 1, then Eq. ( 7) has the bright solitary solution as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where < < m 0 1.
, and , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: ( ) , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: (70) where ≤ ≤ m 0 1.
Through Result (3.2), we are able to find the traveling wave solutions for Eq. ( 7) under the restrictions εα 1 and λ≠0, which is as follows: , and = h m 4 2 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where < ≤ m 0 1.Special case, if = m 1, then Eq. ( 7) has a dark solitary solution as follows: , and = h 1 4 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.

Special case, if
= m 1 or = m 0, then Eq. ( 7) has a singular solitary solution or periodic solutions as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: Strain wave equation in micro-structured solids  9 where < ≤ m 0 1.Special case, if = m 1, then Eq. ( 7) has a bright solitary solution as follows: , and we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ < m 0 1. , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ < m 0 1.
, then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ ≤ m 0 1.

Special case, if
= m 1 or = m 0, then Eq. ( 7) has a singular solitary solution or a periodic wave solution as follows: , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ < m 0 1.
, and , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ < m 0 1.
, then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.
, then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ ≤ m 0 1.
, and , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where L ( ) and ≤ < m 0 1 .
, then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where N ( ) , and ≤ < m 0 1 .
Special case.If = m 0, then Eq. ( 7) has a periodic solution as follows: , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where N ( ) , and ≤ ≤ m 0 1 .
, and 2 , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where < ≤ m 0 1.Special case, if = m 1, then Eq. ( 7) has a dark solitary solution as follows: , and 2 , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where < ≤ m 0 1. , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: = − , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: or where ≤ < m 0 1.
, then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ < m 0 1.Special case, if = m 0, then Eq. ( 7) has a periodic solution as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where < ≤ m 0 1.
Through Result (3.3), we are able to find the travel wave solutions for Eq. ( 7) under the restrictions εα 1 and λ≠0, which is as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ≤ ≤ m 0 1.
, then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ ≤ m 0 1.
2 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows:  , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where ≤ ≤ m 0 1.
, or , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows:  , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows:  , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: and ≤ ≤ m 0 1 .
, and = ± h m 4 4 , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: where ( ) , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: where , and ≤ ≤ m 0 1 .
Special case, if = m 1 or = m 0, then Eq. ( 7) has the combo bright-dark solitary solution or periodic solution as follows: where , and 2 , then we have a Jacobi elliptic function solution for Eq. ( 7) as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows: , then we have the Jacobi elliptic function solutions for Eq. ( 7) as follows:  direct algebraic method.These solutions include bright solitary solutions, dark solitary solutions, singular solitary solutions, singular-dark combo solitary solutions, periodic solutions, Jacobi elliptic function solutions, and other solutions.For the physical illustration, 3D and 2D and contour graphs of ( ) u x t , for some of these solutions are presented in Figures 1-4, by giving suitable values to the involved parameters.Figure 1 represents the 3D, contour and 2D solution of a bright solitary solution for Eq. ( 13) with = r 0.9, = λ 0. .Figure 2 represents the 3D, contour, and 2D solution of a dark solitary solution for Eq. ( 24) with (a)

Results and discussion
The proposed model (7) was studied in the study by Arshad et al. [22] under the constraint = ε δ but in our work, we study (7) without this constraint.In the current work, using the modified extended direct algebraic method, different families of solutions were obtained from Eq. ( 5) by giving specific values to the parameters.Bright solitary solutions, dark solitary solutions, singular solitary solutions, singulardark combo solitary solutions, periodic solutions, and Jacobi elliptic function solutions were obtained.So, several innovative and corrected results have been achieved in this work compared to results in the study by Arshad et al [22].The extracted solutions confirmed the efficacy and strength of the current technique.

Conclusion
In this work, the strain wave model has been studied successfully using the modified extended direct algebraic method.The advantage of the proposed method is that it provides many new exact traveling wave solutions with certain free parameters.Exact solutions are extremely important in interpreting the inner structures of the natural phenomena.The explicit solutions represented several forms of solitary wave solutions based on the variation of the physical parameters.In different articles [6,[21][22][23][24][25], only a few traveling wave solutions are extracted to this model that does not provide a complete representation of the physical phenomena.With the aid of the proposed method, novel exact wave solutions are extracted for this model such as bright solitary solutions, dark solitary solutions, singular solitary solutions, singular-dark combo solitary solutions, periodic solutions, Jacobi elliptic function solutions, and other solutions.The extracted solutions confirmed the efficacy and strength of the current technique.Also, this method can be applied to study many other NLPDEs, which frequently arise in engineering, mathematical physics, and other scientific real-time application fields.Moreover, for the physical illustration of the obtained solutions, 2D, contour, and 3D graphs are presented.
In the future, this work may be extended for stochastic strain wave model in micro-structured solids.

2 ,
then we have the Jacobi elliptic function solutions for Eq.(7) as follows: or