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

  • Wafaa B. Rabie , Hamdy M. Ahmed EMAIL logo and Hisham H. Hussein
Published/Copyright: May 28, 2024
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

In this article, the modified extended direct algebraic method is implemented to investigate the strain wave model that governs the wave propagation in micro-structured 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.

Keywords: stress waves; solitary wave solutions; periodic solutions; Jacobi elliptic function solutions; analytic method

1 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 ABC-fractional 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 bright-dark 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 [26–28]. The modified extended direct algebraic method has been suggested to obtain the exact complex solutions of NLPDEs [29–32].

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 [6,21–25], 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.

2 Method summary

In this section, we describe the modified extended direct algebraic method [29–32].

We assume an NLPDE as follows:

(1) D ( P , P t , P x 1 , P x 1 x 2 , P t x 1 , P t x 2 , … ) = 0 ,

where D denotes a polynomial function in P ( t , x 1 , x 2 , x 3 , … , x n ) and also its partial derivatives.

In order to obtain the solution of Eq. (1) using this method, the following steps are followed:

S t e p 1 ̲ : Suppose that Eq. (1) has a traveling wave solution as follows:

(2) P ( t , x 1 , x 2 , x 3 , … , x n ) = U ( ζ ) , ζ = ∑ i = 1 n x i − μ t ,

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:

(3) Z ( U , U ′ , U ″ , … ) = 0 .

S t e p 2 : ̲ The equation of ancillary for this methodology is as follows:

(4) U ( ζ ) = ∑ i = − m m s i y ( ζ ) i ,

where s i , ( i = − m , … , 0 , 1 , … , m ) are the constants that are found under restrictions s m or s − m ≠ 0 , and y ( ζ ) satisfies the following equation:

(5) y ′ ( ζ ) = ∑ i = 0 6 h i y ( ζ ) i .

S t e p 3 : ̲ The positive integer m is determined by balancing derivatives with the highest order and nonlinear terms in Eq. (3).

S t e p 4 : ̲ By replacing Eq. (4) with Eq. (5) in Eq. (3) and summing 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.

S t e p 5 : ̲ From the different possible values of h i ( i = 0 , 1 , … , 6 ) , Eq. (5) has many fundamental solutions and some of them can be formulated as follows:

Case 1: When h 0 = h 1 = h 3 = h 5 = h 6 = 0 , the following solutions are raised:

y ( ζ ) = − h 2 h 4 sech ( ζ h 2 ) , h 2 > 0 , h 4 < 0 ,

y ( ζ ) = − h 2 h 4 sec ( ξ h 2 ) , h 2 > 0 , h 4 < 0 ,

y ( ζ ) = − h 2 h 4 csc ( ζ h 2 ) , h 2 > 0 , h 4 < 0 .

Case 2: When h 1 = h 3 = h 5 = h 6 = 0 and h 0 = h 2 2 4 h 4 , the following solutions are raised:

y ( ζ ) = ε − h 2 2 h 4 tanh ζ − h 2 2 , h 2 < 0 , h 4 > 0 ,

y ( ζ ) = ε h 2 2 h 4 tan ζ h 2 2 , h 2 > 0 , h 4 > 0 .

Case 3: h 1 = h 3 = h 5 = 0 , h 0 = 8 h 2 2 27 h 4 , and h 0 = h 4 2 4 h 2 , the following solutions are raised:

y ( ζ ) = − 8 h 2 tanh 2 ε ζ − h 2 2 3 h 2 3 + tanh 2 ε ζ − h 2 2 or y ( ζ ) = − 8 h 2 coth 2 ε ζ − h 2 2 3 h 2 3 + coth 2 ε ζ − h 2 2 , h 2 < 0 , h 4 > 0 ,

y ( ζ ) = − 8 h 2 tan 2 ε ζ h 2 2 3 h 2 3 + tan 2 ε ζ h 2 2 or y ( ζ ) = − 8 h 2 cot 2 ε ζ h 2 2 3 h 2 3 + cot 2 ε ζ h 2 2 , h 2 > 0 , h 4 < 0 .

Case 4: h 1 = h 3 = h 5 = h 6 = 0 , the following solutions are raised:

h 0 h 2 h 4 y ( ζ )
1 − ( 1 + m 2 ) m 2 cd ( ζ ) or sn ( ζ )
m 2 − 1 − m 2 + 2 − 1 dn ( ζ )
− m 2 2 m 2 − 1 − m 2 + 1 nc ( ζ )
− 1 − m 2 + 2 m 2 − 1 nd ( ξ )
m 2 − 2 m 3 + m 4 − 4 m − 1 + 6 m − m 2 m dn ( ζ ) cn ( ζ ) 1 + m sn ( ζ ) 2
1 4 1 2 m 2 − 1 m 4 4 sn ( ζ ) 1 + dn ( ζ ) or cn ( ζ ) 1 − m 2 + dn ( ζ )

3 Governing system

The system of stress waves in micro-structured solids is represented as [6,24]

(6) u t t − u x x − ε α 1 ( u 2 ) x x − β α 2 u t x x + δ α 3 u x x x x − ( δ α 4 − β 2 α 7 ) u t t x x + β δ ( α 6 u t t t x x + α 5 u t x x x x ) = 0 ,

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 α i , i = 1 , 2 , … , 7 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]:

(7) u t t − u x x − ε α 1 ( u 2 ) x x + δ α 3 u x x x x − δ α 4 u t t x x = 0 .

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.

4 Dynamical behavior for solutions of strain wave model

To find the solitary wave solutions of Eq. (7), we use the following transformation:

(8) u ( x , t ) = ℛ ( μ ) , μ = λ x − r t ,

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:

(9) δ λ 2 ( α 3 λ 2 − α 4 r 2 ) ℛ ( 4 ) + ( r 2 − λ 2 ) ℛ ″ − ε α 1 λ 2 ( ℛ 2 ) ′ ′ = 0 .

By integrating Eq. (9) twice, we obtain the following:

(10) δ λ 2 ( α 3 λ 2 − α 4 r 2 ) ℛ ″ + ( r 2 − λ 2 ) ℛ − ε α 1 λ 2 ℛ 2 + c = 0 ,

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) as follows:

(11) ℛ ( μ ) = s − 2 y ( μ ) 2 + s − 1 y ( μ ) + s 0 + s 1 y ( μ ) + s 2 y ( μ ) 2 ,

s i , i = − 2 , − 1 , 0 , 1 , 2 are the constants that we can define under the constraint conditions s 2 or s 2 ≠ 0 .

By compensating Eq. (11) with Eq. (5) into Eq. (10), we can obtain the solutions to Eq. (7) as follows:

Case 1: h 1 = h 3 = h 5 = 0 , h 0 = 8 h 2 2 27 h 4 , h 6 = h 4 2 4 h 2 .

s − 1 = s 1 = s 2 = 0 , s − 2 = 16 δ h 2 2 ( α 3 λ 2 − α 4 r 2 ) 9 ε α 1 h 4 , s 0 = r 2 − λ 2 + 4 δ h 2 λ 2 ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 λ 2 ,

c = − 9 ( λ 2 − r 2 ) 2 + 16 δ 2 h 2 2 λ 4 ( α 3 λ 2 − α 4 r 2 ) 2 36 ε α 1 λ 2 .

Through this case, we have two types, namely:

  1. If h 2 < 0 , h 4 > 0 , εα1 and λ≠0 then we have the combo singular and bright solitary solutions for Eq. (7) as follows:

    (12) u 1.1 = r 2 − λ 2 2 ε α 1 λ 2 − δ h 2 ( α 3 λ 2 − α 4 r 2 ) ( 5 + cosh [ 2 ℒ ( λ x − r t ) − h 2 3 ] ) csch 2 [ ℒ ( λ x − r t ) − h 2 3 ] 3 ε α 1 ,

    (13) u 1.2 = r 2 − λ 2 2 ε α 1 λ 2 − δ h 2 ( α 3 λ 2 − α 4 r 2 ) ( − 5 + cosh [ 2 ℒ ( λ x − r t ) − h 2 3 ] ) sech 2 [ ℒ ( λ x − r t ) − h 2 3 ] 3 ε α 1 ,

    where ℒ = ± 1 .

  1. If h 2 > 0 , h 4 < 0 , εα1 and λ≠0 then we have a singular periodic solution for Eq. (7) as follows:

    (14) u 1.3 = r 2 − λ 2 2 ε α 1 λ 2 + δ h 2 ( α 3 λ 2 − α 4 r 2 ) ( 5 + cos [ 2 ℒ ( λ x − r t ) h 2 3 ] ) csc 2 [ ℒ ( λ x − r t ) h 2 3 ] 3 ε α 1 ,

    (15) u 1.4 = r 2 − λ 2 2 ε α 1 λ 2 − δ h 2 ( α 3 λ 2 − α 4 r 2 ) ( − 5 + cos [ 2 ℒ ( λ x − r t ) h 2 3 ] ) sec 2 [ ℒ ( λ x − r t ) h 2 3 ] 3 ε α 1 ,

    where ℒ = ± 1 .

Case 2: h 0 = h 1 = h 3 = h 5 = h 6 = 0

s − 2 = s − 1 = s 1 = 0 , s 0 = r 2 − λ 2 + 4 δ h 2 λ 2 ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 λ 2 , s 2 = 6 δ h 4 ( α 3 λ 2 − α 4 r 2 ) ε α 1 ,

c = − ( λ 2 − r 2 ) 2 − 16 δ 2 h 2 2 λ 4 ( α 3 λ 2 − α 4 r 2 ) 2 4 ε α 1 λ 2 .

Through this case, if h 2 = 1 , h 4 = 1 2 , εα1 and λ≠0 then we have a singular solitary solution for Eq. (7) as follows:

(16) u 2 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 1 + 3 csch 2 [ λ x − r t − B 4 ] ) ε α 1 ,

where B 4 is a constant.

Case 3: h 1 = h 3 = h 5 = h 6 = 0 .

  1. s − 2 = 6 δ h 0 ( α 3 λ 2 − α 4 r 2 ) ε α 1 , s − 1 = s 1 = s 2 = 0 , s 0 = ρ 2 − λ 2 + 4 δ h 2 λ 2 ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 λ 2 ,

    c = − ( λ 2 − r 2 ) 2 − 16 δ 2 λ 4 ( h 2 2 − 3 h 0 h 4 ) ( α 3 λ 2 − α 4 r 2 ) 2 4 ε α 1 λ 2 .

  1. s − 2 = s − 1 = s 1 = 0 , s 0 = r 2 − λ 2 + 4 δ h 2 λ 2 ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 λ 2 , s 2 = 6 δ h 4 ( α 3 λ 2 − α 4 r 2 ) ε α 1 ,

    c = − ( λ 2 − r 2 ) 2 − 16 δ 2 λ 4 ( h 2 2 − 3 h 0 h 4 ) ( α 3 λ 2 − α 4 r 2 ) 2 4 ε α 1 λ 2 .

  1. s − 2 = 6 δ h 0 ( α 3 λ 2 − α 4 r 2 ) ε α 1 , s − 1 = 0 , s 0 = r 2 − λ 2 + 4 δ h 2 λ 2 ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 λ 2 , s 1 = 0 ,

    s 2 = 6 δ h 4 ( α 3 λ 2 − α 4 r 2 ) ε α 1 , c = − ( λ 2 − r 2 ) 2 + 16 δ 2 λ 4 ( h 2 2 + 3 h 0 h 4 ) ( α 3 λ 2 − α 4 r 2 ) 2 4 ε α 1 λ 2 .

Through Result (3.1), we are able to find the traveling wave solutions for Eq. (7) under the restrictions εα1 and λ≠0 as follows:

  • (3.1,1) If h 0 = 1 , h 2 = − ( m 2 + 1 ) , and h 4 = m 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (17) u 3.1 , 1 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 cd 2 ( λ x − r t ) ε α 1 ,

    or

    (18) u 3.1 , 2 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 sn 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has a singular solitary solution or the periodic wave solution as follows:

    (19) u 3.1 , 3 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 2 + 3 coth 2 ( λ x − r t ) ) ε α 1 ,

    or

    (20) u 3.1 , 4 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 3 sec 2 ( λ x − r t ) ) ε α 1 ,

    (21) u 3.1 , 5 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 3 csc 2 ( λ x − r t ) ) ε α 1 .

  • (3.1,2) If h 0 = m 2 , h 2 = − ( m 2 + 1 ) , and h 4 = 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (22) u 3.1 , 6 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 m 2 dc 2 ( λ x − r t ) ε α 1

    or

    (23) u 3.1 , 7 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 m 2 ns 2 ( λ x − r t ) ε α 1 ,

    where 0 < m ≤ 1 .

  • Special case, if m = 1 , then Eq. (7) has a dark solitary solution as follows:

    (24) u 3.1 , 8 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 2 + 3 tanh 2 ( λ x − r t ) ) ε α 1 .

  1. If h 0 = m 2 − 1 , h 2 = 2 − m 2 , and h 4 = − 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (25) u 3.1 , 9 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 + 3 ( m 2 − 1 ) dn 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m < 1 .

  1. If h 0 = 1 − m 2 , h 2 = 2 m 2 − 1 , and h 4 = − m 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (26) u 3.1 , 10 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 + 3 ( 1 − m 2 ) cn 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m < 1 .

  • (3.1,5) If h 0 = − m 2 , h 2 = 2 m 2 − 1 , h 4 = 1 − m 2 , or h 0 − 1 , h 2 = 2 − m 2 , h 4 = m 2 − 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (27) u 3.1 , 11 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 − 3 m 2 nc 2 ( λ x − r t ) ε α 1 ,

    or

    (28) u 3.1 , 12 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 − 3 nd 2 ( λ x − r t ) ε α 1 ,

    where 0 < m ≤ 1 .

  • Special case, if m = 1 , then Eq. (7) has a bright solitary solution as follows:

    (29) u 3.1 , 13 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 1 − 3 sech 2 ( λ x − r t ) ) ε α 1 .

  • (3.1,6) If h 0 = 1 − m 2 , h 2 = 2 − m 2 , and h 4 = 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (30) u 3.1 , 14 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 + 3 ( 1 − m 2 ) cs 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m < 1 .

  • Special case, if m = 0 , then Eq. (7) has a periodic wave solution as follows:

    (31) u 3.1 , 15 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 tan 2 ( λ x − r t ) ) ε α 1 .

  • (3.1,7) If h 0 = 1 , h 2 = 2 − m 2 , and h 4 = 1 − m 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (32) u 3.1 , 16 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 + 3 sc 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has a bright solitary solution or a periodic solution as follows:

    (33) u 3.1 , 17 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 1 + 3 sinh 2 ( λ x − r t ) ) ε α 1 ,

    or

    (34) u 3.1 , 18 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 tan 2 ( λ x − r t ) ) ε α 1 .

  • (3.1,8) If h 0 = 1 , h 2 = 2 m 2 − 1 , and h 4 = m 2 ( m 2 − 1 ) , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (35) u 3.1 , 19 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 + 3 sd 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has a singular solitary solution or a periodic solution as follows:

    (36) u 3.1 , 20 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 1 + 3 cosh 2 ( λ x − r t ) ) ε α 1 ,

    or

    (37) u 3.1 , 21 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 3 csc 2 ( λ x − r t ) ) ε α 1 .

  1. If h 0 = m 2 ( m 2 − 1 ) , h 2 = 2 m 2 − 1 , and h 4 = 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (38) u 3.1 , 22 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 + 3 m 2 ( m 2 − 1 ) ds 2 ( λ x − r t ) ε α 1 ,

    where 0 < m < 1 .

  • (3.1,10) If h 0 = h 4 = 1 4 and h 2 = 1 − 2 m 2 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (39) u 3.1 , 23 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 1 − 2 m 2 + 3 2 ( ns ( λ x − r t ) ± cs ( λ x − r t ) ) 2 ε α 1 ,

    where 0 ≤ m ≤ 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:

    (40) u 3.1 , 24 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( − 5 + cosh [ λ x − r t ] ) sech 2 λ x − r t 2 4 ε α 1 ,

    (41) u 3.1 , 25 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( 5 + cosh [ λ x − r t ] ) csch 2 λ x − r t 2 4 ε α 1 ,

    or

    (42) u 3.1 , 26 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 + 3 tan 2 λ x − r t 2 2 ε α 1 ,

    (43) u 3.1 , 27 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( 5 + cos [ λ x − r t ] ) csc 2 λ x − r t 2 4 ε α 1 .

  • (3.11) If h 0 = h 4 = 1 − m 2 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (44) u 3.1 , 28 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 1 + m 2 − 3 ( m 2 − 1 ) 2 ( nc ( λ x − r t ) ± sc ( λ x − r t ) ) 2 ε α 1 ,

    where 0 ≤ m < 1 .

  • Special case, if m = 0 , then Eq. (7) has a periodic solution as follows:

    (45) u 3.1 , 29 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 1 + 3 2 ( sec ( λ x − r t ) ± tan ( λ x − r t ) ) 2 .

  1. If h 0 = m 4 4 , h 2 = m 2 − 2 2 , and h 4 = 1 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (46) u 3.1 , 30 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) − 2 + m 2 + 3 m 4 2 [ ns ( λ x − r t ) ± ds ( λ x − r t ) ] 2 ε α 1 ,

    where 0 < m ≤ 1 .

  • (3.1,13) If h 0 = h 4 = 1 and h 2 = 2 − 4 m 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (47) u 3.1 , 31 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − 4 m 2 + 3 cn 2 ( λ x − r t ) [ sn ( λ x − r t ) dn ( λ x − r t ) ] 2 ε α 1 ,

    where 0 ≤ m ≤ 1 .

  • Special case. If m = 1 or m = 0 , then Eq. (7) has a singular solitary solution or periodic solution as follows:

    (48) u 3.1 , 32 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 2 + 3 coth 2 [ λ x − r t ] ) ε α 1 ,

    or

    (49) u 3.1 , 33 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 cot 2 [ λ x − r t ] ) ε α 1 .

  • (3.1,14) If h 0 = ( m ∓ 1 ) 2 4 B 1 2 , h 2 = m 2 ± 6 m + 1 2 , and h 4 = B 1 2 ( m ∓ 1 ) 2 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (50) u 3.1 , 34 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( 1 ± 6 m + m 2 + 3 ( m ∓ 1 ) 2 R 1 ( x , t ) ) ε α 1 ,

    where R 1 ( x , t ) = ( 1 + sn [ λ x − r t ] ) 2 ( 1 ± m sn [ λ x − r t ] ) 2 2 cn 2 [ λ x − r t ] dn 2 [ λ x − r t ] , and 0 ≤ m < 1 .

  • Special case, if m = 0 , then Eq. (7) has a periodic solution as follows:

    (51) u 3.1 , 35 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 + 3 ( tan [ λ x − r t ] + sec [ λ x − r t ] ) 2 ] 2 ε α 1 .

  1. If h 0 = m 4 − 2 m 3 + m 2 , h 2 = − m 2 + 6 m − 1 , and h 4 = − 4 m , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (52) u 3.1 , 36 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) [ − 1 + 6 m − m 2 + 3 ( m 2 − 2 m + 1 ) R 3 ( x , t ) ] ε α 1 ,

    where R 3 ( x , t ) = 1 + m sn 2 [ λ x − r t ] cn [ λ x − r t ] dn [ λ x − r t ] 2 , and 0 ≤ m < 1 .

  1. If h 0 = m 4 + 2 m 3 + m 2 , h 2 = − m 2 − 6 m − 1 , and h 4 = 4 m , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (53) u 3.1 , 37 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) [ − 1 − 6 m − m 2 + 3 ( m 2 + 2 m + 1 ) R 4 ( x , t ) ] ε α 1 ,

    where R 4 ( x , t ) = − 1 + m sn 2 [ λ x − r t ] cn [ λ x − r t ] dn [ λ x − r t ] 2 and 0 ≤ m ≤ 1 .

  1. If h 0 = − m 2 ± 2 m + 2 , h 2 = − m 2 ± 6 m + 2 , and h 4 = ± 4 m , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (54) u 3.1 , 38 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) [ − m 2 ± 6 m + 2 + 3 ( − m 2 ± 2 m + 2 ) R 5 ( x , t ) ] ε α 1 ,

    where R 5 ( x , t ) = m ∓ dn 2 [ λ x − r t ] m 2 cn [ λ x − r t ] sn [ λ x − r t ] 2 , and 0 < m ≤ 1 .

  • (3.1,18) If h 0 = m 2 − 1 4 ( B 3 2 m 2 − B 2 2 ) , h 2 = m 2 + 1 2 , and h 4 = ( m 2 − 1 ) ( B 3 2 m 2 − B 2 2 ) 4 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (55) u 3.1 , 39 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 1 + m 2 + 3 ( − 1 + m 2 ) T 1 ( x , t ) ] ε α 1 ,

    where T 1 ( x , t ) = ( B 2 cn [ λ x − r t ] + B 3 dn [ λ x − r t ] ) 2 2 ( B 3 2 m 2 − B 2 2 ) B 2 2 − B 3 2 B 2 2 − B 3 2 m 2 + sn [ λ x − r t ] 2 , and 0 < m ≤ 1 .

  • Special case, if m = 0 , then Eq. (7) has a periodic solution as follows:

    (56) u 3.1 , 40 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 × 1 + 3 ( B 3 + B 2 cos [ λ x − r t ] ) 2 2 ( B 2 2 − B 3 2 + B 2 sin [ λ x − r t ] ) 2 .

  • (3.1,19) If h 0 = m 4 4 ( B 2 2 + B 3 2 ) , h 2 = m 2 2 − 1 , and h 4 = B 2 2 + B 3 2 4 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (57) u 3.1 , 41 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ − 2 + m 2 + 3 m 4 T 2 ( x , t ) ] ε α 1 ,

    where T 2 ( x , t ) = ( B 2 sn [ λ x − r t ] + B 3 cn [ λ x − r t ] ) 2 2 ( B 3 2 + B 2 2 ) B 2 2 + B 3 2 − B 3 2 m 2 B 2 2 + B 3 2 + dn [ λ x − r t ] 2 , and 0 < m ≤ 1 .

  • Special case, if m = 1 , then Eq. (7) has the combo bright-singular solitary solutions as follows:

    (58) u 3.1 , 42 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 × − 1 + 3 ( B 3 + B 2 sinh [ λ x − r t ] ) 2 2 ( B 2 cosh [ λ x − r t ] + B 2 2 + B 3 2 ) 2 .

  • (3.1,20) If h 0 = − m 2 + 2 m − 1 B 2 2 , h 2 = 2 m 2 + 2 , and h 4 = − B 2 2 m 2 − 2 B 2 2 m − B 2 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (59) u 3.1 , 43 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 + 2 m 2 − 3 ( m 2 + 2 m + 1 ) ( − 1 + m sn 2 [ λ x − r t ] ) 2 ( 1 + m sn 2 [ λ x − r t ] ) 2 ε α 1 ,

    where 0 < m ≤ 1 .

  • Special case. If m = 1 , then Eq. (7) has the bright solitary solution as follows:

    (60) u 3.1 , 44 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( − 5 + cosh [ 4 ( λ x − r t ) ] ) sech 2 [ 2 ( λ x − r t ) ] 2 ε α 1 .

  1. If h 0 = h 4 = m 2 − 1 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (61) u 3.1 , 45 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 1 + m 2 + 3 ( m 2 − 1 ) ( 1 ± m sn [ λ x − r t ] ) 2 2 dn 2 [ λ x − r t ] ε α 1 ,

    (62) u 3.1 , 46 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 1 + m 2 + 3 ( m 2 − 1 ) 2 ( m sd [ λ x − r t ] ± nd [ λ x − r t ] ) 2 ε α 1 ,

    where 0 < m < 1 .

  • (3.1,22) If h 0 = h 4 = 1 − m 2 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (63) u 3.1 , 47 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 1 + m 2 + 3 ( 1 − m 2 ) ( 1 ± sn [ λ x − r t ] ) 2 2 cn 2 [ λ x − r t ] ε α 1 ,

    (64) u 3.1 , 48 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 1 + m 2 + 3 ( 1 − m 2 ) 2 ( nc [ λ x − r t ] ± sc [ λ x − r t ] ) 2 ε α 1 ,

    where 0 ≤ m < 1 .

  • Special case, if m = 0 , then Eq. (7) has the periodic solutions as follows:

    (65) u 3.1 , 49 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 tan [ λ x − r t ] + sec [ λ x − r t ] ) 2 2 ε α 1 ,

    (66) u 3.1 , 50 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 ( sin [ λ x − r t ] − 1 ) 2 sec 2 [ λ x − r t ] ) 2 ε α 1 ,

    (67) u 3.1 , 51 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 × 2 + 3 ( sec [ λ x − r t ] ± tan [ λ x − r t ] ) 2 .

  1. If h 0 = ( m 2 − 1 ) 2 4 , h 2 = m 2 + 1 2 , and h 4 = − 1 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (68) u 3.1 , 52 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 × 1 + m 2 + 3 ( m 2 − 1 ) 2 2 ( m cn [ λ x − r t ] ± dn [ λ x − r t ] ) 2 ,

    where 0 ≤ m < 1 .

  1. If h 0 = 1 4 , h 2 = m 2 + 1 2 , and h 4 = ( m 2 − 1 ) 2 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (69) u 3.1 , 53 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 × 1 + m 2 + 3 ( dn [ λ x − r t ] ± cn [ λ x − r t ] ) 2 2 sn 2 [ λ x − r t ] ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = 1 4 , h 2 = m 2 − 2 2 , and h 4 = m 4 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (70) u 3.1 , 54 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 × − 2 + m 2 + 3 ( 1 ± dn [ λ x − r t ] ) 2 2 sn 2 [ λ x − r t ] ,

    (71) u 3.1 , 55 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 × − 2 + m 2 + 3 ( 1 − m 2 ± dn [ λ x − r t ] ) 2 2 cn 2 [ λ x − r t ] ,

    where 0 ≤ m ≤ 1 .

Through Result (3.2), we are able to find the traveling wave solutions for Eq. (7) under the restrictions εα1 and λ≠0, which is as follows:

  • (3.2,1) If h 0 = 1 , h 2 = − ( m 2 + 1 ) , and h 4 = m 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (72) u 3.2 , 1 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) [ − ( m 2 + 1 ) + 3 m 2 cd 2 ( λ x − r t ) ] ε α 1

    or

    (73) u 3.2 , 2 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) [ − ( m 2 + 1 ) + 3 m 2 sn 2 ( λ x − r t ) ] ε α 1 ,

    where 0 < m ≤ 1 .

  • Special case, if m = 1 , then Eq. (7) has a dark solitary solution as follows:

    (74) u 3.2 , 3 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 2 + 3 tanh 2 ( λ x − r t ) ) ε α 1 ,

  • (3.2,2) If h 0 = m 2 , h 2 = − ( m 2 + 1 ) , and h 4 = 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (75) u 3.2 , 4 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − ( m 2 + 1 ) + 3 dc 2 ( λ x − r t ) ) ε α 1

    or

    (76) u 3.2 , 5 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − ( m 2 + 1 ) + 3 ns 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has a singular solitary solution or periodic solutions as follows:

    (77) u 3.2 , 6 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 2 + 3 coth 2 ( λ x − r t ) ) ε α 1 ,

    or

    (78) u 3.2 , 7 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 3 sec 2 ( λ x − r t ) ) ε α 1 ,

    (79) u 3.2 , 8 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 3 csc 2 ( λ x − r t ) ) ε α 1 .

  • (3.2,3) If h 0 = m 2 − 1 , h 2 = 2 − m 2 , h 4 = − 1 , or h 0 = 1 − m 2 , h 2 = 2 m 2 − 1 , h 4 = − m 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (80) u 3.2 , 9 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 − m 2 − 3 dn 2 ( λ x − r t ) ) ε α 1 ,

    (81) u 3.2 , 10 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 2 m 2 − 3 m 2 cn 2 ( λ x − r t ) ) ε α 1 ,

    where 0 < m ≤ 1 .

  • Special case, if m = 1 , then Eq. (7) has a bright solitary solution as follows:

    (82) u 3.2 , 11 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 1 − 3 sech 2 ( λ x − r t ) ) ε α 1 .

  1. If h 0 = − m 2 , h 2 = 2 m 2 − 1 , and h 4 = 1 − m 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (83) u 3.2 , 12 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 2 m 2 + 3 ( 1 − m 2 ) nc 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m < 1 .

  1. If h 0 − 1 , h 2 = 2 − m 2 , and h 4 = m 2 − 1 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (84) u 3.2 , 13 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 − m 2 + 3 ( m 2 − 1 ) nd 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m < 1 .

  • (3.2,6) If h 0 = 1 − m 2 , h 2 = 2 − m 2 , and h 4 = 1 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (85) u 3.2 , 14 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 − m 2 + 3 cs 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has a singular solitary solution or a periodic wave solution as follows:

    (86) u 3.2 , 15 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 1 + 3 csch 2 ( λ x − r t ) ) ε α 1 ,

    or

    (87) u 3.2 , 16 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 cot 2 ( λ x − r t ) ) ε α 1 .

  1. If h 0 = 1 , h 2 = 2 − m 2 , and h 4 = 1 − m 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (88) u 3.2 , 17 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 − m 2 + 3 ( 1 − m 2 ) sc 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m < 1 .

  1. If h 0 = 1 , h 2 = 2 m 2 − 1 , and h 4 = m 2 ( m 2 − 1 ) , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (89) u 3.2 , 18 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 2 m 2 + 3 m 2 ( m 2 − 1 ) sd 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m < 1 .

  1. If h 0 = m 2 ( m 2 − 1 ) , h 2 = 2 m 2 − 1 , and h 4 = 1 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (90) u 3.2 , 19 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 1 + 2 m 2 + 3 ds 2 ( λ x − r t ) ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = h 4 = 1 4 , h 2 = 1 − 2 m 2 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (91) u 3.2 , 20 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( 1 − 2 m 2 ) + 3 ( ns ( λ x − r t ) ± cs ( λ x − r t ) ) 2 ] 2 ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = h 4 = 1 − m 2 4 , h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (92) u 3.2 , 21 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( 1 + m 2 ) − 3 ( m 2 − 1 ) ( nc ( λ x − r t ) ± sc ( λ x − r t ) ) 2 ] 2 ε α 1 ,

    where 0 ≤ m < 1 .

  1. If h 0 = m 4 4 , h 2 = m 2 − 2 2 , and h 4 = 1 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (93) u 3.2 , 22 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( − 2 + m 2 ) + 3 ( ns ( λ x − r t ) ± ds ( λ x − r t ) ) 2 ] 2 ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = h 4 = 1 and h 2 = 2 − 4 m 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (94) u 3.2 , 23 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 2 − 4 m 2 + 3 ( sn ( λ x − r t ) dn ( λ x − r t ) ) 2 cn 2 ( λ x − r t ) ,

    where 0 ≤ m ≤ 1 .

  • (3.2,14) If h 0 = ( m ∓ 1 ) 2 4 B 1 2 , h 2 = m 2 ± 6 m + 1 2 , and h 4 = B 1 2 ( m ∓ 1 ) 2 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (95) u 3.2 , 24 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( 1 ± 6 m + m 2 ) + 3 ( m ∓ 1 ) 2 L 1 ( x , t ) ] 2 ε α 1 ,

    where L 1 ( x , t ) = cn 2 [ λ x − r t ] dn 2 [ λ x − r t ] ( 1 + sn [ λ x − r t ] ) 2 ( 1 ± m sn [ λ x − r t ] ) 2 and 0 ≤ m < 1 .

  • Special case, if m = 0 , then Eq. (7) has a periodic solution as follows:

    (96) u 3.2 , 25 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 + 3 cos 2 ( λ x − r t ) ( 1 + sin ( λ x − r t ) ) 2 .

  1. If h 0 = m 4 − 2 m 3 + m 2 , h 2 = − m 2 + 6 m − 1 , and h 4 = − 4 m , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (97) u 3.2 , 26 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) [ − 1 + 6 m − m 2 − 12 m L 3 ( x , t ) ] ε α 1 ,

    where L 3 ( x , t ) = cn [ λ x − r t ] dn [ λ x − r t ] 1 + m sn 2 [ λ x − r t ] 2 , and 0 < m ≤ 1 .

  1. If h 0 = − m 2 ± 2 m + 2 , h 2 = − m 2 ± 6 m + 2 , and h 4 = ± 4 m , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (98) u 3.2 , 27 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − m 2 ± 6 m + 2 ± 12 m 5 ( cn ( λ x − r t ) sn ( λ x − r t ) ) 2 ( m ∓ dn 2 ( λ x − r t ) ) 2 ε α 1 ,

    where 0 < m ≤ 1 .

  • (3.2,17) If h 0 = m 2 − 1 4 ( B 3 2 m 2 − B 2 2 ) , h 2 = m 2 + 1 2 , and h 4 = ( m 2 − 1 ) ( B 3 2 m 2 − B 2 2 ) 4 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (99) u 3.2 , 28 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( 1 + m 2 ) + 3 ( − 1 + m 2 ) N 1 ( x , t ) ] 2 ε α 1 ,

    where N 1 ( x , t ) = ( B 3 2 m 2 − B 2 2 ) B 2 2 − B 3 2 B 2 2 − B 3 2 m 2 + sn [ λ x − r t ] 2 ( B 2 cn [ λ x − r t ] + B 3 dn [ λ x − r t ] ) 2 , and 0 ≤ m < 1 .

  • Special case. If m = 0 , then Eq. (7) has a periodic solution as follows:

    (100) u 3.2 , 29 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 + 3 ( B 2 2 − B 3 2 + B 2 sin [ λ x − r t ] ) 2 ( B 3 + B 2 cos [ λ x − r t ] ) 2 ,

    where B i , i = 2 , 3 are the constants.

  1. If h 0 = m 4 4 ( B 2 2 + B 3 2 ) , h 2 = m 2 2 − 1 , and h 4 = B 2 2 + B 3 2 4 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (101) u 3.2 , 30 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( − 2 + m 2 ) + 3 N 2 ( x , t ) ] 2 ε α 1 ,

    where N 2 ( x , t ) = ( B 3 2 + B 2 2 ) B 2 2 + B 3 2 − B 3 2 m 2 B 2 2 + B 3 2 + dn [ λ x − r t ] 2 ( B 2 sn [ λ x − ρ t ] + B 3 cn [ λ x − r t ] ) 2 , and 0 ≤ m ≤ 1 .

  • (3.2,19) If h 0 = − m 2 + 2 m − 1 B 2 2 , h 2 = 2 m 2 + 2 , and h 4 = − B 2 2 m 2 − 2 B 2 2 m − B 2 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (102) u 3.2 , 31 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 + 2 m 2 − 3 ( m 2 + 2 m + 1 ) ( − 1 + m sn 2 [ λ x − r t ] ) 2 ( 1 + m sn 2 [ λ x − r t ] ) 2 ε α 1 ,

    where 0 < m ≤ 1 .

  • Special case, if m = 1 , then Eq. (7) has a dark solitary solution as follows:

    (103) u 3.2 , 32 = r 2 − λ 2 2 ε α 1 λ 2 + 8 δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 1 − 3 ( − 1 + tanh [ λ x − r t ] ) 2 ( 1 + tanh [ λ x − r t ] ) 2 .

  1. If h 0 = − m 2 + 2 m − 1 B 2 2 , h 2 = 2 m 2 + 2 , and h 4 = − B 2 2 m 2 − 2 B 2 2 m − B 2 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (104) u 3.2 , 33 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 + 2 m 2 − 3 ( m 2 + 2 m + 1 ) ( 1 + m sn 2 [ λ x − r t ] ) 2 ( − 1 + m sn 2 [ λ x − r t ] ) 2 ε α 1 ,

    where 0 < m ≤ 1 .

  1. If h 0 = h 4 = m 2 − 1 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (105) u 3.2 , 34 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( m 2 + 1 ) + 3 ( − 1 + m 2 ) dn 2 ( λ x − r t ) ( 1 ± m sn ( λ x − r t ) ) 2 2 ε α 1 ,

    or

    (106) u 3.2 , 35 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( m 2 + 1 ) + 3 ( m 2 − 1 ) ( m sd ( λ x − r t ) ± nd ( λ x − r t ) ) 2 ] 2 ε α 1 ,

    where 0 < m < 1 .

  1. If h 0 = h 4 = 1 − m 2 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (107) u 3.2 , 36 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 + m 2 ) + 3 ( 1 − m 2 ) cn 2 [ λ x − r t ] ( 1 ± sn [ λ x − r t ] ) 2 2 ε α 1 ,

    or

    (108) u 3.2 , 37 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ 2 ( 1 + m 2 ) + 3 ( 1 − m 2 ) ( nc [ λ x − r t ] ± sc [ λ x − r t ] ) 2 ] 2 ε α 1 ,

    where 0 ≤ m < 1 .

  1. If h 0 = ( m 2 − 1 ) 2 4 , h 2 = m 2 + 1 2 , and h 4 = − 1 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (109) u 3.2 , 38 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 [ 2 ( 1 + m 2 ) − 3 ( m cn [ λ x − r t ] ± dn [ λ x − r t ] ) 2 ] ,

    where 0 < m ≤ 1 .

  • (3.2,24) If h 0 = 1 4 , h 2 = m 2 + 1 2 , and h 4 = ( m 2 − 1 ) 2 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (110) u 3.2 , 39 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 ( 1 + m 2 ) + 3 ( m 2 − 1 ) 2 sn 2 [ λ x − r t ] ( dn [ λ x − r t ] ± cn [ λ x − r t ] ) 2 ,

    where 0 ≤ m < 1 .

  • Special case, if m = 0 , then Eq. (7) has a periodic solution as follows:

    (111) u 3.2 , 40 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ( 5 + cos [ λ x − r t ] ) csc 2 λ x − r t 2 4 ε α 1 .

  1. If h 0 = 1 4 , h 2 = m 2 − 2 2 , and h 4 = m 4 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (112) u 3.2 , 41 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 ( − 2 + m 2 ) + 3 m 4 sn 2 [ λ x − r t ] ( 1 ± dn [ λ x − r t ] ) 2

    or

    (113) u 3.2 , 42 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 ( − 2 + m 2 ) + 3 m 4 cn 2 [ λ x − r t ] ( 1 − m 2 ± dn [ λ x − r t ] ) 2 ,

    where 0 < m ≤ 1 .

Through Result (3.3), we are able to find the travel wave solutions for Eq. (7) under the restrictions εα1 and λ≠0, which is as follows:

  1. If h 0 = 1 , h 2 = − ( m 2 + 1 ) , h 4 = m 2 , or h 0 = m 2 , h 2 = − ( m 2 + 1 ) , h 4 = 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (114) u 3.3 , 1 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 m 2 cd 2 ( λ x − r t ) + 3 cd 2 ( λ x − r t ) ε α 1 ,

    (115) u 3.3 , 2 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 m 2 sn 2 ( λ x − r t ) + 3 sn 2 ( λ x − r t ) ε α 1 ,

    or

    (116) u 3.3 , 3 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 dc 2 ( λ x − r t ) + 3 m 2 dc 2 ( λ x − r t ) ε α 1 ,

    (117) u 3.3 , 4 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − ( m 2 + 1 ) + 3 ns 2 ( λ x − r t ) + 3 m 2 ns 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = m 2 − 1 , h 2 = 2 − m 2 , and h 4 = − 1 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (118) u 3.3 , 5 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 − 3 dn 2 ( λ x − r t ) + 3 ( m 2 − 1 ) dn 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = 1 − m 2 , h 2 = 2 m 2 − 1 , h 4 = − m 2 , or h 0 = − m 2 , h 2 = 2 m 2 − 1 , h 4 = 1 − m 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (119) u 3.3 , 6 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 − 3 m 2 cn 2 ( λ x − r t ) + 3 ( 1 − m 2 ) cn 2 ( λ x − r t ) ε α 1 ,

    or

    (120) u 3.3 , 7 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 + 3 ( 1 − m 2 ) nc 2 ( λ x − r t ) − 3 m 2 nc 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 − 1 , h 2 = 2 − m 2 , and h 4 = m 2 − 1 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (121) u 3.3 , 8 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 + 3 ( m 2 − 1 ) nd 2 ( λ x − r t ) − 3 nd 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = 1 − m 2 , h 2 = 2 − m 2 , and h 4 = 1 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (122) u 3.3 , 9 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 + 3 cs 2 ( λ x − r t ) + 3 ( 1 − m 2 ) cs 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = 1 , h 2 = 2 − m 2 , and h 4 = 1 − m 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (123) u 3.3 , 10 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 − m 2 + 3 ( 1 − m 2 ) sc 2 ( λ x − r t ) + 3 sc 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = 1 , h 2 = 2 m 2 − 1 , h 4 = m 2 ( m 2 − 1 ) , or h 0 = m 2 ( m 2 − 1 ) , h 2 = 2 m 2 − 1 , h 4 = 1 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (124) u 3.3 , 11 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 + 3 m 2 ( m 2 − 1 ) sd 2 ( λ x − r t ) + 3 sd 2 ( λ x − r t ) ε α 1 ‘

    or

    (125) u 3.3 , 12 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 + 2 m 2 + 3 ds 2 ( λ x − r t ) + 3 m 2 ( m 2 − 1 ) ds 2 ( λ x − r t ) ε α 1 ,

    where 0 ≤ m ≤ 1 .

  1. If h 0 = h 4 = 1 4 and h 2 = 1 − 2 m 2 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (126) u 3.3 , 13 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 − 2 m 2 ) + 3 ℳ ( x , t ) + 3 ℳ ( x , t ) 2 ε α 1 ,

    where ℳ ( x , t ) = ( ns [ λ x − r t ] ± cs [ λ x − r t ] ) 2 and 0 ≤ m ≤ 1 .

  1. If h 0 = h 4 = 1 − m 2 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (127) u 3.3 , 14 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 + m 2 ) + 3 ( 1 − m 2 ) ℒ ( x , t ) + 3 ( 1 − m 2 ) ℒ ( x , t ) 2 ε α 1 ,

    where ℒ ( x , t ) = ( nc [ λ x − r t ] ± sc [ λ x − r t ] ) 2 and 0 ≤ m < 1 .

  1. If h 0 = m 4 4 , h 2 = m 2 − 2 2 , and h 4 = 1 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (128) u 3.3 , 15 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( − 2 + m 2 ) + 3 m 4 R ( x , t ) + 3 R ( x , t ) 2 ε α 1 ,

    where R ( x , t ) = ( ns [ λ x − r t ] ± ds [ λ x − r t ] ) 2 and 0 ≤ m ≤ 1 .

  • (3.3,11) If h 0 = h 4 = 1 and h 2 = 2 − 4 m 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (129) u 3.3 , 16 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 2 − 4 m 2 + 3 g ( x , t ) + 3 g ( x , t ) ,

    where g ( x , t ) = ( dn [ λ x − r t ] sn [ λ x − r t ] ) 2 cn 2 [ λ x − r t ] and 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has the combo singular-dark solitary solution or periodic solution as follows:

    (130) u 3.3 , 17 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( − 2 + 3 ( 1 + coth 4 [ λ x − r t ] ) tanh 2 [ λ x − r t ] ) ε α 1 ,

    or

    (131) u 3.3 , 18 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) ( 2 + 3 ( 1 + cot 4 [ λ x − r t ] ) tan 2 [ λ x − r t ] ) ε α 1 .

  1. If h 0 = ( m ∓ 1 ) 2 4 B 1 2 , h 2 = m 2 ± 6 m + 1 2 , and h 4 = B 1 2 ( m ∓ 1 ) 2 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (132) u 3.3 , 19 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 ± 6 m + m 2 ) + 3 ( m − 1 ) 2 G ( x , t ) + 3 ( m ∓ 1 ) 2 G ( x , t ) 2 ε α 1 ,

    where G ( x , t ) = cn 2 [ λ x − r t ] dn 2 [ λ x − r t ] ( 1 + sn [ λ x − r t ] ) 2 ( 1 ± m sn [ λ x − r t ] ) 2 , and 0 ≤ m ≤ 1 .

  1. If h 0 = m 4 ∓ 2 m 3 + m 2 , h 2 = − m 2 ± 6 m − 1 , and h 4 = ∓ 4 m , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (133) u 3.3 , 20 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − 1 ± 6 m − m 2 + 3 ( m 2 ∓ 2 m + 1 ) A ( x , t ) ∓ 12 m A ( x , t ) ε α 1 ,

    where A ( x , t ) = cn [ λ x − r t ] dn [ λ x − r t ] ± 1 + m sn 2 [ λ x − r t ] 2 , and 0 ≤ m ≤ 1 .

  1. If h 0 = − m 2 ± 2 m + 2 , h 2 = − m 2 ± 6 m + 2 , and h 4 = ± 4 m , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (134) u 3.3 , 21 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) − m 2 ± 6 m + 2 + 3 ( − m 2 + 2 m + 2 ) m 4 K ( x , t ) ± 12 m 5 K ( x , t ) ε α 1 ,

    where K ( x , t ) = cn ( λ x − r t ) sn ( λ x − r t ) m ∓ dn ( λ x − r t ) 2 , and 0 < m ≤ 1 .

  1. If h 0 = m 2 − 1 4 ( B 3 2 m 2 − B 2 2 ) , h 2 = m 2 + 1 2 , and h 4 = ( m 2 − 1 ) ( B 3 2 m 2 − B 2 2 ) 4 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (135) u 3.3 , 22 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 + m 2 ) + 3 ( m 2 − 1 ) N ( x , t ) + 3 ( m 2 − 1 ) N ( x , t ) 2 ε α 1 ,

    where N ( x , t ) = ( B 3 2 m 2 − B 2 2 ) B 2 2 − B 3 2 B 2 2 − B 3 2 m 2 + sn [ λ x − r t ] 2 ( B 2 cn [ λ x − r t ] + B 3 d n [ λ x − r t ] ) 2 , B i ( i = 2 , 3 ) are the constants and 0 ≤ m < 1 .

  • (3.3,16) If h 0 = m 4 4 ( B 2 2 + B 3 2 ) , h 2 = m 2 2 − 1 , and h 4 = B 2 2 + B 3 2 4 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (136) u 3.3 , 23 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( − 2 + m 2 ) + 3 m 4 A ( x , t ) + 3 A ( x , t ) 2 ε α 1 ,

    where A ( x , t ) = ( B 3 2 + B 2 2 ) B 2 2 + B 3 2 − B 3 2 m 2 B 2 2 + B 3 2 + dn [ λ x − r t ] 2 ( B 2 sn [ λ x − r t ] + B 3 cn [ λ x − r t ] ) 2 , and 0 ≤ m ≤ 1 .

  • Special case, if m = 1 or m = 0 , then Eq. (7) has the combo bright- dark solitary solution or periodic solution as follows:

    (137) u 3.3 , 24 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) [ − 2 + 3 S ( x , t ) ] 2 ε α 1 ,

    or

    (138) u 3.3 , 25 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 − 1 + 3 ( B 2 2 + B 3 2 ) ( B 2 sin [ λ x − r t ] + B 3 cos [ λ x − r t ] ) 2 ,

    where

    S ( x , t ) = ( B 2 2 + B 2 2 + B 3 2 sech ( λ x − r t ) ) 4 + ( B 2 tanh ( λ x − r t ) + B 3 sech ( λ x − r t ) ) 4 ( B 2 2 + B 2 2 + B 3 2 sech ( λ x − r t ) ) 2 ( B 2 tanh ( λ x − r t ) + B 3 sech ( λ x − r t ) ) 2 .

  1. If h 0 = − m 2 + 2 m − 1 B 2 2 , h 2 = 2 m 2 + 2 , and h 4 = − B 2 2 m 2 − 2 B 2 2 m − B 2 2 , then we have a Jacobi elliptic function solution for Eq. (7) as follows:

    (139) u 3.3 , 26 = r 2 − λ 2 2 ε α 1 λ 2 + 2 δ ( α 3 λ 2 − α 4 r 2 ) 2 + 2 m 2 − 3 ( m 2 − 2 m + 1 ) ℬ ( x , t ) − 3 ( m 2 + 2 m + 1 ) ℬ ( x , t ) ε α 1 ,

    where ℬ ( x , t ) = ( m sn 2 ( λ x − r t ) − 1 ) 2 ( m sn 2 ( λ x − r t ) + 1 ) 2 , and 0 < m ≤ 1 .

  1. If h 0 = h 4 = m 2 − 1 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (140) u 3.3 , 27 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( m 2 + 1 ) + 3 ( m 2 − 1 ) ℛ 1 ( x , t ) + 3 ( m 2 − 1 ) ℛ 1 ( x , t ) 2 ε α 1 ,

    or

    (141) u 3.3 , 28 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( m 2 + 1 ) + 3 ( m 2 − 1 ) ℛ 2 ( x , t ) + 3 ( m 2 − 1 ) ℛ 2 ( x , t ) 2 ε α 1 ,

    where ℛ 1 ( x , t ) = dn 2 ( λ x − r t ) ( 1 ± m sn ( λ x − r t ) ) 2 , ℛ 2 ( x , t ) = ( m sd ( λ x − r t ) ± nd ( λ x − r t ) ) 2 and 0 ≤ m < 1 .

  1. If h 0 = h 4 = 1 − m 2 4 and h 2 = m 2 + 1 2 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (142) u 3.3 , 29 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 + m 2 ) + 3 ( 1 − m 2 ) K 1 ( x , t ) + 3 ( 1 − m 2 ) K 1 ( x , t ) 2 ε α 1 ,

    or

    (143) u 5.30 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ( 1 + m 2 ) + 3 ( 1 − m 2 ) K 2 ( x , t ) + 3 ( 1 − m 2 ) K 2 ( x , t ) 2 ε α 1 ,

    where K 1 ( x , t ) = cn 2 [ λ x − r t ] ( 1 ± sn [ λ x − r t ] ) 2 , K 2 ( x , t ) = ( nc [ λ x − r t ] ± sc [ λ x − r t ] ) 2 , and 0 ≤ m < 1 .

  1. If h 0 = ( m 2 − 1 ) 2 4 , h 2 = m 2 + 1 2 , and h 4 = − 1 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (144) u 3.3 , 31 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) ε α 1 2 ( 1 + m 2 ) + 3 ( m 2 − 1 ) 2 T ( x , t ) − 3 T ( x , t ) ,

    where T ( x , t ) = ( m cn [ λ x − r t ] ± dn [ λ x − r t ] ) 2 and 0 ≤ m ≤ 1 .

  1. If h 0 = 1 4 , h 2 = m 2 + 1 2 , and h 4 = ( m 2 − 1 ) 2 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (145) u 3.3 , 32 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 ( 1 + m 2 ) + 3 T 1 ( x , t ) + 3 ( m 2 − 1 ) 2 T 1 ( x , t ) ,

    where T 1 ( x , t ) = sn 2 [ λ x − r t ] ( dn [ λ x − r t ] ± cn [ λ x − r t ] ) 2 and 0 ≤ m ≤ 1 .

  1. If h 0 = 1 4 , h 2 = m 2 − 2 2 , and h 4 = m 4 4 , then we have the Jacobi elliptic function solutions for Eq. (7) as follows:

    (146) u 3.3 , 33 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 ( − 2 + m 2 ) + 3 V 1 ( x , t ) + 3 m 4 V 1 ( x , t ) ,

    or

    (147) u 3.3 , 34 = r 2 − λ 2 2 ε α 1 λ 2 + δ ( α 3 λ 2 − α 4 r 2 ) 2 ε α 1 2 ( − 2 + m 2 ) + 3 V 2 ( x , t ) + 3 m 4 V 2 ( x , t ) ,

    where V 1 ( x , t ) = sn 2 ( λ x − r t ) ( 1 ± dn ( λ x − r t ) ) 2 , V 2 = cn 2 [ λ x − r t ] ( 1 − m 2 ± dn [ λ x − r t ] ) 2 , and 0 ≤ m ≤ 1 .

5 Graphical representation

This study successfully established new solutions to the strain wave model with the aid of the modified extended 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, 2, 3, 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 = 0.7 , ε = 0.7 , α 1 = 0.8 , δ = 0.9 , h 2 = − 0.9 , α 3 = 0.5 , and α 4 = 0.5 . Figure 2 represents the 3D, contour, and 2D solution of a dark solitary solution for Eq. (24) with r = 0.7 , λ = 0.5 , ε = 1 , α 1 = 0.7 , δ = 0.8 , α 3 = 1.4 , and α 4 = 0.5 . Figure 3 represents the 3D, contour, and 2D solution of a bright solitary solution for Eq. (82) with r = 1.2 , λ = 0.8 , ε = 0.8 , α 1 = 0.8 , δ = 0.7 , α 3 = 0.9 , and α 4 = 0.5 . Figure 4 represents the 3D, contour and 2D solution of a periodic solution for Eq. (96) with r = 1.5 , λ = 0.9 , ε = 0.8 , α 1 = 0.9 , δ = 1.2 , α 3 = 2.2 , and α 4 = 0.5 .

Figure 1 Bright solitary solution of (13): (a) 3D, (b) contour, and (c) 2D.
Figure 1

Bright solitary solution of (13): (a) 3D, (b) contour, and (c) 2D.

Figure 2 Dark solitary solution (24): (a) 3D, (b) contour, and (c) 2D.
Figure 2

Dark solitary solution (24): (a) 3D, (b) contour, and (c) 2D.

Figure 3 Bright solitary solution of (82): (a) 3D, (b) contour, and (c) 2D.
Figure 3

Bright solitary solution of (82): (a) 3D, (b) contour, and (c) 2D.

Figure 4 Periodic solution of (96): (a) 3D, (b) contour, and (c) 2D.
Figure 4

Periodic solution of (96): (a) 3D, (b) contour, and (c) 2D.

6 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, singular-dark 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.

7 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 [6,21–25], 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.

  1. Funding information: Not available.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors have no conflict of interest.

  4. Data availability statement: All data are included inside the manuscript.

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Received: 2022-12-19
Revised: 2023-04-05
Accepted: 2024-04-11
Published Online: 2024-05-28

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0293
Keywords for this article
stress waves; solitary wave solutions; periodic solutions; Jacobi elliptic function solutions; analytic method
Creative Commons
BY 4.0

Articles in the same Issue

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
  3. Research Articles
  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
  5. Robust passivity-based nonlinear controller design for bilateral teleoperation system under variable time delay and variable load disturbance
  6. A physically consistent AI-based SPH emulator for computational fluid dynamics
  7. Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption
  8. A novel framework for effective structural vulnerability assessment of tubular structures using machine learning algorithms (GA and ANN) for hybrid simulations
  9. Flow and irreversible mechanism of pure and hybridized non-Newtonian nanofluids through elastic surfaces with melting effects
  10. Stability analysis of the corruption dynamics under fractional-order interventions
  11. Solutions of certain initial-boundary value problems via a new extended Laplace transform
  12. Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method
  13. Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system
  14. Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method
  15. Analysis of nonlinear dynamics of RC slabs under blast loads: A hybrid machine learning approach
  16. On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations
  17. Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation
  18. Influence of damage on large displacement buckling analysis of beams
  19. Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines
  20. Mathematical analysis of a combustible viscoelastic material in a cylindrical channel taking into account induced electric field: A spectral approach
  21. A new operational matrix method to solve nonlinear fractional differential equations
  22. New solutions for the generalized q-deformed wave equation with q-translation symmetry
  23. Optimize the corrosion behaviour and mechanical properties of AISI 316 stainless steel under heat treatment and previous cold working
  24. Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena
  25. Investigation of the lubrication performance of a marine diesel engine crankshaft using a thermo-electrohydrodynamic model
  26. Modeling credit risk with mixed fractional Brownian motion: An application to barrier options
  27. Method of feature extraction of abnormal communication signal in network based on nonlinear technology
  28. An innovative binocular vision-based method for displacement measurement in membrane structures
  29. An analysis of exponential kernel fractional difference operator for delta positivity
  30. Novel analytic solutions of strain wave model in micro-structured solids
  31. Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model
  32. Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel
  33. Non-linear influences of track dynamic irregularities on vertical levelling loss of heavy-haul railway track geometry under cyclic loadings
  34. Fast analysis approach for instability problems of thin shells utilizing ANNs and a Bayesian regularization back-propagation algorithm
  35. Validity and error analysis of calculating matrix exponential function and vector product
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  39. Discovering optical solutions to a nonlinear Schrödinger equation and its bifurcation and chaos analysis
  40. New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation
  41. Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation
  42. Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input
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  44. Solitary and periodic pattern solutions for time-fractional generalized nonlinear Schrödinger equation
  45. An unconditionally stable numerical scheme for solving nonlinear Fisher equation
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  47. Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity
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  52. Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems
  53. Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation
  54. Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering
  55. Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method
  56. Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme
  57. Mathematical model based on nonlinear differential equations and its control algorithm
  58. Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation
  59. Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative
  60. Minimal universal laser network model: Synchronization, extreme events, and multistability
  61. Valuation of forward start option with mean reverting stock model for uncertain markets
  62. Geometric nonlinear analysis based on the generalized displacement control method and orthogonal iteration
  63. Fuzzy neural network with backpropagation for fuzzy quadratic programming problems and portfolio optimization problems
  64. B-spline curve theory: An overview and applications in real life
  65. Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise
  66. Quantitative analysis and modeling of ride sharing behavior based on internet of vehicles
  67. Review Article
  68. Bond performance of recycled coarse aggregate concrete with rebar under freeze–thaw environment: A review
  69. Retraction
  70. Retraction of “Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning”
  71. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part II
  72. Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems
  73. Anti-control of Hopf bifurcation for a chaotic system
  74. Special Issue: Decision and Control in Nonlinear Systems - Part I
  75. Addressing target loss and actuator saturation in visual servoing of multirotors: A nonrecursive augmented dynamics control approach
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  78. Special Issue: Unleashing the Power of AI and ML in Dynamical System Research
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  81. Research on the role of multi-sensor system information fusion in improving hardware control accuracy of intelligent system
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