Aly R. Seadawy, Syed Tahir Raza Rizvi, Sarfraz Ahmad, Muhammad Younis and Dumitru Baleanu

Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation

De Gruyter | 2021

Abstract

The aim of this article was to address the lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation with the aid of Hirota bilinear technique. This model concerns in a massive nematic liquid crystal director field. By choosing the function f in Hirota bilinear form, as the general quadratic function, trigonometric function and exponential function along with appropriate set of parameters, we find the lump, lump-one stripe, multiwave and breather solutions successfully. We also interpreted some three-dimensional and contour profiles to anticipate the wave dynamics. These newly obtained solutions have some arbitrary constants and so can be applicable to explain diversity in qualitative features of wave phenomena.

1 Introduction

In waves theory, nonlinear partial differential equations(NPDEs), which explain nonlinear aspects, appear in an extensive diversity of scientific and engineering applications, for example, plasma physics, fluid dynamics, hydrodynamics, acoustics, solid-state physics, hydrodynamics and theory of turbulence, optics, optical fibers, chemical physics, chaos theory and many other applications. The study of NPDEs becomes increasingly significant because of their prominent features. A main attachment of scientific work has been perceived in the last few decades on NPDEs such as efficient integration method [1], improved modified Kudryashov method [2], asymptotic method [3], geometric singular perturbation [4], Lie symmetry analysis [5], Painleve expansion procedure [6], G G expansion approach, highly optical solitons, homotopy perturbation method, the semi-inverse method [7,8, 9,10], simple-equation method, logistic function, Backlund transformation technique, asymptotic method, integrability method [11,12,13, 14,15], extended mapping method, non-perturbative method, nonlinearity and conservation, Hirota bilinear method [16,17, 18,19], extended and modified direct algebraic method, extended mapping method and Seadawy techniques [20,21,22, 23,24,25, 26,27] and so on.

There are many renowned models, such as Vakhnenko dynamical equation [35], nonlinear Schrodinger equation [36], KdV equation [37], Camassa-Holm equation [38], sine-Gordon equation [39] and Biswas-Milovic equation [40], but here we will obtain the exact solutions of the Hunter–Saxton (HS) equation [41],

(1) u t x x 2 k u x + 2 u x u x x + u u x x x = 0 .
This equation is used for propagation of orientation waves in a massive nematic liquid crystal director field. The HS equation can be used as a short wave limit of the Camassa–Holm equation:
(2) m t + 2 m u x + u m x = 0 , m = k + u u x x .
The content of this article is organized as follows: in Section 2, we evaluate the lump solutions via some three-dimensional (3D) and contour shapes. In Section 3, we find out lump-one stripe interactional solutions and some physical 3D shapes. In Section 4, the brief discussion of multiwave solutions for the proposed model is given. In Section 5, we find breather solutions. In Section 6, there are results and discussion about our newly obtained solutions and comparison with already published work, and in Section 7, we give concluding remarks.

2 Lump solution

In the direction to find lump solutions of equation (1), we apply the transformation [42],

(3) u = m + 2 b ln f x ,
which transforms equation (1),
(4) 4 k b f 3 f x 2 12 b f f y f x 3 + + 12 b m f 4 f x x x x + 4 b 2 f 3 f x f x x x x = 0 .
Now the function f in equation (5) can be assumed as [ 36],
(5) f = a 7 + g 2 + h 2 ,
where g 2 =  a 1 x +  a 2 t +  a 3h 2 =  a 4 x +  a 5 t +  a 6. However, a i (1 ≤  i ≤ 7) are all real parameters to be measured. Now, substituting f into equation (5) and associating the coefficients of x and t imply us the subsequent result on parameters:

Set I.

(6) a 1 = 3 m 50 b , a 2 = 1 5 3 m 50 b m , a 3 = a 3 , a 4 = 0 , a 5 = 0 .
The parameters in equation (6) prevent the lump solutions to equation (1)
(7) u ( x , t ) = m + 2 6 b m b a 3 1 25 3 m 2 b m t + 1 5 3 m 2 b x 5 a 6 2 + a 7 a 3 1 25 3 m 2 b m t + 1 5 3 m 2 b x 2 .

Set II.

(8) a 1 = a 1 , a 2 = a 2 , a 3 = a 3 , a 4 = 1 a 1 , a 5 = 1 a 2 .
The parameters in equation (8) imply the lump solutions to equation (1)
(9) u ( x , t ) = m + 2 b 2 i a 1 a 6 + i a 2 t + i a 1 x + 2 a 1 a 3 + a 2 t + a 1 x a 7 + a 6 + i a 2 t + i a 1 x 2 + a 3 + a 2 t + a 1 x 2 .

3 Lump-one stripe soliton interaction solution

To this aim, f in the bilinear equation can be assumed as [43],

(10) f = g 2 + h 2 + b 1 exp ( f 1 ) + a 7 ,
where g 2 =  a 1 x +  a 2 t +  a 3h 2 =  a 4 x +  a 5 t +  a 6f 1 =  k 1 x +  k 2 t. However, a i  (1 ≤  i ≤ 7), k 1 and k 2 are all real parameters to be found. Now, inserting f in equation (3) and relating the coefficients of x and t give us:

Set I.

(11) a 1 = 1 a 4 , a 2 = 1 5 m k 1 2 + 2 k k 1 a 4 , a 5 = 2 k m k 1 2 2 k 1 a 4 , k 2 = 2 k m k 1 2 k 1 a 1 .
The parameters in equation (11) create the required solutions to equation (1)
(12) u ( x , t ) = m + 2 b b 1 Δ 1 k 1 + 2 i a 4 a 3 + i μ + i a 4 x + 2 a 4 a 6 + μ + a 4 x a 7 + b 1 Δ 1 + a 3 + i μ + i a 4 x 2 + a 6 + μ + a 4 x 2 ,
where Δ 1 = e i a 4 2 k m k 1 2 t k 1 + k 1 x and μ = i ( 2 k m k 1 2 ) 2 a 4 k 1 .

Set II.

(13) a 1 = 1 a 4 , a 2 = 1 a 4 a 5 k 1 + m k 1 2 2 k k 1 a 4 , a 5 = 2 k m k 1 2 2 k 1 a 4 , k 2 = 2 k m k 1 2 k 1 a 1 .
The parameters in equation (13) reveal the required solutions to equation (1)
(14) u ( x , t ) = m + 2 b b 1 Δ 1 k 1 + 2 i a 4 a 3 + i ν + i a 4 x + 2 a 4 a 6 + a 5 t + a 4 x a 7 + b 1 Δ 1 + a 3 + i ν + i a 4 x 2 + a 6 + a 5 t + a 4 x 2 ,
where Δ 1 = e i a 4 ( 2 k m k 1 2 ) t k 1 + k 1 x and ν = i ( 2 k + a 5 a 4 k 1 m k 1 2 ) t a 4 k 1 .

4 Multiwave solutions

For finding multiwave solutions, we use the succeeding transformation in equation (1) [44],

(15) u ( x , t ) = ψ ( ξ ) , ξ = k 1 x c 1 t .
With the help of the above transformation, we obtain:
(16) 2 k k 1 ψ + 2 k 1 3 k 2 ψ ψ c 1 k 1 2 ψ 3 + k 1 3 ψ ψ 3 = 0 .
Now with the aid of the following assumption in equation (16)
(17) ψ = 2 ln f ξ ,
we get,
(18) 2 k 1 20 k 1 2 f ξ 5 + 2 k 1 f f ξ 3 3 c 1 f ξ + 22 k 1 f ξ ξ 6 k 1 f 2 f ξ + 2 k 1 f ξ 2 c 1 f ξ ξ ξ + k 1 f ξ ξ ξ ξ = 0 .
To find the multiwave solutions of equation (18), we apply the subsequent hypothesis [ 40]:
(19) f = b 0 cosh ( a 1 ξ + a 2 ) + b 1 cos ( a 3 ξ + a 4 ) + b 2 cosh ( a 5 ξ + a 6 ) ,
where a 1a 2a 3a 4a 5 and a 6 are any constants to be examined. Substituting equation (19) into equation (18) via symbolic computation and collecting the coefficients of all powers of sinh ( a 1 ξ + a 2 ) and sinh ( a 5 ξ + a 6 ) , cos ( a 3 ξ + a 4 ) , cosh ( a 1 ξ + a 2 ) , cosh ( a 5 ξ + a 6 ) , sin ( a 3 ξ + a 4 ) , functions to be zero, we get a system of equations. After solving this algebraic system, we obtain some different parametric values:

Set I.

(20) a 1 = a 1 , b 0 = 0 , a 2 = a 2 , c 1 = 24 k 1 2 a 3 2 a 5 2 a 3 2 3 a 5 2 , a 3 = a 3 , a 4 = a 4 .
By substituting equation (20) into equation (19), we get
(21) f = b 0 cosh ( a 1 ξ + a 2 ) + b 1 cos ( a 3 ξ + a 4 ) + b 2 cosh ( a 5 ξ + a 6 ) .

As f solves equation (18), then ψ solves equation (16) via ψ = 2 ln f ξ we obtain

(22) u ( x , t ) = 2 a 3 b 1 cos a 4 + a 3 Ω 1 sin a 4 + a 3 Ω 1 + a 5 b 2 cos a 6 + a 5 Ω 1 sin a 6 + a 5 Ω 1 b 1 cos a 4 + a 3 Ω 1 + b 2 cos a 6 + a 5 Ω 1 ,
where Ω 1 = 24 a 3 2 a 5 2 k 1 2 t a 3 2 3 a 5 2 + k 1 x .

5 Breather solutions

For finding the breathers of equation (18), we assume the successive transformation [45]:

(23) f = e p a 2 + a 1 ξ + b 1 e p a 4 + a 3 ξ + b 0 cos p 1 a 6 + a 5 ξ ,
where a s are any real constants to be found. Inserting equation (23) into (18) via computational Mathematica and collecting all the coefficients of trigonometric and exponential functions to be zero we get a system of equations. After solving this system, we found:

Set I.

(24) a 2 = a 2 , a 3 = 4 1 3 k k 1 2 1 3 p , a 4 = a 4 , a 5 = 0 , c 1 = 4 1 3 k k 1 2 1 3 2 k 1 , b 0 = b 0 .
As f solves equation (18), after that ψ solves equation (16) through applying ψ = 2 ln f ξ , we get
(25) u ( x , t ) = 2 2 2 3 b 1 e p ω k k 1 2 1 3 b 0 + e a 2 p + b 1 e p ω k k 1 2 1 3 ,
where ω = a 4 + 2 2 3 k k 1 2 1 3 k k 1 2 1 3 k 1 t 2 1 3 + k 1 x p .

6 Results and discussion

In this section, we have made a detailed comparison of our accomplished results with the earlier literature. Many researchers used various methods for calculating solitary wave solutions of the Hunter–Saxton equation. Particularly, Beals et al. applied inverse scattering technique [46], Alberto et al. used distance functional [47], Lenells applied properties of the Riemannian [48], Lenells et al. applied a geometric approach [49], Bressan et al. utilized Lipschitz metric [50], Zhao et al. applied conservation laws [51], Korpinar used symmetry analysis [51] and Zhao applied conservation laws to obtain the exact solutions for the presented model [52]. But here, in this work we have found the lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation with the aid of Hirota bilinear approach. These types of solutions have been utilized in many fields of science, for example, physics, chemistry, biology, finance, oceanographic engineering, capillary flow and nonlinear optics [42,43, 44,45]. Now we will notice how our obtained results alter their shapes via different values of a 3 from Figure 1(i), and we can see at a 3 = 0.2 that u has a maximum value at some points, expressing 3D shape of u making two bright lump solutions. Similarly, in Figure 1(ii) at a 3 = 0.8 the same features are observed. But at a 3 = 2 in Figure 1(iii), we obtain three bright lump solutions. In the same way, for a 3 = 5 in Figure 2(i) we have achieved seven bright lump solutions and the process was repeated for gradually increasing values of a 3, for instance, a 3 = 8, a 3 = 15 (Figure 2(ii) and (iii)) respectively. Figures 3 and 4 express the relating contour profiles for Figures 1 and 2, respectively. Now we have observed how our obtained solutions change their wave structure via appropriate choices of a 3 from Figure 5(i), and we can notice a lump-one stripe soliton at a 3 = 0.2. Similarly, Figures 5(ii), 5(iii) and 6(i)–(iii) show how a lump-one stripe soliton rises or descends for different values of a 3. Figures 7 and 8 present the associating contour graphs for Figures 5 and 6, respectively. Also, Figures 9(i)–(iii) and 10(i)–(iii) show kink wave and their changes in wave shape via a 3 = −1, a 3 = 0.2, a 3 = 0.5, a 3 = 0.8, a 3 = 2 and a 3 = 5, respectively. Similarly, Figures 11(i)–(iii) and 12(i)–(iii) show solitary wave and the changes in their structure through a 2 = −1, a 2 = 0.2, a 2 = 0.5, a 2 = 0.8, a 2 = 2 and a 2 = 5, respectively. Finally, Figures 13 and 14 present the concerning contour graphs for Figures 11 and 12, respectively.

Figure 1 
               The graphs of the solution u(x, t) in equation (7) are shown via suitable parameters m = 5, b = −5, a
                  7 = 7, a
                  6 = 5. 3D graphs at (i) a
                  3 = 0.2, (ii) a
                  3 = 0.8 and (iii) a
                  3 = 2, respectively.

Figure 1

The graphs of the solution u(xt) in equation (7) are shown via suitable parameters m = 5, b = −5, a 7 = 7, a 6 = 5. 3D graphs at (i) a 3 = 0.2, (ii) a 3 = 0.8 and (iii) a 3 = 2, respectively.

Figure 2 
               The profiles of the solution u(x, t) in equation (7) are shown by different choices of parameters m = 5, b = −5, a
                  7 = 7, a
                  6 = 5. 3D graphs at (i) a
                  3 = 5, (ii) a
                  3 = 8 and (iii) a
                  3 = 15, respectively.

Figure 2

The profiles of the solution u(xt) in equation (7) are shown by different choices of parameters m = 5, b = −5, a 7 = 7, a 6 = 5. 3D graphs at (i) a 3 = 5, (ii) a 3 = 8 and (iii) a 3 = 15, respectively.

Figure 3 
               The corresponding contour profiles for Figure 1.

Figure 3

The corresponding contour profiles for Figure 1.

Figure 4 
               The associating contour graphs for Figure 2.

Figure 4

The associating contour graphs for Figure 2.

Figure 5 
               The profiles of the solution u(x, t) in equation (14) via various choices of parameters m = 1, b = 5, a
                  3 = 0.2, a
                  4 = 1, a
                  5 = 2, a
                  6 = 5, b
                  1 = 2, k
                  1 = −1, k = 1. Contour profiles at (i) a
                  3 = 0.2, (ii) a
                  3 = 0.8 and (iii) a
                  3 = 2, respectively.

Figure 5

The profiles of the solution u(xt) in equation (14) via various choices of parameters m = 1, b = 5, a 3 = 0.2, a 4 = 1, a 5 = 2, a 6 = 5, b 1 = 2, k 1 = −1, k = 1. Contour profiles at (i) a 3 = 0.2, (ii) a 3 = 0.8 and (iii) a 3 = 2, respectively.

Figure 6 
               The shapes of the solution u(x, t) in equation (14) are shown by various choices of parameters m = 1, b = 5, a
                  3 = 0.2, a
                  4 = 1, a
                  5 = 2, a
                  6 = 5, b
                  1 = 2, k
                  1 = −1, k = 1. 3D graphs at (i) a
                  3 = 5, (ii) a
                  3 = 10 and (iii) a
                  3 = 15, respectively.

Figure 6

The shapes of the solution u(xt) in equation (14) are shown by various choices of parameters m = 1, b = 5, a 3 = 0.2, a 4 = 1, a 5 = 2, a 6 = 5, b 1 = 2, k 1 = −1, k = 1. 3D graphs at (i) a 3 = 5, (ii) a 3 = 10 and (iii) a 3 = 15, respectively.

Figure 7 
               The relating contour graphs for Figure 5.

Figure 7

The relating contour graphs for Figure 5.

Figure 8 
               The associating contour profiles for Figure 6.

Figure 8

The associating contour profiles for Figure 6.

Figure 9 
               The profiles of the solution u(x, t) in equation (22) via various choices of parameters m = 1, b = 5, a
                  4 = 1, a
                  5 = 2, a
                  6 = 5, a
                  7 = 1, b
                  1 = 2, b
                  2 = 1, k
                  1 = −1, k = 1. 3D graphs at (i) a
                  3 = 0.2, (ii) a
                  3 = 0.8 and (iii) a
                  3 = 2, respectively.

Figure 9

The profiles of the solution u(xt) in equation (22) via various choices of parameters m = 1, b = 5, a 4 = 1, a 5 = 2, a 6 = 5, a 7 = 1, b 1 = 2, b 2 = 1, k 1 = −1, k = 1. 3D graphs at (i) a 3 = 0.2, (ii) a 3 = 0.8 and (iii) a 3 = 2, respectively.

Figure 10 
               The profiles of the solution u(x, t) in equation (22) via various choices of parameters m = 1, b = 5, a
                  4 = 1, a
                  5 = 2, a
                  6 = 5, a
                  7 = 1, b
                  1 = 2, b
                  2 = 1, k
                  1 = −1, k = 1. 3D graphs at (i) a
                  3 = 5, (ii) a
                  3 = 10 and (iii) a
                  3 = 15, respectively.

Figure 10

The profiles of the solution u(xt) in equation (22) via various choices of parameters m = 1, b = 5, a 4 = 1, a 5 = 2, a 6 = 5, a 7 = 1, b 1 = 2, b 2 = 1, k 1 = −1, k = 1. 3D graphs at (i) a 3 = 5, (ii) a 3 = 10 and (iii) a 3 = 15, respectively.

Figure 11 
               The graphs of the solution u(x, t) in equation (25) through different choices of parameters m = 1, p = 5, a
                  4 = 1, b
                  0 = 1, b
                  1 = 2, k
                  1 = −1, k = 1. 3D graphs at (i) a
                  2 = −1, (ii) a
                  2 = 0.2 and (iii) a
                  2 = 0.4, respectively.

Figure 11

The graphs of the solution u(xt) in equation (25) through different choices of parameters m = 1, p = 5, a 4 = 1, b 0 = 1, b 1 = 2, k 1 = −1, k = 1. 3D graphs at (i) a 2 = −1, (ii) a 2 = 0.2 and (iii) a 2 = 0.4, respectively.

Figure 12 
               The shapes of the solution u(x, t) in equation (25) via appropriate choices of parameters m = 1, p = 5, a
                  4 = 1, b
                  0 = 1, b
                  1 = 2, k
                  1 = −1, k = 1. 3D graphs at (i) a
                  2 = 0.8, (ii) a
                  2 = 2 and (iii) a
                  2 = 15, respectively.

Figure 12

The shapes of the solution u(xt) in equation (25) via appropriate choices of parameters m = 1, p = 5, a 4 = 1, b 0 = 1, b 1 = 2, k 1 = −1, k = 1. 3D graphs at (i) a 2 = 0.8, (ii) a 2 = 2 and (iii) a 2 = 15, respectively.

Figure 13 
               The corresponding contour shapes of Figure 11.

Figure 13

The corresponding contour shapes of Figure 11.

Figure 14 
               The relating contour profiles of Figure 12.

Figure 14

The relating contour profiles of Figure 12.

7 Concluding remarks

The purpose of this article is to accumulate the lump, lump-one stripe, multi wave and breather solutions for the Hunter–Saxton equation by way of Hirota bilinear scheme and through defining appropriate transformations. We have successfully generated some new exact solutions to the concerning model. The 3D and contour graphs mapped different numeric values, to observe the physical behavior of the system. For better understanding and more effectiveness, we have also explained the geometry of the graphs. The attained solutions show that the proposed method is very reliable, aggressive and simple, and so, the recommended idea could be extended for further nonlinear models in mathematical physics.

References

[1] Ghanbari B, Nisar KS, Aldhaifallah M. Abundant solitary wave solutions to an extended nonlinear Schrödingers equation with conformable derivative using an efficient integration method. Adv Diff Equ. 2020;2020(1):1–25. Search in Google Scholar

[2] Hyder AA, Barakat MA. General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics. Phys Scr. 2020;95(4):045212. Search in Google Scholar

[3] Bildik N, Deniz S. Comparative study between optimal homotopy asymptotic method and perturbation-iteration technique for different types of nonlinear equations. Iran J Sci Technol A. 2018:42(2):647–54 Search in Google Scholar

[4] Ge J, Du Z. The solitary wave solutions of the nonlinear perturbed shallow water wave model. Appl Math Lett. 2020;103:106202. Search in Google Scholar

[5] Tian SF. Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation. Appl Math Lett. 2020;100:106056. Search in Google Scholar

[6] Farah N, Seadawy AR, Ahmad S, Rizvi STR, Younis M. Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model. Opt Quantum Electron. 2020;52(7):1–15. Search in Google Scholar

[7] Seadawy Aly R, El-Rashidy K. Application of the extension exponential rational function method for higher dimensional Broer-Kaup-Kupershmidt dynamical system. Mod Phys Lett A. 2020;35(1):1950345 (14 pages). Search in Google Scholar

[8] Kudryashov NA. Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations. Appl Math Comput. 2020;371:124972. Search in Google Scholar

[9] Younas U, Seadawy Aly R, Younis M, Rizvi STR. Dispersive of propagation wave structures to the Dullin-Gottwald-Holm dynamical equation in a shallow water waves. Chin J Phys. 2020;68:348–64. Search in Google Scholar

[10] Younas U, Younis M, Seadawy AR, Rizvi STR. Rizvi, Optical solitons and closed form solutions to (3 + 1)-dimensional resonant Schrodinger equation. Int J Mod Phys B. 2020;34(30):2050291. 10.1142/S0217979220502914. Search in Google Scholar

[11] Bilal M, Seadawy AR, Younis M, Rizvi STR, Zahed Hanadi. Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin-Gottwald-Holm system and modulation instability analysis. Math Methods Appl Sci. 2020. 10.1002/mma.7013. Search in Google Scholar

[12] Kudryashov NA. Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik. 2020;312:163550. Search in Google Scholar

[13] Zhao Z. Bäcklund transformations, rational solutions and soliton-cnoidal wave solutions of the modified Kadomtsev-Petviashvili equation. Appl Math Lett. 2019;8:103–10. Search in Google Scholar

[14] Raza Rizvi Syed Tahir, Seadawy AR, Ali Ijaz, Bibi Ishrat, Muhammad Younis. Chirp-free optical dromions for the presence of higher order spatio-temporal dispersions and absence of self-phase modulation in birefringent fibers. Mod Phys Lett B. 2020;34(35):2050399. 10.1142/S0217984920503996. Search in Google Scholar

[15] Liu JG, Yang XJ, Feng YY. On integrability of the time fractional nonlinear heat conduction equation. J Geom Phys. 2019;144:190–8. Search in Google Scholar

[16] Seadawy AR, Cheemaa N. Applications of extended modified auxiliary equation mapping method for high-order dispersive extended nonlinear Schrödinger equation in nonlinear optics. Mod Phys Lett B. 2019;33(18):1950203. Search in Google Scholar

[17] James AJ, Konik RM, Lecheminant P, Robinson NJ, Tsvelik AM. Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods. Rep Prog Phys. 2018;81(4):046002. Search in Google Scholar

[18] Kumar S. A new analytical modelling for fractional telegraph equation via Laplace transform. Appl Math Model. 38(13):3154–63. Search in Google Scholar

[19] Kumar S, Ghosh S, Samet B, Goufo EFD. An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator. Math Methods Appl Sci. 2020;43(9):6062–80. Search in Google Scholar

[20] Kumar S, Kumar R, Cattani C, Samet B. Chaotic behaviour of fractional predator-prey dynamical system. Chaos, Solitons Fractals. 2020;135:109811. Search in Google Scholar

[21] Kumar S, Nisar KS, Kumar R, Cattani C, Samet B. A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force. 2020;43(7):4460–71. Search in Google Scholar

[22] Goufo EFD, Kumar S, Mugisha SB. Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos, Solitons Fractals. 2019;130:109467. Search in Google Scholar

[23] Wang KL, Yao SW, Liu YP, Zhang LN. A fractal variational principle for the telegraph equation with fractal derivatives. Fractals. 2020;28(4):2050058. Search in Google Scholar

[24] Wang KL, Wang KJ. A modification of the reduced differential transform method for fractional calculus. Therm Sci. 2018;22(4):1871–75. Search in Google Scholar

[25] Djilali S. Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability. J Appl Math Compu. 2017;58(1–2):125–49. Search in Google Scholar

[26] Djilali S., Touaoula TM, Miri SEH. A Heroin epidemic model: very general non linear incidence, treat-age, and global stability. Acta Appl Math. 2017;152(1):171–94. Search in Google Scholar

[27] Djilali S. Impact of prey herd shape on the predator-prey interaction. Chaos, Solitons Fractals. 2019;120:139–48. Search in Google Scholar

[28] Djilali S., Bentout S. Spatiotemporal patterns in a diffusive predator-prey model with prey social behavior. Acta Appl Math. 2020;169:125–43. Search in Google Scholar

[29] Djilali S. Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition. Math Meth Appl Scien. 2020;43(5):2233–50. Search in Google Scholar

[30] Ahmad H, Seadawy AR, Khan TA, Thounthong P. Analytic approximate solutions Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J Taibah Univ Sci. 2020;14(1):346–58. Search in Google Scholar

[31] Ozkan YG, Yasar E, Seadawy AR A third-order nonlinear Schrodinger equation: the exact solutions, group-invariant solutions and conservation laws. J Taibah Univ Sci. 2020;14(1):585–97. Search in Google Scholar

[32] Helal MA, Seadawy AR, Zekry MH. Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation. Appl Math Comput. 2014;232:1094–103. Search in Google Scholar

[33] Iqbal M., Seadawy AR, Lu D. Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods. Mod Phys Lett A. 2018;33(32):1850183. Search in Google Scholar

[34] Seadawy AR, Lu D, Yue C. Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability. J Taibah Univ Sci. 2017;11:623–33. Search in Google Scholar

[35] Seadawy AR, Asghar A, Baleanu D. Transmission of high-frequency waves in a tranquil medium with general form of the Vakhnenko dynamical equation. Phys Scr. 2020;95:095208. Search in Google Scholar

[36] Islam W, Younis M, Rizvi STR. Optical solitons with time fractional nonlinear Schrödinger equation and competing weakly nonlocal nonlinearity. Optik. 2017;130:562–67. Search in Google Scholar

[37] Wang XB, Tian SF, Xua MJ, Zhang TT. On integrability and quasi-periodic wave solutions to a (3 + 1)-dimensional generalized KdV-like model equation. Appl Math Comput. 2016;283:216–33. Search in Google Scholar

[38] Dehghan M, Abbaszadeh M, Mohebbi A. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations. Eng Anal Bound Elem. 2015;50:412–34. Search in Google Scholar

[39] Ghaffar A, Ali A, Ahmed S, Akram S, Baleanu D, Nisar KS. A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order. Adv Differ Equ. 2020;2020(1):1–15. Search in Google Scholar

[40] Rizvi STR, Ali K, Ahmad M. Optical solitons for Biswas-Milovic equation by new extended auxiliary equation method. Optik. 2020;204:164181. Search in Google Scholar

[41] Lou S, Feng BF, Yao R. Multi-soliton solution to the two-component Hunter-Saxton equation. Wave Motn. 2016;65:17–28. Search in Google Scholar

[42] WaZhou Y, Manukure S, Ma WX. Lump and lump-soliton solutions to the Hirota Satsuma equation. Commun Nonlinear Sci Numer Simul. 2019;68:56–62. Search in Google Scholar

[43] Wang H. Lump and interaction solutions to the (2 + 1)-dimensional Burgers equation. Appl Math Lett. 2018;85:27–34. Search in Google Scholar

[44] Ahmed I, Seadawy AR, Lu D. Kinky breathers, W-shaped and multi-peak solitons interaction in (2 + 1)-dimensional nonlinear Schrödinger equation with Kerr law of nonlinearity. Eur Phys J Plus. 2019;134(3):1–10. Search in Google Scholar

[45] Ahmed I, Seadawy AR, Lu D. Mixed lump-solitons, periodic lump and breather soliton solutions for (2 + 1) dimensional extended Kadomtsev Petviashvili dynamical equation. Int J Mod Phys B. 2019;33(05):1950019. Search in Google Scholar

[46] Beals R, Sattinger DH, Szmigielski J. Inverse scattering solutions of the Hunter-Saxton equation. Appl Anal. 2001;78(3):255–69. Search in Google Scholar

[47] Bressan A, Constantin A. Global solutions of the Hunter-Saxton equation. SIAM J Math Anal. 2005;37(3):996–1026. Search in Google Scholar

[48] Lenells J. The Hunter-Saxton equation describes the geodesic flow on a sphere. J Geom Phys. 2007;57(10):2049–64. Search in Google Scholar

[49] Lenells J. The Hunter-Saxton equation: a geometric approach. SIAM J Math Anal. 2008;40(1):266–77. Search in Google Scholar

[50] Bressan A, Holden H, Raynaud X. Lipschitz metric for the Hunter-Saxton equation. J Math Pures Appl. 2010;94(1):68–92. Search in Google Scholar

[51] Korpinar Z. On characterization of invariant and exact solutions of Hunter saxton equation. J Sci Arts. 2018;94(1):18(3):603–10. Search in Google Scholar

[52] Zhao Z. Conservation laws and nonlocally related systems of the Hunter-Saxton equation for liquid crystal. Anal Math Phys. 2019;9(4):2311–27. Search in Google Scholar