Investigation of interactional phenomena and multi wave solutions of the quantum hydrodynamic Zakharov–Kuznetsov model

Khaled El-Rashidy; Aly R. Seadawy; Saad Althobaiti; M. M. Makhlouf

doi:10.1515/phys-2021-0009

Article Open Access

Investigation of interactional phenomena and multi wave solutions of the quantum hydrodynamic Zakharov–Kuznetsov model

Khaled El-Rashidy , Aly R. Seadawy , Saad Althobaiti and M. M. Makhlouf

Published/Copyright: March 9, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 19 Issue 1

Abstract

The symbolic computation with the ansatz function and the logarithmic transformation method are used to obtain a formula for certain exact solutions of the ( 3 + 1 ) Zakharov–Kuznetsov (Z–K) equation. We use homoclinic breather, three waves method, and double exponential. There is a conflict of results with considerably known results, which indicates the solutions found in this study are new. By selecting appropriate parameter values, 3d representations are plotted to establish W-shaped, multi-peak, and kinky breathers solutions.

Keywords: 02.30Xx soliton solutions; (3 + 1) Zakharov–Kuznetsov equation; the logarithmic transformation method

1 Introduction

Many physical phenomena originate in various fields of engineering and science, such as fluid dynamics, fiber optics, quantum mechanics, plasma, and physics. These physical phenomena were constructed in the form of nonlinear equations [1,2, 3,4,5, 6,7]. To understand the behavior of a phenomenon, we need to solve nonlinear equations that describe the phenomenon, which is often challenging. Many methods have been developed over the years to solve such nonlinear equations, many of which are based on assumptions. Investigating the exact solutions for these nonlinear equations is a major area of concern for many mathematicians and physicists because of their important role in understanding the behavior of nonlinear physical phenomena. The Zakharov–Kuznetsov (Z–K) equation supports stable solitary waves, see ref. [8,9]. This makes the Z–K equation a very attractive model equation for investigating of vortices in geophysical owe. The Z–K equation determines weak non-linear behavior of ion sound waves that contain electrons with hot and cold temperatures in a regular magnetic field, see ref. [9]. With regard to the exact solutions of the Z–K and ( 3 + 1 ) -dimensional Z–K equations, many solving methods have been used to solve it, see [2,9, 10,11,12, 13,14], and have many applications in numerical and analytical approaches in physical sciences [15,16, 17,18,19, 20,21]. An efficient analytical technique for fractional partial differential equations occurs in ion-acoustic waves in plasma and warm plasma [37,38].

Some new methods and important developments in the search for analysis have been done to investigate wave solutions for partial non-linear differential equations. The results of this manuscript may be completely complementary to in existence manuscripts of literature such as: direct algebraic method that is extended and modified; techniques’ Seadawy and extended mapping method to find solutions for some nonlinear partial differential equations [5]; showing the bi-directional propagation of small-amplitude long-capillary gravity waves on the surface of shallow water [22]; bright and dark solitons, solitary wave solutions of higher-order non-linear differential equations and the elliptic function [23]; the higher-order non-linear differential equations with cubic quintic nonlinearity [24]; solitary wave solutions to the nonlinear-modified KdV dynamical equation [25]; modified equal–width equations and dispersive traveling wave solutions of the equal–width [26]; the exact travelling wave solutions of the SRLW equation and modified Liouville equation [27]; fourth-order nonlinear Ablowitz–Kaup–Newell–Segur water wave dynamical equation [28]; nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation [29]; the third-order NLS equation [30]; approximate solutions of nonlinear Parabolic equation by using modified variational iteration algorithm [31]; new solitary wave solutions of nonlinear Nizhnik–Novikov–Vesselov equation [32]; and Hirota bilinear method was used to study multiple soliton interactions [33].

The purpose of this work is to find traveling wave solutions of ( 3 + 1 ) equation-dimensional Z–K using logarithmic transformation and symbolic computation by using the function method, see [10,34, 35,36]:

(1) u t + μ u u x + 1 2 u x x x + 1 2 ( δ + 1 ) u x y y + u x z z = 0 ,

the function u is a function of the variables x , y , z and the temporal variable t which is the electrostatic wave potential in plasmas, where μ and δ are real constants coefficient.

Consider the solution in its complex form as follows:

(2) u ( x , y , z , t ) = ϕ ( ξ ) , ξ = k x + ν y + λ z + ω t ,

where k , ν , λ , and ω are real constants wave numbers and frequency.

Substituting from equation (2) into equation (1) and separating the imaginary and real parts lead to the ordinary differential equation as:

(3) k ( δ + 1 ) ν 2 + k 2 + 2 λ 2 ϕ ″ ( ξ ) + k μ ϕ ( ξ ) 2 + 2 ω ϕ ( ξ ) = 0

For exact solutions to equation (1), we use the logarithmic transformation and function method for equation (3). Introduction to the set of basic equations is procured in Section 3. A conclusion is carried out in Section 4.

2 Interactional phenomena and multi-waves solutions

Solutions for multiple waves and fractional phenomena logarithmic transformation can be shown as

(4) ϕ ( ξ ) = 2 log ( f ( ξ ) ) ξ

equation (3) has the form:

(5) k ( δ + 1 ) ν 2 + k 2 f ( ξ ) 2 f ( 3 ) ( ξ ) + 2 k ( δ + 1 ) ν 2 + k 2 + 2 λ 2 f ′ ( ξ ) 3 + 2 k μ f ( ξ ) f ′ ( ξ ) 2 + 2 ω f ( ξ ) 2 f ′ ( ξ ) − 3 k ( δ + 1 ) ν 2 + k 2 + 2 λ 2 f ( ξ ) f ′ ( ξ ) f ″ ( ξ ) = 0

2.1 Type (I): Three waves hypothesis

In this subsection, for the next three waves hypothesis, multiple wave solutions can be expressed for nonlinear equation (5) as follows:

(6) f ( ξ ) = 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 the real constants a j , ( j = 1 , … , 6 ) are determined later. Substituting (6) into (5) with the help of collecting all the parameters for all the exponents of cosh ( a 1 ξ + a 2 ) , cos ( a 3 ξ + a 4 ) , cosh ( a 5 ξ + a 6 ) , sinh ( a 1 ξ + a 2 ) , sin ( a 3 ξ + a 4 ) and sinh ( a 5 ξ + a 6 ) functions be zero, which establishes a set of algebraic equations with respect to above constants. With help of the Wolfram Mathematica program software, solving the determination algebraic equations, we conclude the following cases:

Case 1

(7) a 1 = ω 2 λ k 2 − 6 λ 2 4 , a 5 = a 1 , a 3 = 0 , b 1 = b 0 2 + b 2 2 , ν = 4 λ 2 − k 2 δ + 1 ,

substituting from (6), (7), and (4). By taking γ 1 = ω 2 λ k 2 − 6 λ 2 4 and τ = 2 ω λ k 2 − 6 λ 2 4 , we obtain,

(8) u 11 ( x , y , z , t ) = τ b 0 sinh a 2 + γ ( k x + t ω + ν y + λ z ) + b 2 sinh a 6 + γ ( k x + t ω + ν y + λ z ) b 0 cosh a 2 + γ ( k x + t ω + ν y + λ z ) + b 2 cosh a 6 + γ ( k x + t ω + ν y + λ z ) + b 0 2 + b 2 2 cos a 4

Figure 1

$Figure 1 This figure for wave solutions (8) using parameters: k = 3 k=3 , λ = 1 \lambda =1 , δ = 5 \delta =5 , ω = 4 \omega =4 , a 2 = 1 {a}_{2}=1 , a 4 = 30 {a}_{4}=30 , a 6 = 2 {a}_{6}=2 , b 0 = 1 {b}_{0}=1 , b 2 = 1 {b}_{2}=1 , z = 1 z=1 , y = 3 5 y=\sqrt{\frac{3}{5}} . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.$

Figure 1

This figure for wave solutions (8) using parameters: k = 3 , λ = 1 , δ = 5 , ω = 4 , a 2 = 1 , a 4 = 30 , a 6 = 2 , b 0 = 1 , b 2 = 1 , z = 1 , y = 3 5 . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.

Case 2

(9) a 1 = 1 2 , a 3 = − i 2 , a 5 = 0 , ω = − λ 2 k 2 − 6 λ 2 , δ = − k 2 + 4 λ 2 − ν 2 ν 2 , b 0 = i b 1 ,

substituting from (6), (9), and (4), we have:

(10) u 12 ( x , y , z , t ) = 2 b 1 i sinh a 2 + k x + t ω + ν y + λ z 2 + sinh k x + t ω + ν y + λ z 2 + i a 4 b 2 cosh a 6 + b 1 i cosh a 2 + k x + t ω + ν y + λ z 2 + cosh k x + t ω + ν y + λ z 2 + i a 4

Figure 2

$Figure 2 This figure for wave solutions (10) using parameters: a 2 = 1 {a}_{2}=1 , a 4 = 1 {a}_{4}=1 , a 6 = 2 {a}_{6}=2 , b 2 = 2 2 {b}_{2}=2\sqrt{2} , b 1 = 2 2 {b}_{1}=2\sqrt{2} , y = 2 y=2 , k = 4 k=4 , ν = 2 \nu =2 . Wave solutions appear in (a) and (b) as three dimensions but (c) and (d) represented wave solutions in two dimensions and (e) and (f) a contour solutions.$

Figure 2

This figure for wave solutions (10) using parameters: a 2 = 1 , a 4 = 1 , a 6 = 2 , b 2 = 2 2 , b 1 = 2 2 , y = 2 , k = 4 , ν = 2 . Wave solutions appear in (a) and (b) as three dimensions but (c) and (d) represented wave solutions in two dimensions and (e) and (f) a contour solutions.

Case 3

(11) a 1 = 0 , a 3 = k δ λ 2 ν 2 + 2 λ 4 + λ 2 ν 2 , a 5 → i k δ λ 2 ν 2 + 2 λ 4 + λ 2 ν 2 , b 1 = i b 2 , ω = − 4 k 2 δ ν 2 + 2 λ 2 + ν 2

substituting from (6), (11), and (4). By taking γ 2 = k ( δ + 1 ) λ 2 ν 2 + 2 λ 4 , we obtain

(12) u 13 ( x , y , z , t ) 2 i b 2 γ 3 sin a 4 + γ 2 ( k x + t ω + ν y + λ z ) − i sin γ 2 ( k x + t ω + ν y + λ z ) − i a 6 b 0 cosh a 2 + b 2 i cos a 4 + γ 2 ( k x + t ω + ν y + λ z ) + cos γ 2 ( k x + t ω + ν y + λ z ) − i a 6

Figure 3

$Figure 3 This figure for wave solutions (12) using parameters: k = − 4 k=-4 , λ = 2 \lambda =2 , δ = 1 \delta =1 , a 2 = 2 {a}_{2}=2 , a 4 = 1 {a}_{4}=1 , a 6 = 2 {a}_{6}=2 , b 2 = 2 {b}_{2}=2 , z = 1 z=1 , y = 2 y=2 , ν = 2 \nu =2 , b 0 = 1 {b}_{0}=1 . (a) and (b) Ploted as wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.$

Figure 3

This figure for wave solutions (12) using parameters: k = − 4 , λ = 2 , δ = 1 , a 2 = 2 , a 4 = 1 , a 6 = 2 , b 2 = 2 , z = 1 , y = 2 , ν = 2 , b 0 = 1 . (a) and (b) Ploted as wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.

2.2 Type (II): Interactional phenomena and exponential form

The hypothesis of a double exponential function takes the form:

(13) f ( ξ ) = b 1 e a 1 ξ + a 2 + b 2 e a 3 ξ + a 4 ,

where the real constants a j ( j = 1 , … , 6 ) are determined later. By substituting (13) into (5) with the help of symbolic accounts having all the coefficients of all powers of exponential functions zero, we have a system of algebraic equations. By resolving this system, we get

(14) a 1 = 2 3 , a 3 = 2 3 , μ = − 6 3 ν 2 + 12 ν 2 − 11 3 + 22 ,

substitution from (13), (14) and (4), we obtain

(15) u 2 ( x , y , z , t ) = 2 2 e a 2 b 1 + e a 4 b 2 e 2 ( k x + t ω + ν y + λ z ) 3 3 e a 2 b 1 + e a 4 b 2 e 2 ( k x + t ω + ν y + λ z ) 3

Figure 4

$Figure 4 This figure for wave solutions (15) using parameters: a 1 = 2 3 {a}_{1}=\frac{2}{\sqrt{3}} , a 3 = 2 3 {a}_{3}=\frac{2}{\sqrt{3}} . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.$

Figure 4

This figure for wave solutions (15) using parameters: a 1 = 2 3 , a 3 = 2 3 . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.

2.3 Type (III): Homoclinic breather approach

The Homoclinic breather approach function takes the form:

(16) f ( ξ ) = e ξ 1 ( − p ) + b 0 cos ξ 3 p 1 + b 1 e ξ 2 p ,

where the real constants a j ( j = 1 , … , 6 ) are determined later. Substituting (16) into (4) and by resolving the algebraic equations, we obtain:

(17) a 5 → i 6 μ λ 2 p 1 , a 1 → 3 μ λ 2 p , a 3 → − 3 μ λ 2 p , ω → 9 k μ 2 λ 2 ,

substitution from (16) and (17) using (4). By taking γ 3 = μ λ 2 , the wave solution takes the form:

(18) u 3 ( x , y , z , t ) = 2 γ 3 6 b 0 e a 2 p + 3 γ 3 ( k x + t ω + ν y + λ z ) sinh 6 γ 3 ( k x + t ω + ν y + λ z ) − i a 6 p 1 − 3 b 1 e a 2 + a 4 p − 3 b 0 e a 2 p + 3 γ 3 ( k x + t ω + ν y + λ z ) cosh 6 γ 3 ( k x + t ω + ν y + λ z ) − i a 6 p 1 + b 1 e a 2 + a 4 p + 1

Figure 5

$Figure 5 This figure for wave solutions (18) using parameters: k = 1 k=1 , z = 1 z=1 , y = 1 y=1 , p 1 = 1 {p}_{1}=1 , p = 1 p=1 , λ = 1 \lambda =1 , μ = 1 3 \mu =\frac{1}{3} , ν = 1 \nu =1 , a 2 = 1 {a}_{2}=1 , a 4 = 1 {a}_{4}=1 , a 6 = − i {a}_{6}=-i , b 0 = 1 {b}_{0}=1 , b 1 = 1 {b}_{1}=1 . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.$

Figure 5

This figure for wave solutions (18) using parameters: k = 1 , z = 1 , y = 1 , p 1 = 1 , p = 1 , λ = 1 , μ = 1 3 , ν = 1 , a 2 = 1 , a 4 = 1 , a 6 = − i , b 0 = 1 , b 1 = 1 . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.

3 Conclusion

In this work, the multi-wave solution method has been used successfully to obtain solution the ( 3 + 1 ) -dimensions (Z–K) equation by helping of logarithmic transformation and symbolic computation. This methods is important from an applied and theoretical point of view, and an additional family of the solutions has been found, which is the best thing about this method. Novel solutions include the generalized solutions to ( 2 + 1 ) -dimensional (Z–K) equation and represent their graphs by determining the appropriate values of the parameters involved.

Acknowledgments

Taif University Researchers Supporting Project number (TURSP-2020/165), Taif University, Taif, Saudi Arabia.

Conflict of interest: Authors state no conflict of interest.

References

[1] Seadawy AR, El-Rashidy K. Traveling wave solutions for some coupled nonlinear evolution equations. Math Comput Model. 2013;57:1371–9. 10.1016/j.mcm.2012.11.026Search in Google Scholar

[2] Seadawy AR, El-Rashidy K. Water wave solutions of the coupled system Zakharov–Kuznetsov and generalized coupled KdV equations. Sci World J. 2014;2014:724759, 6 pages. 10.1155/2014/724759Search in Google Scholar PubMed PubMed Central

[3] Seadawy AR, El-Rashidy K. Classification of multiply travelling wave solutions for coupled burgers. Abstr Appl Anal. 2015;2015:369294, 7 pages. 10.1155/2015/369294Search in Google Scholar

[4] Seadawy AR, El-Rashidy K. Nonlinear Rayleigh Taylor in stability of the cylindrical fluid flow with mass and heat transfer. Pramana-J Phys. 2016;87:20. 10.1007/s12043-016-1222-xSearch in Google Scholar

[5] Seadawy AR, El-Rashidy K. Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in un-magnetized dust plasma. Results Phys. 2018;8:1216–22. 10.1016/j.rinp.2018.01.053Search in Google Scholar

[6] Seadawy AR, El-Rashidy K. Application of the extension exponential rational function method for higher-dimensional Broer Kaup Kupershmidt dynamical system. Mod Phys Lett A. 2019;34:14. 10.1142/S0217732319503450Search in Google Scholar

[7] El-Rashidy K. New traveling wave solutions for the higher Sharma-Tasso-Olver equation by using extension exponential rational function method. Results Phys. 2020;20:103066. 10.1016/j.rinp.2020.103066Search in Google Scholar

[8] Zakharov VE, Kuznetsov EA. On three-dimensional solitons. Soviet Phys. 1974;39:285–8. Search in Google Scholar

[9] Seadawy AR. Stability analysis for Zakharov–Kuznetsov equation of weaklynonlinear ion-acoustic waves in a plasma. Comput Math Appl. 2014;67:172–80. 10.1016/j.camwa.2013.11.001Search in Google Scholar

[10] Wazwaz AM. The extended tanh method for the Zakharov–Kuznetsov (Z–K) equation, the modified (Z–K) equation, and its generalized forms. Comm Nonlinear Sci Numer Simulat. 2008;13:1039–47. 10.1016/j.cnsns.2006.10.007Search in Google Scholar

[11] Hesam S, Nazemi A, Haghbin A. Analytical solution for the Zakharov–Kuznetsov equations by differential transform method. World Acad Sci. 2011;51:372–84.Search in Google Scholar

[12] Guner O, Aksoy E, Bekir A, Cevikel AC. Different methods for (3+1)-dimensional space-time fractional modified KdV-Zakharov–Kuznetsov equation. Comput Math Appl. 2016;71:1259–69. 10.1016/j.camwa.2016.02.004Search in Google Scholar

[13] Korkmaz A. Exact solutions (3+1) conformable time fractional jimbo-miwa, Zakharov–Kuznetsov and modified Zakharov–Kuznetsov equations. Commun Theor Phys. 2017;67:16665. 10.1088/0253-6102/67/5/479Search in Google Scholar

[14] Sahoo S, Saha Ray S. Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV–Zakharov–Kuznetsov equations. Comput Math Appl. 2015;70:158–66. 10.1016/j.camwa.2015.05.002Search in Google Scholar

[15] Rabczuk T, Ren H, Zhuang X. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Comput Mater Continua. 2019;59(1):31–55. 10.32604/cmc.2019.04567Search in Google Scholar

[16] Goswami A, Sushila, Singh J, Kumar D. Numerical computation of fractional Kersten-Krasilashchik coupled KdV-mKdV system arising in multi-component plasmas. AIMS Mathematics. 2020;5(3):2346–68. 10.3934/math.2020155Search in Google Scholar

[17] Goswami A, Singh J, Kumar D, Sushila S. An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A. 2019;524:563–75. 10.1016/j.physa.2019.04.058Search in Google Scholar

[18] Goswami A, Singh J, Kumar D. Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves. Ain Shams Eng J. 2018;9:2265–73. 10.1016/j.asej.2017.03.004Search in Google Scholar

[19] Kumar D, Singh J, Purohit SD, Swroop R. A hybrid analytical algorithm for nonlinear fractional wave-like equations. Math Model Nat Phenom. 2019;14:304. 10.1051/mmnp/2018063Search in Google Scholar

[20] Singh J, Jassim HK, Kumar D. An efficient computational technique for local fractional Fokker Planck equation. Physica A. 2020;555:124525. 10.1016/j.physa.2020.124525Search in Google Scholar

[21] Bhatter S, Mathur A, Kumar D, Singh J. A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory. Physica A. 2020;537:122578. 10.1016/j.physa.2019.122578Search in Google Scholar

[22] 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. 10.1016/j.amc.2014.01.066Search in Google Scholar

[23] Arshad M, Seadawy AR, Lu D. Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability. Eur Phys J Plus. 2017;132:371. 10.1140/epjp/i2017-11655-9Search in Google Scholar

[24] Arshad M, Seadawy AR, Lu D. Modulation stability and optical soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms and its applications. Superlattices Microstruct. 2017;112:422–34. 10.1016/j.spmi.2017.09.054Search in Google Scholar

[25] 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. 10.1142/S0217732318501833Search in Google Scholar

[26] Lu D, Seadawy AR, Ali A. Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications. Results Phys. 2018;9:313–20. 10.1016/j.rinp.2018.02.036Search in Google Scholar

[27] Lu D, Seadawy AR, Ali A. Applications of exact traveling wave solutions of Modified Liouville and the Symmetric Regularized Long Wave equations via two new techniques. Results Phys. 2018;9:1403–10. 10.1016/j.rinp.2018.04.039Search in Google Scholar

[28] Arshad M, Seadawy AR, Lu D. Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications. Open Phys. 2018;16:219–26. 10.1515/phys-2018-0032Search in Google Scholar

[29] Seadawy AR, Ahmed HM, Rabie WB, Biswas A. Chirp-free optical solitons in fiber Bragg gratings with dispersive reflectivity having polynomial law of nonlinearity. Optik. 2021;225:165681. 10.1016/j.ijleo.2020.165681Search in Google Scholar

[30] Özkan YS, Yaşar E, Seadawy AR. A third-order nonlinear Schrödinger equation: the exact solutions, group-invariant solutions and conservation laws. J Taibah Univ Sci. 2020;14(1):585–97. 10.1080/16583655.2020.1760513Search in Google Scholar

[31] El-Sheikh MMA, Seadawy AR, Ahmed HM, Arnous AH, Rabie WB. Dispersive and propagation of shallow water waves as a higher order nonlinear Boussinesq-like dynamical wave equations. Physica A: Stat Mech Appl. 2020;537:12266. 10.1016/j.physa.2019.122662Search in Google Scholar

[32] Iqbal M, Seadawy AR. Khalil OH, Lu D. Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear Nizhnik-Novikov-Vesselov dynamical equation. Results Phys. 2020;16:102838. 10.1016/j.rinp.2019.102838Search in Google Scholar

[33] Farah N, Seadawy AR, Ahmad S, Rizvi STR, Younis M. Interaction properties of soliton molecules and painleve analysis for nano bioelectronics transmission model. Opt Quant Electron. 2020;52:329, 1–15. 10.1007/s11082-020-02443-0Search in Google Scholar

[34] Seadawy AR. Three-dimensional nonlinear modified Zakharov–Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Comput Math Appl. 2016;71:201–12. 10.1016/j.camwa.2015.11.006Search in Google Scholar

[35] El-Rashidy K, Seadawy AR. Kinky breathers, multi-peak and multi-wave soliton solutions for the nonlinear propagation of Kundu Eckhaus dynamical model. Int J Mod Phys B. 2020;2050317. 10.1142/S0217979220503178Search in Google Scholar

[36] El-Rashidy K, Seadawy AR, Althobaiti S, Makhlouf MM. Multiwave, Kinky breathers and multi-peak soliton solutions for the nonlinear Hirota dynamical system. Results Phys 2020;19:103678. 10.1016/j.rinp.2020.103678Search in Google Scholar

[37] Goswami A, Singh J, Kumar D. A reliable algorithm for KdV equations arising in warm plasma. Nonlinear Engineering: Modeling and Application. 2016;5(1):7–16. 10.1515/nleng-2015-0024Search in Google Scholar

[38] Goswami A, J, Singh, D, Kumar, S, Gupta, Sushila. An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma. J Ocean Engg Sci. 2019;4:85–99. 10.1016/j.joes.2019.01.003Search in Google Scholar

Received: 2020-09-30

Revised: 2021-01-28

Accepted: 2021-02-09

Published Online: 2021-03-09

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

Articles in the same Issue

https://doi.org/10.1515/phys-2021-0009

Keywords for this article

02.30Xx soliton solutions; (3 + 1) Zakharov–Kuznetsov equation; the logarithmic transformation method

Creative Commons

BY 4.0