On the G'/G Expansion Method Applied to (2+1)-Dimensional Asymmetric-Nizhnik-Novikov-Veselov Equation
Abstract - 71


ANNV equation
Optical soliton solutions
G^\prime/G expansion method

How to Cite

Mabrouk, S., & Rashed, A. S. (2023). On the G’/G Expansion Method Applied to (2+1)-Dimensional Asymmetric-Nizhnik-Novikov-Veselov Equation. Journal of Advances in Applied & Computational Mathematics, 10, 39–49. https://doi.org/10.15377/2409-5761.2023.10.4


In this paper, the  G'/G expansion method is applied to the (2+1)-dimensional Asymmetric-Nizhnik-Novikov-Veselov equation (ANNV). The motivation is creating new families of solitary waves. The system of equations has been combined in one partial differential equation (PDE) and the traveling wave variable has been applied to transform the resultant equation into an ordinary differential equation (ODE). The homogenous balance condition has been applied to determine the truncation variable of the G'/G expansion. Four cases are created according to the appropriate choice of the arbitrary constants relations. For each case, some new solitary wave solutions including solitons and kinks represented by trigonometric, hyperbolic, logarithmic, polynomial, and combinations of these functions.



Columbu A, Frassu S, Viglialoro G. Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions. Appl Anal. 2023: 1-17. https://doi.org/10.1080/00036811.2023.2187789

Li T, Frassu S, Viglialoro G. Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption. Zeitschrift Für Angewandte Mathematik Und Physik. 2023; 74: Article number 109. https://doi.org/10.1007/s00033-023-01976-0

Li T, Pintus N, Viglialoro G. Properties of solutions to porous medium problems with different sources and boundary conditions. Zeitschrift Für Angewandte Mathematik Und Physik. 2019; 70: Article number 86. https://doi.org/10.1007/s00033-019-1130-2

Li T, Viglialoro G. Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differential and Integral Equations. 2021; 34: 315-36. https://doi.org/10.57262/die034-0506-315

Boiti M, Leon JJ-P, Manna M, Pempinelli F. On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Probl. 1986; 2: 271-9. https://doi.org/10.1088/0266-5611/2/3/005

Estévez PG, Leble SB. A kdv equation in 2+1 dimensions: Painlevé analysis, solutions, and similarity reductions. Acta Appl Math. 1995; 39: 277-94. https://doi.org/10.1007/BF00994637

Lou S-Y. (2+1)-Dimensional Integrable Models from the Constraints of the KP Equation. Commun Theor Phys. 1997; 27: 249-52. https://doi.org/10.1088/0253-6102/27/2/249

Lou S-Y, Hu X-B. Infinitely many Lax pairs and symmetry constraints of the KP equation. J Math Phys. 1997; 38: 6401-27. https://doi.org/10.1063/1.532219

Clarkson PA, Mansfield EL. On a shallow water wave equation. Nonlinearity. 1994; 7: 975-1000. https://doi.org/10.1088/0951-7715/7/3/012

Li-Hua Z, Xi-Qiang L, Cheng-Lin B. Symmetry, reductions and new exact solutions of ANNV equation through lax pair. Commun Theor Phys. 2008; 50: 1-6. https://doi.org/10.1088/0253-6102/50/1/01

Ma H-C, Lou S-Y. Finite symmetry transformation groups and exact solutions of lax integrable systems. Commun Theor Phys. 2005; 44: 193-6. https://doi.org/10.1088/6102/44/2/193

Lü Z-S. Special bi-solitons for asymmetric nizhnik-novikov-veselov equation. Commun Theor Phys. 2011; 55: 85-8. https://doi.org/10.1088/0253-6102/55/1/17

Ling W, Zhong-Zhou D, Xi-Qiang L. Symmetry reductions, exact solutions and conservation laws of asymmetric nizhnik-novikov-veselov equation. Commun Theor Phys. 2008; 49: 1-8. https://doi.org/10.1088/0253-6102/49/1/01

Yu G-F, Tam H-W. A vector asymmetrical nnv equation: Soliton solutions, bilinear bäcklund transformation and lax pair. J Math Anal Appl. 2008; 344: 593-600. https://doi.org/10.1016/j.jmaa.2008.02.057

Bai C-L, Zhao H. The study of soliton fission and fusion in (2+1)-dimensional nonlinear system. Eur Phys J D. 2006; 39: 93-9. https://doi.org/10.1140/epjd/e2006-00080-8

Hang-Yu R, Zhi-Fang L. Interaction between line soliton and algebraic soliton for asymmetric nizhnik novikov-veselov-equation. Commun Theor Phys. 2008; 49: 1547-52. https://doi.org/10.1088/0253-6102/49/6/41

Ruan H, Chen Y. Interaction between a line soliton and a y-periodic soliton in the (2+1)-dimensional nizhnik-novikov-veselov equation. Zeitschrift Für Naturforschung A. 2002; 57: 948-54. https://doi.org/10.1515/zna-2002-1207

Qian X, Lou S, Hu X. Variable separation approach for a differential-difference asymmetric nizhnik-novikov-veselov equation. Zeitschrift Für Naturforschung A. 2004; 59: 645-58. https://doi.org/10.1515/zna-2004-1005

Lou SY, Ruan HY. Revisitation of the localized excitations of the (2 + 1)-dimensional KdV equation. J Phys A Math Gen. 2001; 34: 305. https://doi.org/10.1088/0305-4470/34/2/307

Lin L. Quasi-periodic waves and asymptotic property for boiti-leon-manna-pempinelli equation. Commun Theor Phys. 2010; 54: 208-14. https://doi.org/10.1088/0253-6102/54/2/02

Li Y, Li D. New exact solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Appl Math Sci. 2012; 6(9-12): 579-87.

Luo L. New exact solutions and bäcklund transformation for boiti-leon-manna-pempinelli equation. Phys Lett A. 2011; 375: 1059-63. https://doi.org/10.1016/j.physleta.2011.01.009

Song-Hua M, Jian-Ping F. Multi dromion-solitoff and fractal excitations for (2+1)-dimensional boiti-leon-manna-pempinelli system. Commun Theor Phys. 2009; 52: 641-5. https://doi.org/10.1088/0253-6102/52/4/18

Rashed AS, Kassem MM. Hidden symmetries and exact solutions of integro-differential Jaulent–Miodek evolution equation. Appl Math Comput. 2014; 247: 1141-55. https://doi.org/10.1016/j.amc.2014.09.025

Kassem MM, Rashed AS. N-solitons and cuspon waves solutions of (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations via hidden symmetries of Lie optimal system. Chinese J Phys. 2019; 57: 90-104. https://doi.org/10.1016/j.cjph.2018.12.007

Mabrouk SM, Rashed AS. N-Solitons, kink and periodic wave solutions for (3 + 1)-dimensional Hirota bilinear equation using three distinct techniques. Chinese J Phys. 2019; 60: 48-60. https://doi.org/10.1016/j.cjph.2019.02.032

Rashed AS. Analysis of (3+1)-dimensional unsteady gas flow using optimal system of lie symmetries. Math Comput Simul. 2019; 156: 327-46. https://doi.org/10.1016/j.matcom.2018.08.008

Saleh R, Rashed AS. New exact solutions of (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili equation using a combination of lie symmetry and singular manifold methods. Math Methods Appl Sci. 2020; 43: 2045-55. https://doi.org/10.1002/mma.6031

Saleh R, Rashed AS, Wazwaz A-M. Plasma-waves evolution and propagation modeled by sixth order Ramani and coupled Ramani equations using symmetry methods. Phys Scr. 2021; 96: 085213. https://doi.org/10.1088/1402-4896/ac0075

Rashed AS. Interaction of two long waves in shallow water using Hirota-Satsuma model and similarity transformations. Delta University Sci J. 2022; 5: 93-101. https://doi.org/10.21608/dusj.2022.233925

Rashed AS, Mabrouk SM, Wazwaz A-M. Forward scattering for non-linear wave propagation in (3 + 1)-dimensional Jimbo-Miwa equation using singular manifold and group transformation methods. Waves Random Complex Media. 2022; 32: 663-75. https://doi.org/10.1080/17455030.2020.1795303

Rashed AS, Inc M, Saleh R. Extensive novel waves evolution of three-dimensional Yu–Toda–Sasa–Fukuyama equation compatible with plasma and electromagnetic applications. Mod Phys Lett B. 2023; 37: 22501950. https://doi.org/10.1142/S0217984922501950

Rashed AS, Mostafa ANM, Wazwaz AM, Mabrouk SM. Dynamical behavior and soliton solutions of the jumarie’s space-time fractional modified benjamin-bona-mahony equation in plasma physics. Rom Rep Phys. 2023; 75: Article no.104.

Mabrouk SM, Rashed AS. Analysis of (3 + 1)-dimensional boiti - leon -manna-pempinelli equation via lax pair investigation and group transformation method. Comput Math Appl. 2017; 74: 2546-56. https://doi.org/10.1016/j.camwa.2017.07.033

Shang Y. Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation. Appl Math Comput. 2007; 187: 1286-97. https://doi.org/10.1016/j.amc.2006.09.038

He Y, Tam H-W. Bilinear backlund transformation and lax pair for a coupled ramani equation. J Math Anal Appl. 2009; 357: 132-6. https://doi.org/10.1016/j.jmaa.2009.04.006

Cheng-Lin B. Extended homogeneous balance method and lax pairs, backlund transformation. Commun Theor Phys. 2002; 37: 645-8. https://doi.org/10.1088/0253-6102/37/6/645

Mabrouk S, Kassem M. Group similarity solutions of (2 + 1) boiti-leon-manna-pempinelli lax pair. Ain Shams Eng J. 2014; 5: 227-35. https://doi.org/10.1016/j.asej.2013.06.004

Mohamed NA, Rashed AS, Melaibari A, Sedighi HM, Eltaher MA. Effective numerical technique applied for Burgers’ equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms. Math Methods Appl Sci. 2021; 44: 10135-53. https://doi.org/10.1002/mma.7395

Arora G, Joshi V. A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers’ equation in one and two dimensions. Alex Eng J. 2018; 57: 1087-98. https://doi.org/10.1016/j.aej.2017.02.017

Biazar J, Aminikhah H. Exact and numerical solutions for non-linear Burger’s equation by VIM. Math Comput Model. 2009; 49: 1394-400. https://doi.org/10.1016/j.mcm.2008.12.006

Shi F, Zheng H, Cao Y, Li J, Zhao R. A fast numerical method for solving coupled burgers’ equations fast numerical method, Numer. Numer Methods Partial Differ Equ. 2017; 33: 1823-38. https://doi.org/10.1002/num.22160

Fang J, Nadeem M, Habib M, Akgül A. Numerical investigation of nonlinear shock wave equations with fractional order in propagating disturbance. Symmetry. 2022; 14(6), 1179. https://doi.org/10.3390/sym14061179

Kaya D, El-Sayed SM. A numerical method for solving jaulent-miodek equation. Phys Lett A. 2003; 318: 345-53. https://doi.org/10.1016/j.physleta.2003.08.033

Wazwaz A-M. The tanh and the sine–cosine methods for a reliable treatment of the modified equal width equation and its variants. Commun Nonlinear Sci Numer Simul. 2006; 11: 148-60. https://doi.org/10.1016/j.cnsns.2004.07.001

Wazwaz A-M. The sine–cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations. Appl Math Comput. 2006; 173: 150-64. https://doi.org/10.1016/j.amc.2005.02.047

Wazwaz A-M. Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method. Appl Math Comput. 2007; 190: 633-40. https://doi.org/10.1016/j.amc.2007.01.056

Wazwaz A-M. The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Appl Math Comput. 2007; 188: 1467-75. https://doi.org/10.1016/j.amc.2006.11.013

Wazwaz A-M. The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the pochhammer-chree equations. Appl Math Comput. 2008; 195: 24-33. https://doi.org/10.1016/j.amc.2007.04.066

Darvishi MT, Najafi M. A modification of extended homoclinic test approach to solve the (3+1)-dimensional potential-ytsf equation. Chin Phys Lett. 2011; 28: 040202. https://doi.org/10.1088/0256-307X/28/4/040202

Darvishi MT, Najafi M, Najafi M. Application of multiple exp-function method to obtain multi-soliton solutions of (2 + 1)- and (3 + 1)-dimensional breaking soliton equations. Am J Comput Appl Math. 2012; 1: 41-7. https://doi.org/10.5923/j.ajcam.20110102.08

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