Structure of New Solitary Solutions for The Schwarzian Korteweg De Vries Equation And (2+1)-Ablowitz-Kaup-Newell-Segur Equation

  • Raghda A.M. Attia
  • Dianchen Lu
  • Mostafa M. A. Khater Jiangsu University
  • Omnia M. Abd elazeem
Keywords: Schwarzian Korteweg de Vries equation, (2 1)-Ablowitz-Kaup-Newell-Segur equation, Modified Khater method, ptical traveling wave solutions, Exact, solitary and approximate solutions


In this research, we introduce and represent the modified Khater method on two basic models in the optical fiber. These two models describe the dynamics of the wave movement in the optical fiber.  It is a new modification of new recent method which developed by Mostafa M. A. Khater in 2017. We implement this new modified technique on Schwarzian Korteweg de Vries equation and (2+1)-Ablowitz-Kaup-Newell-Segur equation. This modification of Khater method produces more closed solutions than many other methods. Schwarzian Korteweg de Vries (SKdV) equation has a closed relationship with (2+1)-Ablowitz- Kaup-Newell-Segur equation. Schwarzian Korteweg de Vries equation prescribes the location in a micro-segment of space and motion of the isolated waves in varied fields which localized in a tiny portion of space. It is a great and basic system in fluid mechanics, nonlinear optics, plasma physics, and quantum field theory.


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Yang, Xiao-Jun, Feng Gao, and Hari M. Srivastava. "Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations." Computers & Mathematics with Applications 73, no. 2 (2017): 203-210.

Seadawy, Aly R., Dianchen Lu, and Mostafa MA Khater. "Bifurcations of solitary wave solutions for the three dimensional Zakharov–Kuznetsov–Burgers equation and Boussinesq equation with dual dispersion." Optik-International Journal for Light and Electron Optics 143 (2017): 104-114.

Seadawy, Aly R. "Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron–positron plasma." Physica A: Statistical Mechanics and its Applications 455 (2016): 44-51.

Khater, Mostafa MA, Dianchen Lu, and Emad HM Zahran. "Solitary wave solutions of the Benjamin–Bona–Mahoney–Burgers equation with dual power-law nonlinearity." Appl. Math. Inf. Sci 11, no. 5 (2017): 1-5.

Cheemaa, Nadia, and Muhammad Younis. "New and more exact traveling wave solutions to integrable (2+ 1)-dimensional Maccari system." Nonlinear Dynamics 83, no. 3 (2016): 1395-1401.

Lu, Dianchen, Aly R. Seadawy, and Mostafa MA Khater. "Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods." Results in Physics 7 (2017): 2028-2035.

Wang, Haiyan, and Xiang-Sheng Wang. "Traveling wave phenomena in a Kermack–McKendrick SIR model." Journal of Dynamics and Differential Equations 28, no. 1 (2016): 143-166.

Seadawy, Aly R., Dianchen Lu, and Mostafa MA Khater. "Bifurcations of traveling wave solutions for Dodd–Bullough–Mikhailov equation and coupled Higgs equation and their applications." Chinese Journal of Physics 55, no. 4 (2017): 1310-1318.

Khater, Mostafa MA. "Exact traveling wave solutions for the generalized Hirota-Satsuma couple KdV system using the exp (− φ (ξ))-expansion method." Cogent Mathematics 3, no. 1 (2016): 1172397.

Khater, Mostafa MA, Aly R. Seadawy, and Dianchen Lu. "Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Time-fractional Cahn-Allen equation." Results in Physics 7 (2017): 2325-2333.

Darvishi, MTa, Sb Arbabi, Mb Najafi, and AMc Wazwaz. "Traveling wave solutions of a (2+ 1)-dimensional Zakharov-like equation by the first integral method and the tanh method." Optik-International Journal for Light and Electron Optics 127, no. 16 (2016): 6312-6321.

Seadawy, Aly R., Dianchen Lu, and Mostafa MA Khater. "New wave solutions for the fractional-order biological population model, time fractional burgers, Drinfel’d–Sokolov–Wilson and system of shallow water wave equations and their applications." European Journal of Computational Mechanics (2017): 1-17.

Zahran, Emad HM, and Mostafa MA Khater. "Modified extended tanh-function method and its applications to the Bogoyavlenskii equation." Applied Mathematical Modelling 40, no. 3 (2016): 1769-1775.

Khater, Mostafa MA, Aly R. Seadawy, and Dianchen Lu. "Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method." Superlattices and Microstructures (2017).

Cheemaa, Nadia, and Muhammad Younis. "New and more general traveling wave solutions for nonlinear Schrödinger equation." Waves in Random and Complex Media 26, no. 1 (2016): 30-41.

Khater, Mostafa MA, Aly R. Seadawy, and Dianchen Lu. "Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications." Journal of King Saud University-Science (2017).

Khater, Mostafa MA, Emad HM Zahran, and Maha SM Shehata. "Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system." Journal of the Egyptian Mathematical Society 25, no. 1 (2017): 8-12.

Khater, Mostafa MA, Aly R. Seadawy, and Dianchen Lu. "Optical soliton and rogue wave solutions of the ultra-short femto-second pulses in an optical fiber via two different methods and its applications." Optik-International Journal for Light and Electron Optics (2017).

Gao, Feng, Xiao-Jun Yang, and Hari Mohan Srivastava. "Exact-Traveling-Wave Solutions for Linear and Nonlinear Heat-Transfer Equations." Thermal Science (2017).

Khater, Mostafa MA, and Dipankar Kumar. "New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water." Journal of Ocean Engineering and Science 2, no. 3 (2017): 223-228.

Seadawy, Aly. "Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations." Applied Mathematics & Information Sciences 10, no. 1 (2016): 209.

Khater, Mostafa MA, and Dipankar Kumar. "Implementation of three reliable methods for finding the exact solutions of (2+ 1) dimensional generalized fractional evolution equations." Optical and Quantum Electronics 49, no. 12 (2017): 427.

Yang, Xiao-Jun, J. A. Tenreiro Machado, Dumitru Baleanu, and Carlo Cattani. "On exact traveling-wave solutions for local fractional Korteweg-de Vries equation." Chaos: An Interdisciplinary Journal of Nonlinear Science 26, no. 8 (2016): 084312.

Baskonus, Haci Mehmet, and Hasan Bulut. "Exponential prototype structures for (2+ 1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics." Waves in Random and Complex Media 26, no. 2 (2016): 189-196.

Yang, Xiao-Jun, JA Tenreiro Machado, and Dumitru Baleanu. "Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain." Fractals 25, no. 04 (2017): 1740006.

Krichever, Igor Moiseevich, and Sergei Petrovich Novikov. "Holomorphic bundles over algebraic curves and non-linear equations." Russian Mathematical Surveys 35, no. 6 (1980): 53.

Bruzón, M. S., M. L. Gandarias, Concepción Muriel, J. Ramirez, and F. R. Romero. "Traveling-Wave Solutions of the Schwarz-Korteweg-de Vries Equation in 2+ 1 Dimensions and the Ablowitz-Kaup-Newell-Segur Equation Through Symmetry Reductions." Theoretical and mathematical physics 137, no. 1 (2003): 1378-1389.

Toda, Kouichi, and Song-Ju Yu. "The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in (2+ 1) dimensions." Journal of Mathematical Physics 41, no. 7 (2000): 4747-4751.

Aslan, İsmail. "The Exp-function approach to the Schwarzian Korteweg–de Vries equation." Computers & Mathematics with Applications 59, no. 8 (2010): 2896-2900.

Xiao-Yan, Tang, and Hu Heng-Chun. "Characteristic manifold and Painlevé integrability: Fifth-order Schwarzian Korteweg-de Vries type equation." Chinese physics letters 19, no. 9 (2002): 1225.

Weiss, John. "The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative." Journal of Mathematical Physics 24, no. 6 (1983): 1405-1413.

Chun-Long, Zheng, and Zhang Jie-Fang. "General solution and fractal localized structures for the (2+ 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system." Chinese physics letters 19, no. 10 (2002): 1399.

Ishimori, Yuji. "A relationship between the Ablowitz-Kaup-Newell-Segur and Wadati-Konno-Ichikawa schemes of the inverse scattering method." Journal of the Physical Society of Japan 51, no. 9 (1982): 3036-3041.

Najafi, Mohammad, Maliheh Najafi, and M. T. Darvishi. "New Exact Solutions to the (2+ 1)-Dimensional Ablowitz-Kaup-Newell-Segur Equation: Modification of the Extended Homoclinic Test Approach." Chinese Physics Letters 29, no. 4 (2012): 040202.

Chun-Long, Zheng, Zhang Jie-Fang, Wu Feng-Min, Sheng Zheng-Mao, and Chen Li-Qun. "Solitons in a generalized (2+ 1)-dimensional Ablowitz-Kaup-Newell-Segur system." Chinese Physics 12, no. 5 (2003): 472.

How to Cite
Attia, R., Lu, D., Khater, M., & Abd elazeem, O. (2018). Structure of New Solitary Solutions for The Schwarzian Korteweg De Vries Equation And (2+1)-Ablowitz-Kaup-Newell-Segur Equation. To Physics Journal, 1(3), 234-254. Retrieved from
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