Solutions of Kudryashov - Sinelshchikov equation and generalized Radhakrishnan-Kundu-Lakshmanan equation by the first integral method

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    This paper shows the applicability of the First Integral Method in obtaining solutions of Nonlinear Partial Differential Equations (NLPDEs). The method is applied in constructing solutions of Kudryashov-Sinelshchikov equation (KSE) and Generalized Radhakrishnan-Kundu-Lakshmanan Equation (GRKLE). The First Integral Method, which is based on the Ring Theory of Commutative Algebra, is a direct algebraic method for obtaining exact solutions of NLPDEs. This method is applicable to integrable as well as nonintegrable NLPDEs. The method is an efficient method for obtaining exact solutions of many Nonlinear Evolution Equations (NLEEs).


  • Keywords


    Division Theorem; First Integral Method; Generalized Radhakrishnan - Kundu- Lakshmanan Equation (GRKLE); Kudryashov – Sinelshchikov Equation (KSE); NLEEs; Optical Solitons.

  • References


      [1] Feng, Z.S., The first integral method to study the Burgers Korteweg-de Vries equation, J. Phys. A, 2002, 35 (2), 343 – 349. http://dx.doi.org/10.1088/0305-4470/35/2/312.

      [2] Feng, Z. S.,On explicit exact solutions to the compound Burg ers- Korteweg-de Vries equation, Phys. Lett. A. 2002, 293, 57 – 66. http://dx.doi.org/10.1016/S0375-9601(01)00825-8.

      [3] Feng, Z. S., Travelling wave behavior for a generalized Fisher equation, Chaos, Soliton.Fract. 2008, 38, 481 – 488. http://dx.doi.org/10.1016/j.chaos.2006.11.031.

      [4] Feng, Z. S., Exact solution to an approximate sine- Gordon equation in (n + 1) – dimensional space, Phys. Lett. A, 2002, 302, 64 – 76. http://dx.doi.org/10.1016/S0375-9601(02)01114-3.

      [5] Feng, Z. S. and Wang, X. H., The first integral method to the two-dimensional Burgers- KdV equation, Phys. Lett. A 2002, 308, 173 – 178. http://dx.doi.org/10.1016/S0375-9601(03)00016-1.

      [6] Feng, Z. S. and Knobel, R., Travelling waves to a Burgers- Korteweg – de Vries equation with higher order nonlineari ties, J. Math. Anal. Appl. 2007, 328 (2), 1435 – 1450. http://dx.doi.org/10.1016/j.jmaa.2006.05.085.

      [7] Raslan, R. K., The first integral method for solving some im portant nonlinear partial differential equations, Nonlinear Dynam. 2008, 53 (4), 281 – 286. http://dx.doi.org/10.1007/s11071-007-9262-x.

      [8] Taghizadeh, N. Mirzazadeh, M. and Farahrooz, F., Exact solu tions of the nonlinear Schrodinger equation by the first inte gral method, J. Math. Anal. Appl. 2011, 374, 549 – 553. http://dx.doi.org/10.1016/j.jmaa.2010.08.050.

      [9] Abbasbandy, S. and Shirzadi, A., The first integral method for modified Benjamin- Bona-Mahony equation, Commun. Nonlinear Sci. Numer.Simul. 2010, 15 (7), 1759 – 1764. http://dx.doi.org/10.1016/j.cnsns.2009.08.003.

      [10] Jafari, H. Sooraki, A. Telabi Y. and Biswas, A. , The first inte gral method and traveling wave solutions to Davey- Stewartson equation, Nonlinear Anal., Model. Control 2012, 17 (2) 182 -193.

      [11] Taghizadeh, N., Mirzazadeh M. and Paghaleh, A. S., The First Integral Method to Nonlinear Partial differential Equations, Appl. Appl. Math. 2012, 7 (1), 117 – 132.

      [12] Taghizadeh, N. Mirzazadeh, M. and Paghaleh, A. S., Exact solu tions for the nonlinear Schrodinger equation with power law nonlinearity, Math. Sci. Lett. 2012, 1 (1), 7 – 15. http://dx.doi.org/10.12785/msl/010102.

      [13] El-Sabbagh, M. F. and El- Ganaini, S.I.,The First Integral Method and its Applications to Nonlinear Equations, Appl.Math. Sci., 2012, 6(78), 3893 – 3906.

      [14] El- Dabe, N. T. M., Moussa, M. H. M., El- Shiekh R. M. and Hamdy, H. A., New Solutions for the Higher-Order Nonlinear Schrodinger Equation Using Integral Methods, American Jour nal of Computational and Applied Mathemattics, 2012,2 (2), 25 – 28.

      [15] El-Sabbagh , M. F. and El- Ganaini,, S.I., New Exact Solutions of Broer- Kaup (BK) and Whitham – Broer - Kaup (BWK) systems via the first integral method, Int. Journal of Math. Analysis 2012, 61(46), 2287 – 2298.

      [16] Jafari, , H. Soltani, , R. Khalique ,C. M. and Baleanu, D., Exact solutions of two nonlinear partial differential equations by using the first integral method, Boundary Value Problems, 2013: 117. http://dx.doi.org/10.1186/1687-2770-2013-117.

      [17] N. Taghizadeh, M. Mirzazadeh, Filiz Tascan, The first integral method applied to the Eckhaus equation, Appl. Math. Lett. 2012, 25,798 -802. http://dx.doi.org/10.1016/j.aml.2011.10.021.

      [18] Taghizadeh, N., Mirzazadeh, M., Farahrooz F., Exact soliton solutions of the modified KdV – KP equation and the Burgers- KP equation by using the first integral method, Appl. Math. Model. 2011, 35(8), 3991-3997. http://dx.doi.org/10.1016/j.apm.2011.02.001.

      [19] Kudryashov, N. A. and Sinelshchikov, D. I. Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer, Phys. Lett. A 2010, 374, 2011 – 2016. http://dx.doi.org/10.1016/j.physleta.2010.02.067.

      [20] Kudryashov, N. A. and Sinelshchikov, D. I., Nonlinear evolu tion equations for describing waves in bubbly liquids with viscosity and heat transfer consideration, Appl. Math. Comput. 2010, 217, 414 – 421. http://dx.doi.org/10.1016/j.amc.2010.05.033.

      [21] Kudryashov, N. A. and Sinelshchikov, D. I., Nonlinear waves in liquids with gas bubbles with account of viscosity and heat transfer, Fluid Dynam. 2010, 45, 96 – 112. http://dx.doi.org/10.1134/S0015462810010114.

      [22] Mohammad Mirzazadeh, Mostafa Eslami, Exact solutions of the Kudryashov – Sinelshchikov equation and Nonlinear Tele graph equation via the first integral method, Nonlinear Anal. Model. 2012, 17(4), 481 – 488.

      [23] Sturdevant, b. lott d.a. and biswas, a., topological 1- soliton solution of the generalized radha krishnan kundu lakshmanan equation with nonlinear dispersion, modern physics let ters b, 2010, 24 (16), 1825 – 1831. http://dx.doi.org/10.1142/S0217984910024109.

      [24] Anjan Biswas, (). 1 – Soliton solution of the generalized Radha krishnan, Kundu, Lakshmanan equation,Phys. Lett. A 2009, 373 (30), 2546 – 2548. http://dx.doi.org/10.1016/j.physleta.2009.05.010.

      [25] Anjan Biswas, Kaiser R. Khan, Mohammad F. Mahmood & Milivoj Belic,. Bright and Dark solitons in optical metamaterials, Optik 2014, 125(13), 3299 – 3302. http://dx.doi.org/10.1016/j.ijleo.2013.12.061.


 

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Article ID: 6202
 
DOI: 10.14419/ijpr.v4i2.6202




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