A Study of Nonlinear Riccati Equation and Its Applications to Multi-dimensional Nonlinear Evolution Equations

• Lanre Akinyemi ^[4] ; Francis Erebholo ^[1] ; Valerio Palamara ^[2] ; Kayode Oluwasegun ^[3]
  1. [1] Virginia State University

     Virginia State University

     Estados Unidos

     
  2. [2] Hampton University

     Hampton University

     Estados Unidos

     
  3. [3] Drexel University

     Drexel University

     City of Philadelphia, Estados Unidos

     
  4. [4] Prairie View A&M University

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01137-2
• Enlaces
  • Texto completo
• Resumen

  • In this paper, an improved nonlinear Riccati equation method is established and then used to study nonlinear evolution equations with variable coefficients. The idea behind this current study is to use the nonlinear Riccati equation with constant coefficients and convert it to second-order linear ordinary differential equations using appropriate transformations. The solutions of these second-order linear ordinary differential equations are then used to propose the solutions to the nonlinear Riccati equation.

    This improved method consists of the well-known (G/G)-expansion and (1/G)- expansionmethods as a special case. It also consists of a newly proposed (in this current study) (G/G)-expansion method. As a result, this improved method is employed to investigate the traveling wave solutions of (2 + 1)-dimensional models with variable coefficients, such as the extended Kadomtsev–Petviashvili equation, nonlinear Schrödinger equation, and Davey–Stewartson equations. Solitary waves are obtained by taking the parameters involving these derived traveling wave solutions as special values. Furthermore, the constraint conditions for the existence of solutions to these variable coefficient equations are also revealed. This improved method has a wider range of applications for addressing nonlinear wave equations.

• Referencias bibliográficas
  • 1. Kudryashov, N.A.: On types of nonlinear nonintegrable equations with exact solutions. Phys. Lett. A 155(4–5), 269–275 (1991)
  • 2. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (1991)
  • 3. Arnous, A.H., Mirzazadeh,M., Akbulut, A., Akinyemi, L.: Optical solutions and conservation laws of the Chen-Lee-Liu equation with Kudryashov’s...
  • 4. AlQahtani, S.A., Shohib, R.M., Alngar, M.E., Alawwad, A.M.: High-stochastic solitons for the eighthorderNLSE through Itô calculus and STDwith...
  • 5. Durur, H., Ta¸sbozan, O., Kurt, A., ¸ Senol, M.: New wave solutions of time fractional Kadomtsev– Petviashvili equation arising in the...
  • 6. Akinyemi, L., Manukure, S., Houwe, A., Abbagari, S.: A study of (2 + 1)-dimensional variable coefficients equation: its oceanic solitons...
  • 7. Palamara, V., Neal, B., Akinyemi, L., Erebholo, F., Bogale, M.: Shallow-water waves through two new generalized multi-dimensional variable...
  • 8. Zhou, Q., Triki, H., Xu, J., Zeng, Z., Liu,W., Biswas, A.: Perturbation of chirped localized waves in a dual-power law nonlinear medium....
  • 9. Wu, J., Yang, Z.: Global existence and boundedness of chemotaxis-fluid equations to the coupled Solow–Swan model. AIMS Math. 8(8), 17914–17942...
  • 10. Li, Z., Liu, C.: Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized...
  • 11. Alquran, M., Al-deiakeh, R.: Lie-Backlund symmetry generators and a variety of novel periodicsoliton solutions to the complex-mode of...
  • 12. Ilhan, O.A., Manafian, J., Lakestani, M., Singh, G.: Some novel optical solutions to the perturbed nonlinear Schrödinger model arising...
  • 13. Liu, C., Li, Z.: The dynamical behavior analysis of the fractional perturbed Gerdjikov–Ivanov equation. Res. Phys. 59, 107537 (2024)
  • 14. Alquran,M.: Dynamic behavior of explicit elliptic and quasi periodic-wave solutions to the generalized (2 + 1)-dimensional Kundu–Mukherjee–Naskar...
  • 15. Ilhan, O.A., Manafian, J.: Periodic type and periodic cross-kink wave solutions to the (2 + 1)- dimensional breaking soliton equation...
  • 16. Alquran,M.: Necessary conditions for convex-periodic, elliptic-periodic, inclined-periodic, and rogue wave-solutions to exist for the...
  • 17. Zhao, N., Manafian, J., Ilhan, O.A., Singh, G., Zulfugarova, R.: Abundant interaction between lump and k-kink, periodic and other analytical...
  • 18. Li, Z., Hussain, E.: Qualitative analysis and optical solitons for the (1+1)-dimensionalBiswas–Milovic equation with parabolic law...
  • 19. AlQahtani, S.A., Alngar, M.E., Shohib, R.M., Pathak, P.: Highly dispersive embedded solitons with quadratic χ(2) and cubic χ(3) non-linear...
  • 20. Wang, K.J., Wang, G.D., Shi, F.: Nonlinear dynamics of soliton molecules, hybrid interactions and other wave solutions for the (3 +...
  • 21. Zhou, Q.: Influence of parameters of optical fibers on optical soliton interactions. Chin. Phys. Lett. 39(1), 010501 (2022)
  • 22. Li, C., Manafian, J., Eslami, B.,Mahmoud, K.H., Abass, R.R., Bashar, B.S., Ilhan, O.A.: A generalized trial equation scheme: a tool for...
  • 23. Arnous, A.H., Nofal, T.A., Biswas, A., Yıldırım, Y., Asiri, A.: Cubic-quartic optical solitons of the complex Ginzburg–Landau equation:...
  • 24. Samir, I., Nofal, T.A., Arnous, A.H., Eldidamony, H.A.: Traveling wave solutions for the Radhakrishnan–Kundu–Lakshmanan equation with...
  • 25. Hashemi, M.S.: A variable coefficient third degree generalized Abel equation method for solving stochastic Schrödinger–Hirota model. Chaos...
  • 26. Wang, M., Zhou, Y., Li, Z.: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics....
  • 27. Conte, R.,Musette, M.: Link between solitary waves and projective Riccati equations. J. Phys. AMath. Gen. 25(21), 5609 (1992)
  • 28. Vahidi, J., Zekavatmand, S.M., Rezazadeh, H., Inc, M., Akinlar, M.A., Chu, Y.M.: New solitary wave solutions to the coupled Maccari’s...
  • 29. Biswas, A., Yildirim, Y., Yasar, E., Triki, H., Alshomrani, A.S., Ullah,M.Z., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton perturbation...
  • 30. Wang, Y.Y., Zhang, Y.P., Dai, C.Q.: Re-study on localized structures based on variable separation solutions from the modified tanh-function...
  • 31. Arnous, A.H., Mirzazadeh, M., Hashemi, M.S., Shah, N.A., Chung, J.D.: Three different integration schemes for finding soliton solutions...
  • 32. Ghanbari, B., Inc, M., Rada, L.: Solitary wave solutions to the Tzitzeica type equations obtained by a new efficient approach. J. Appl....
  • 33. Akpan, U., Akinyemi, L., Ntiamoah, D., Houwe, A., Abbagari, S.: Generalized stochastic Korteweg-de Vries equations, their Painlevè integrability,...
  • 34. Inc, M., Rezazadeh, H., Vahidi, J., Eslami, M., Akinlar, M.A., Ali, M.N., Chu, Y.M.: New solitary wave solutions for the conformable Klein-Gordon...
  • 35. Rezazadeh, H.: New solitons solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity. Optik 167, 218–227 (2018)
  • 36. Wang, M., Li, X., Zhang, J.: The (G /G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical...
  • 37. Yoku¸s, A.: Comparison of Caputo and conformable derivatives for time-fractional Korteweg-de Vries equation via the finite difference...
  • 38. Alquran, M.: Derivation of some bi-wave solutions for a new two-mode version of the combined Schamel and KdV equations. Partial Differ....
  • 39. Wang, K.J., Shi, F., Xu, P.: Multiple soliton, soliton molecules and the other diverse wave solutions to the (2 + 1)-dimensional Kadomtsev-Petviashvili...
  • 40. Zhou, Q., Huang, Z., Sun, Y., Triki, H., Liu,W., Biswas, A.: Collision dynamics of three-solitons in an optical communication system with...
  • 41. Zhou, Q., Xu,M., Sun, Y., Zhong, Y.,Mirzazadeh,M.: Generation and transformation of dark solitons, anti-dark solitons and dark double-hump...
  • 42. Wang, K., Wang, G.D., Shi, F.: Sub-picosecond pulses in single-mode optical fibres with the Kaup– Newell model via two innovative methods....
  • 43. Ding, C.C., Zhou, Q., Xu, S.L., Sun,Y.Z., Liu,W.J., Mihalache, D., Malomed, B.A.: Controlled nonautonomous matter-wave solitons in spinor...
  • 44. Akinyemi, L., ¸ Senol, M., Akpan, U., Oluwasegun, K.: The optical soliton solutions of generalized coupled nonlinear Schrödinger-Korteweg-de...
  • 45. Akinyemi, L., Veeresha, P., Darvishi,M.T., Rezazadeh, H., ¸ Senol,M., Akpan, U.: A novel approach to study generalized coupled cubic Schrödinger-Korteweg-de...
  • 46. Bilal, M., Haris, H., Waheed, A., Faheem, M.: The analysis of exact solitons solutions in monomode optical fibers to the generalized nonlinear...
  • 47. AlQahtani, S.A., Al-Rakhami,M.S., Shohib, R.M.,Alngar, M.E., Pathak, P.: Dispersive optical solitons with Schrödinger–Hirota equation...
  • 48. Arnous, A.H., Biswas, A., Kara, A.H., Yıldırım,Y., Moraru, L., Iticescu, C., Moldovanu, S., Alghamdi, A.A.: Optical solitons and conservation...
  • 49. Mirzazadeh, M., Hashemi, M.S., Akbulu, A., Ur Rehman, H., Iqbal, I., Eslami, M.: Dynamics of optical solitons in the extended (3+1)-dimensional...
  • 50. Khalifa, A.S., Rabie,W.B., Badra, N.M., Ahmed, H.M., Mirzazadeh,M., Hashemi, M.S., Bayram,M.: Discovering novel optical solitons of two...
  • 51. Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 24(5),...
  • 52. Kudryashov, N.A.: A note on the G /G-expansion method. Appl. Math. Comput. 217(4), 1755–1758 (2010)
  • 53. Sugai, I.: Riccati’s nonlinear differential equation. Am. Math. Mon. 67(2), 134–139 (1960)
  • 54. Haaheim, D.R., Stein, F.M.: Methods of solution of the Riccati differential equation. Math.Mag. 42(5), 233–240 (1969)
  • 55. Xie, F., Zhang, Y., Lü, Z.: Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili...
  • 56. Akinyemi, L., ¸ Senol,M., Rezazadeh, H., Ahmad, H.,Wang, H.: Abundant optical soliton solutions for an integrable (2 + 1)-dimensional...
  • 57. Xiao-Ping, L., Chun-Ping, L.: Relationship among solutions of a generalized Riccati equation. Commun. Theor. Phys. 48(4), 610 (2007)
  • 58. Bekir, A., Guner, O.: Exact solutions of nonlinear fractional differential equations by G /G-expansion method. Chin. Phys. B 22(11), 1–6...
  • 59. Zhang, S., Tong, J.L., Wang, W.: A generalized (G /G)-expansion method for the mKdV equation with variable coefficients. Phys. Lett. A...
  • 60. Bekhouche, F., Alquran, M., Komashynska, I.: Explicit rational solutions for time-space fractional nonlinear equation describing the propagation...
  • 61. Zhang, S., Manafian, J., Ilhan, O.A., Jalil, A.T., Yasin, Y., Abdulfadhil Gatea, M.: Nonparaxial pulse propagation to the cubic-quintic...
  • 62. Zaitsev, V.F., Polyanin, A.D.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press (2002)
  • 63. Hille, E.:OrdinaryDifferential Equations in theComplex Domain.Dover Publications,Mineola (1997)
  • 64. Malfliet,W.: Solitary wave solutions of nonlinear wave equations. Am. J. of Phys. 60, 650–654 (1992)
  • 65. Wazwaz, A.M.: Bright and dark optical solitons for (2+1)-dimensional Schrödinger (NLS) equations in the anomalous dispersion regimes...
  • 66. Akinyemi, L., ¸ Senol,M., Osman, M.S.: Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in...
  • 67. Davey, A., Stewartson, K.: On three-dimensional packets of surface waves. Proc. R. Soc. Lond. Math. Phys. Sci. 338(1613), 101–110 (1974)
  • 68. Baldwin, D.E.: Nonlinear Waves. https://douglasbaldwin.com/nl-waves.html