Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation

  • Run-Fa Zhang
  • Sudao BiligeEmail author
Original Paper


A new method named bilinear neural network is introduced in this paper, and the corresponding tensor formula is proposed to obtain the exact analytical solutions of nonlinear partial differential equations (PDEs). This is the first time that the neural network model is used to find the exact analytical solution, and this method covers almost all methods of constructing a function after bilinearization to solve nonlinear PDEs. Furthermore, this method is most likely a universal method to obtain the exact analytical solutions of nonlinear PDEs. Abundant arbitrary functions solutions of the reduced p-gBKP equation are obtained by using this method. Various beautiful plots of the presented solutions, which show diversity of exact solutions to PDEs, are made. By choosing appropriate values and functions, the fractal solitons waves are obtained and the self-similar characteristics of these waves are observed by reducing the observation range and magnifying local images. Via various three-dimensional plots, the evolution characteristics of these waves are exhibited.


Bilinear neural network method Universal tensor formula Exact analytical solution Fractal soliton waves 



This work is supported by the National Natural Science Foundation of China (11661060) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (2018LH01013).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


  1. 1.
    Liu, X.Y., Triki, H., Zhou, Q., Liu, W.J., Biswas, A.: Analytic study on interactions between periodic solitons with controllable parameters. Nonlinear Dyn. 94, 703–709 (2018)Google Scholar
  2. 2.
    Zhang, Y., Liu, Y.P., Tang, X.Y.: M-lump and interactive solutions to a (3+1)-dimensional nonlinear system. Nonlinear Dyn. 93, 2533–2541 (2018)zbMATHGoogle Scholar
  3. 3.
    Zhang, Y.J., Yang, C.Y., Yu, W.T., Mirzazadeh, M., Zhou, Q., Liu, W.J.: Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dyn. 94, 1351–1360 (2018)Google Scholar
  4. 4.
    Ankiewicz, A., Akhmediev, N.: Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions. Nonlinear Dyn. 91, 1931–1938 (2018)Google Scholar
  5. 5.
    Carboni, B., Lacarbonara, W.: Nonlinear dynamic characterization of a new hysteretic device: experiments and computations. Nonlinear Dyn. 83, 23–39 (2016)Google Scholar
  6. 6.
    Tang, Y.N., Tao, S.Q., Guan, Q.: Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Nonlinear Dyn. 89, 429–442 (2017)Google Scholar
  7. 7.
    Arena, A., Lacarbonara, W.: Nonlinear parametric modeling of suspension bridges under aeroelastic forces: torsional divergence and flutter. Nonlinear Dyn. 70, 2487–2510 (2012)MathSciNetGoogle Scholar
  8. 8.
    Zhang, R.F., Bilige, S.D., Bai, Y.X., Lü, J.Q., Gao, X.Q.: Interaction phenomenon to dimensionally reduced p-gBKP equation. Mod. Phys. Lett. B. 32(6), 1850074 (2018)MathSciNetGoogle Scholar
  9. 9.
    Lü, J.Q., Bilige, S.D., Chaolu, T.: The study of lump solution and interaction phenomenon to (2+1)-dimensional generalized fifth-order kdv equation. Nonlinear Dyn. 91(2), 1669–1676 (2018)Google Scholar
  10. 10.
    Lü, J.Q., Bilige, S.D., Gao, X.Q., Bai, Y.X., Zhang, R.F.: Abundant lump solutions and interaction phenomena to the Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. J. Appl. Math. Phys. 6, 1733–1747 (2018)Google Scholar
  11. 11.
    Lv, J.Q., Bilige, S.D.: Lump solutions of a (2+1)-dimensional bSK equation. Nonlinear Dyn. 90, 2119–2124 (2017)MathSciNetGoogle Scholar
  12. 12.
    Lü, J.Q., Bilige, S.D.: Diversity of interaction solutions to the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation. Mod. Phys. Lett. B. 13, 1850311 (2018)MathSciNetGoogle Scholar
  13. 13.
    Wang, X.M., Bilige, S.D., Bai, Y.X.: A general sub-equation method to the burgers-like equation. Therm. Sci. 21(4), 1681–1687 (2017)Google Scholar
  14. 14.
    Lü, J.Q., Bilige, S.D.: The study of lump solution and interaction phenomenon to (2+1)-dimensional potential Kadomstev-Petviashvili equation. Math. Phys. Anal. (2018).
  15. 15.
    Liu, J.G., Du, J.Q., Zeng, Z.F., Nie, B.: New three-wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 88(1), 655–661 (2017)MathSciNetGoogle Scholar
  16. 16.
    Liu, J.G.: Lump-type solutions and interaction solutions for the (2+1)-dimensional generalized fifth-order KdV equation. Appl. Math. Lett. 86, 36–41 (2018)MathSciNetGoogle Scholar
  17. 17.
    Li, Y., Liu, J.G.: New periodic solitary wave solutions for the new (2+1)-dimensional Korteweg–de Vries equation. Nonlinear Dyn. 91(1), 497–504 (2018)MathSciNetGoogle Scholar
  18. 18.
    Lü, Z.S., Chen, Y.N.: Construction of rogue wave and lump solutions for nonlinear evolution equations. Eur. Phys. J. B 88(7), 88–187 (2015)MathSciNetGoogle Scholar
  19. 19.
    Lü, Z.S., Chen, Y.N.: Constructing rogue wave prototypes of nonlinear evolution equations via an extended tanh method. Chaos Solitons Fractals 81, 218–223 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhao, Z.L., Chen, Y., Han, B.: Lump soliton, mixed lump stripe and periodic lump solutions of a (2+1)-dimensional asymmetrical Nizhnik Novikov Veselov equation. Mod. Phys. Lett. B 31(14), 1750157 (2017)MathSciNetGoogle Scholar
  21. 21.
    Zhang, X.E., Chen, Y.: Deformation rogue wave to the (2+1)-dimensional KdV equation. Nonlinear Dyn. 90(2), 755–763 (2017)MathSciNetGoogle Scholar
  22. 22.
    Zhang, X.E., Chen, Y.: Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo–Miwa equation. Commun. Nonlinear Sci. Numer. Simul. 52, 24–31 (2017)MathSciNetGoogle Scholar
  23. 23.
    Xu, T., Chen, Y.: Mixed interactions of localized waves in the three-component coupled derivative nonlinear Schr\(\ddot{o}\)dinger equations. Nonlinear Dyn. 92, 2133–2142 (2018)Google Scholar
  24. 24.
    Yang, B., Chen, Y.: Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations. Nonlinear Dyn. 94, 489–502 (2018)Google Scholar
  25. 25.
    Zhang, X.E., Chen, Y.: General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis. Nonlinear Dyn. 93, 2169–2184 (2018)zbMATHGoogle Scholar
  26. 26.
    Wazwaz, A.M.: Two-mode fifth-order kdv equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. 87(3), 1685–1691 (2017)MathSciNetGoogle Scholar
  27. 27.
    Osman, M.S., Wazwaz, A.M.: An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-dimensional kdv equation with variable coefficients. Appl. Math. Comput. 321, 282–289 (2018)MathSciNetGoogle Scholar
  28. 28.
    Wazwaz, A.M.: Compact and noncompact physical structures for the ZK–BBM equation. Appl. Math. Comput. 169(1), 713–725 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sun, Y., Tian, B., Xie, X.Y., Chai, J., Yin, H.H.: Rogue waves and lump solitons for a-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics. Waves Random Complex Media 28(3), 544–552 (2018)MathSciNetGoogle Scholar
  30. 30.
    Dong, M.J., Tian, S.F., Wang, X.B., Zhang, T.T.: Lump-type solutions and interaction solutions in the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation. Anal. Math. Phys. (2018).
  31. 31.
    Yong, X.L., Li, X.J., Huang, Y.H.: General lump-type solutions of the (3+1)-dimensional Jimbo–Miwa equation. Appl. Math. Lett. 86, 222–228 (2018)MathSciNetGoogle Scholar
  32. 32.
    Jia, S.L., Gao, Y.T., Hu, L., Huang, Q.M., Hu, W.Q.: Soliton-like periodic wave and rational solutions for a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation in the incompressible fluid. Superlattices Microstruct. 102, 273–283 (2017)Google Scholar
  33. 33.
    Zhang, Y., Dong, H.H., Zhang, X.E., Yang, H.W.: Rational solutions and lump solutions to the generalized (3+1)-dimensional shallow water-like equation. Comput. Math. Appl. 73, 246–252 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ma, W.X., Qin, Z.Y., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Yang, J.Y., Ma, W.X., Qin, Z.Y.: Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation. Anal. Math. Phys. 8, 427–436 (2018)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Ma, W.X., Yong, X., Zhang, H.Q.: Diversity of interaction solutions to the (2+1)-dimensional Ito equation. Comput. Math. Appl. 75, 289–295 (2018)MathSciNetGoogle Scholar
  39. 39.
    Ma, W.X.: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2(4), 140–144 (2011)Google Scholar
  40. 40.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379(36), 1975–1978 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principal. J. Phys. 411, 12021 (2013)Google Scholar
  42. 42.
    Ma, W.X., Yong, X., Zhang, H.Q.: Diversity of interaction solutions to the (2+1)-dimensional ito equation. Comput. Math. Appl. 75(1), 289–295 (2018)MathSciNetGoogle Scholar
  43. 43.
    Zhang, H.Q., Ma, W.X.: Lump solutions to the (2+1)-dimensional Sawada–Kotera equation. Nonlinear Dyn. 87(4), 2305–2310 (2017)MathSciNetGoogle Scholar
  44. 44.
    Ma, W.X.: Lump-type solutions to the (3+1)-dimensional Jimbo–Miwa equation. Int. J. Nonlinear Sci. Numer. 17, 355–359 (2016)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Lü, X., Ma, W.X., Khalique, C.M.: A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model. Appl. Math. Lett. 50, 37–42 (2015)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Lü, X., Ma, W.X., Yu, J., Khalique, C.M.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 31, 40–46 (2016)MathSciNetGoogle Scholar
  47. 47.
    Lü, X., Ma, W.X., Yu, J., Lin, Fh, Khalique, C.M.: Envelope bright-soliton and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)MathSciNetGoogle Scholar
  48. 48.
    Lü, X., Ma, W.X., Zhou, Y., Khalique, C.M.: Rational solutions to an extended Kadomtsev–Petviashvili-like equation with symbolic computation. Comput. Math. Appl. 71, 1560–1567 (2016)MathSciNetGoogle Scholar
  49. 49.
    Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Lü, X., Ma, W.X., Chen, S.T., Chaudry, M.K.: A note on rational solutions to a Hirota–Satsuma-like equation. Appl. Math. Lett. 58, 13–18 (2016)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 86, 523–534 (2016)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Gao, L.N., Zhao, X.Y., Zi, Y.Y., Lü, X.: Resonant behavior of multiple wave solutions to a Hirota bilinear equation. Comput. Math. Appl. 72, 1225–1229 (2016)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Gao, L.N., Zi, Y.Y., Yin, Y.H., Ma, W.X., Lü, x: Bäcklund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 89, 2233–2240 (2017)Google Scholar
  54. 54.
    Lü, X., Lin, F.H.: Soliton excitations and shape-changing collisions in alphahelical proteins with interspine coupling at higher order. Commun. Nonlinear Sci. 32, 241–261 (2016)MathSciNetGoogle Scholar
  55. 55.
    Lin, F.H., Chen, S.T., Qu, Q.X., Wang, J.P., Zhou, X.W., Lü, X.: Resonant multiple wave solutions to a new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation: linear superposition principle. Appl. Math. Lett. 78, 112–117 (2018)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Lü, X., Wang, J.P., Lin, F.H., Zhou, X.W.: Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water. Nonlinear Dyn. 91(2), 1249–1259 (2018)Google Scholar
  57. 57.
    Yin, Y.H., Ma, W.X., Liu, J.G., Lü, X.: Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction. Comput. Math. Appl. 76, 1275–1283 (2018)MathSciNetGoogle Scholar
  58. 58.
    Batwa, S., Ma, W.X.: A study of lump-type and interaction solutions to a (3+1)-dimensional Jimbo–Miwa-like equation. Comput. Math. Appl. 76, 1576–1582 (2018)MathSciNetGoogle Scholar
  59. 59.
    Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4, 251–257 (1991)Google Scholar
  60. 60.
    Chen, S.T., Ma, W.X.: Lump solutions to a generalized Bogoyavlensky–Konopelchenko equation. Front. Math. China 13(3), 525–534 (2018)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Chen, S.T., Ma, W.X.: Lump solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation. Comput. Math. Appl. 76, 1680–1685 (2018)MathSciNetGoogle Scholar
  62. 62.
    Manukure, S., Zhou, Y., Ma, W.X.: Lump solutions to a (2+1)-dimensional extended KP equation. Comput. Math. Appl. 75, 2414–2419 (2018)MathSciNetGoogle Scholar
  63. 63.
    Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Yang, J.Y., Ma, W.X., Qin, Z.Y.: Abundant mixed lump-soliton solutions to the BKP equation. East Asian J. Appl. Math. 8(2), 224–232 (2018)Google Scholar
  65. 65.
    Ma, W.X.: Lump and interaction solutions of linear PDEs in (3+1)-dimensions. East Asian J. Appl. Math.
  66. 66.
    Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsInner Mongolia University of TechnologyHohhotChina

Personalised recommendations