Skip to main content
SpringerLink
Log in

Bilinear Auto-Bäcklund Transformations and Similarity Reductions for a (3+1)-dimensional Generalized Yu-Toda-Sasa-Fukuyama System in Fluid Mechanics and Lattice Dynamics

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Recent investigations on the liquids and lattices are both active. In this paper, with symbolic computation, we consider a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system for the interfacial waves in a two-layer liquid or elastic waves in a lattice, with two sets of the bilinear auto-Bäcklund transformations hereby built up. Moreover, we construct one set of the similarity reductions, from that system to a known ordinary differential equation. As for the amplitude or elevation of the relevant wave, our results rely on the coefficients in that system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availibility

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Other fluid-mechanics studies have been shown, e.g., in Refs. [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34].

  2. Other nonlinear-optics investigations have been found, e.g., in Refs. [35,36,37,38,39,40,41,42,43,44,45,46,47,48].

  3. Other auto-Bäcklund transformations have been reported, e.g., in Refs. [49,50,51]. Besides, hetero-Bäcklund transformations, also named the non-auto-Bäcklund transformations, could been seen, e.g., in Refs. [52, 53].

  4. similar to those in Refs. [63, 64]

  5. with F and G as the real differentiable functions of x, y, z and t

  6. The plural form is used here, because of the existence of \(\mu _1\) (which is as-yet-undetermined).

  7. The plural form is used here, because of the existence of \(\sigma _1\) and \(\delta _1\) (which are as-yet-undetermined) and of the fact that we get a family of the solutions.

References

  1. Souza, R.R., Vargas, V.: Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure. Qual. Theory Dyn. Syst. 21, 5 (2022)

    MathSciNet  MATH  Google Scholar 

  2. Bhatti, M.M., Lu, D.Q.: Head-on Collision Between Two Hydroelastic Solitary Waves in Shallow Water. Qual. Theory Dyn. Syst. 17, 103 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Wannan, R.T., Abdallah, A.Y.: Long-Time Behavior of Non-Autonomous FitzHugh-Nagumo Lattice Systems. Qual. Theory Dyn. Syst. 19, 78 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Gao, X.Y., Guo, Y.J., Shan, W.R.: Auto-Bäcklund transformation, similarity reductions and solitons of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics. Qual. Theory Dyn. Syst. 21, 60 (2022)

    MATH  Google Scholar 

  5. Abdallah, A.Y.: Dynamics of Second Order Lattice Systems with Almost Periodic Nonlinear Part. Qual. Theory Dyn. Syst. 20, 58 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Wang, M., Tian, B., Qu, Q.X., Du, X.X., Zhang, C.R., Zhang, Z.: Lump, lumpoff and rogue waves for a (2+1)-dimensional reduced Yu-Toda-Sasa-Fukuyama equation in a lattice or liquid. Eur. Phys. J. Plus 134, 578 (2019)

    Google Scholar 

  7. Shen, Y., Tian, B., Zhao, X., Shan, W.R., Jiang, Y.: Bilinear form, bilinear auto-Bäcklund transformation, breather and lump solutions for a (3+1)-dimensional generalised Yu-Toda-Sasa-Fukuyama equation in a two-layer liquid or a lattice. Pramana-J. Phys. 95, 137 (2021)

    Google Scholar 

  8. Deng, G.F., Gao, Y.T., Su, J.J., Ding, C.C.: Multi-breather wave solutions for a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a two-layer liquid. Appl. Math. Lett. 98, 177 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. SIAM, Phil (1981)

    MATH  Google Scholar 

  10. Hirota, R.: The Direct Method in Soliton Theory. Cambridge Univ. Press, New York (2004)

    MATH  Google Scholar 

  11. Hadac, M., Herr, S., Koch, H.: Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincare Anal. Non Lineaire 26, 917 (2009)

  12. Senatorski, A., Infeld, E.: Simulations of two-dimensional Kadomtsev-Petviashvili soliton dynamics in three-dimensional space. Phy. Rev. Lett. 77, 2855 (1996)

    Google Scholar 

  13. Esfandyari, A.R., Khorram, S., Rostami, A.: Ion-acoustic solitons in a plasma with a relativistic electron beam. Phys. Plasmas 8, 4753 (2001)

    Google Scholar 

  14. Wazwaz, A.M.: Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method. Chaos Solitons Fract. 12, 2283 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Xue, L., Gao, Y.T., Zuo, D.W., Sun, Y.H., Yu, X.: Multi-Soliton Solutions and Interaction for a Generalized Variable-Coefficient Calogero-Bogoyavlenskii-Schiff Equation. Z. Naturforsch. A 69, 239 (2014)

    Google Scholar 

  16. Bruzón, M.S., Gandarias, M.L., Muriel, C., Saez, S., Romero, F.R.: The Calogero-Bogoyavlenskii-Schiff equation in 2+1 dimensions. Theor. Math. Phys. 137, 1367 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Yin, H.M., Tian, B., Chai, J., Wu, X.Y., Sun, W.R.: Solitons and bilinear Bäcklund transformations for a (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a liquid or lattice. Appl. Math. Lett. 58, 178 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Liu, F.Y., Gao, Y.T., Yu, X., Hu, L., Wu, X.H.: Hybrid solutions for the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. Chaos Solitons Fract. 152, 111355 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Shen, Y., Tian, B., Liu, S.H., Zhou, T.Y.: Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Nonlinear Dyn. 108, 2447 (2022)

    Google Scholar 

  20. Hu, L., Gao, Y.T., Jia, S.L., Su, J.J., Deng, G.F.: Solitons for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique. Mod. Phys. Lett. B 33, 1950376 (2019)

    MathSciNet  Google Scholar 

  21. Wang, M., Tian, B., Sun, Y., Yin, H.M., Zhang, Z.: Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev-Petviashvili equation in fluid mechanics. Chin. J. Phys. 60, 440 (2019)

    MathSciNet  Google Scholar 

  22. Yu, X., Sun, Z.Y.: Parabola solitons for the nonautonomous KP equation in fluids and plasmas. Ann. Phys.-New York 367, 251 (2016)

  23. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: Bilinear auto-Bäcklund transformation, soliton and periodic-wave solutions for a (\(2+1\))-dimensional generalized Kadomtsev-Petviashvili system in fluid mechanics and plasma physics. Chin. J. Phys. 77, 2698 (2022)

    Google Scholar 

  24. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C., Deng, G.F., Jia, T.T.: Painlevé analysis, Lie group analysis and soliton-cnoidal, resonant, hyperbolic function and rational solutions for the modified Korteweg-de Vries-Calogero Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics. Chaos Solitons Fract. 144, 110559 (2021)

    MATH  Google Scholar 

  25. Wang, M., Tian, B.: Soliton, multiple-lump, and hybrid solutions for a (\(3+1\))-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in plasma physics, fluid mechanics, and ocean dynamics. Rom. Rep. Phys. 73, 127 (2021)

    Google Scholar 

  26. Yu, X., Sun, Z.Y.: Unconventional characteristic line for the nonautonomous KP equation. Appl. Math. Lett. 100, 106047 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Guan, S.N., Wei, G.M., Li, Q.: Lie symmetry analysis, optimal system and conservation law of a generalized (\(2+1\))-dimensional Hirota-Satsuma-Ito equation. Mod. Phys. Lett. B 35, 2150515 (2021)

    MathSciNet  Google Scholar 

  28. Li, L.Q., Gao, Y.T., Yu, X., Deng, G.F., Ding, C.C.: Gramian solutions and solitonic interactions of a (\(2+1\))-dimensional Broer-Kaup-Kupershmidt system for the shallow water. Int. J. Numer. Method. H. 32, 2282 (2022)

    Google Scholar 

  29. Hu, L., Gao, Y.T., Jia, T.T., Deng, G.F., Li, L.Q.: Higher-order hybrid waves for the (\(2+1\))-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique. Z. Angew. Math. Phys. 72, 75 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: Shallow-water-wave studies on a (\(2+1\))-dimensional Hirota-Satsuma-Ito system: X-type soliton, resonant Y-type soliton and hybrid solutions. Chaos Solitons Fract. 157, 111861 (2022)

    MathSciNet  MATH  Google Scholar 

  31. Wang, M., Tian, B., Qu, Q.X., Zhao, X.H., Zhang, Z., Tian, H.Y.: Lump, lumpoff, rogue wave, breather wave and periodic lump solutions for a (\(3+1\))-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics. Int. J. Comput. Math. 97, 2474 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Hu, C.C., Tian, B., Zhao, X.: Rogue and lump waves for the (\(3+1\))-dimensional Yu-Toda-Sasa-Fukuyama equation in a liquid or lattice. Int. J. Mod. Phys. B 35, 2150320 (2021)

    Google Scholar 

  33. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C.: Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (\(3+1\))-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Nonlinear Dyn. 108, 1599 (2022)

    Google Scholar 

  34. Ding, C.C., Gao, Y.T., Hu, L., Deng, G.F., Zhang, C.Y.: Vector bright soliton interactions of the two-component AB system in a baroclinic fluid. Chaos Solitons Fract. 142, 110363 (2021)

    MathSciNet  MATH  Google Scholar 

  35. Yang, D.Y., Tian, B., Wang, M., Zhao, X., Shan, W.R., Jiang, Y.: Lax pair, Darboux transformation, breathers and rogue waves of an N-coupled nonautonomous nonlinear Schrödinger system for an optical fiber or plasma. Nonlinear Dyn. 107, 2657 (2022)

    Google Scholar 

  36. Tian, H.Y., Tian, B., Zhang, C.R., Chen, S.S.: Darboux dressing transformation and superregular breathers for a coupled nonlinear Schrödinger system with the negative coherent coupling in a weakly birefringent fiber. Int. J. Comput. Math. 98, 2445 (2021)

    MathSciNet  MATH  Google Scholar 

  37. Wang, M., Tian, B.: Darboux transformation, generalized Darboux transformation and vector breather solutions for the coupled variable-coefficient cubic-quintic nonlinear Schrödinger system in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide. Wave. Random Complex (2022). https://doi.org/10.1080/17455030.2021.1986649

    Article  Google Scholar 

  38. Yang, D.Y., Tian, B., Hu, C.C., Liu, S.H., Shan, W.R., Jiang, Y.: Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber. Wave. Random Complex (2022). https://doi.org/10.1080/17455030.2021.1983237

    Article  Google Scholar 

  39. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Liu, F.Y., Jia, T.T.: Darboux transformation, bright and dark-bright solitons of an N-coupled high-order nonlinear Schrödinger system in an optical fiber. Mod. Phys. Lett. B (2022). https://doi.org/10.1142/s0217984921505680

    Article  Google Scholar 

  40. Lu, Y.L., Wei, G.M., Liu, X.: Lax Pair, improved \(\Gamma \)-Riccati Backlund transformation and soliton-like solutions to variable-coefficient higher-order nonlinear Schrodinger equation in optical fibers. Acta Appl. Math. 164, 185 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Yang, D.Y., Tian, B., Qu, Q.X., Yuan, Y.Q., Zhang, C.R., Tian, H.Y.: Generalized Darboux transformation and the higher-order semirational solutions for a nonlinear Schrodinger system in a birefringent fiber. Mod. Phys. Lett. B 34, 2150013 (2020)

    Google Scholar 

  42. Wang, M., Tian, B.: Lax pair, generalized Darboux transformation and solitonic solutions for a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber. Rom. J. Phys. 66, 119 (2021)

    Google Scholar 

  43. Wei, G.M., Lu, Y.L., Xie, Y.Q., Zheng, W.X.: Lie symmetry analysis and conservation law of variable-coefficient Davey-Stewartson equation. Comput. Math. Appl. 75, 3420 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Yang, D.Y., Tian, B., Tian, H.Y., Wei, C.C., Shan, W.R., Jiang, Y.: Darboux transformation, localized waves and conservation laws for an M-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber. Chaos Solitons Fract. 156, 111719 (2022)

    Google Scholar 

  45. Tian, H.Y., Tian, B., Sun, Y., Zhang, C.R.: Three-component coupled nonlinear Schrödinger system in a multimode optical fiber: Darboux transformation induced via a rank-two projection matrix. Commun. Nonlinear Sci. Numer. Simul. 107, 106097 (2022)

    MATH  Google Scholar 

  46. Wang, M., Tian, B.: In an inhomogeneous multicomponent optical fiber: Lax pair, generalized Darboux transformation and vector breathers for a three-coupled variable-coefficient nonlinear Schrödinger system. Eur. Phys. J. Plus 136, 1002 (2021)

    Google Scholar 

  47. Yang, D.Y., Tian, B., Qu, Q.X., Zhang, C.R., Chen, S.S., Wei, C.C.: Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber. Chaos Solitons Fract. 150, 110487 (2021)

    MathSciNet  MATH  Google Scholar 

  48. Wang, M., Tian, B., Hu, C.C., Liu, S.H.: Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrödinger system in a birefringent optical fiber. Appl. Math. Lett. 119, 106936 (2021)

    MATH  Google Scholar 

  49. Zhou, T.Y., Tian, B., Chen, S.S., Wei, C.C., Chen, Y.Q.: Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves. Mod. Phys. Lett. B 35, 2150421 (2021)

    Google Scholar 

  50. Gao, X.Y., Guo, Y.J., Shan, W.R.: Optical waves/modes in a multicomponent inhomogeneous optical fiber via a three-coupled variable-coefficient nonlinear Schrödinger system. Appl. Math. Lett. 120, 107161 (2021)

    MATH  Google Scholar 

  51. Zhou, T.Y., Tian, B., Chen, Y.Q., Shen, Y.: Painlevé analysis, auto-Bäcklund transformation and analytic solutions of a (2+1)-dimensional generalized Burgers system with the variable coefficients in a fluid. Nonlinear Dyn. 108, 2417 (2022)

    Google Scholar 

  52. Gao, X.Y., Guo, Y.J., Shan, W.R.: Looking at an open sea via a generalized (2+1)-dimensional dispersive long-wave system for the shallow water: hetero-Bäcklund transformations, bilinear forms and N solitons. Eur. Phys. J. Plus 136, 893 (2021)

    Google Scholar 

  53. Gao, X.Y., Guo, Y.J., Shan, W.R.: Symbolic computation on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system for water waves. Chaos Solitons Fract. 150, 111066 (2021)

    MathSciNet  MATH  Google Scholar 

  54. Wang, M., Tian, B., Zhou, T.Y.: Darboux transformation, generalized Darboux transformation and vector breathers for a matrix Lakshmanan-Porsezian-Daniel equation in a Heisenberg ferromagnetic spin chain. Chaos Solitons Fract. 152, 111411 (2021)

    MathSciNet  MATH  Google Scholar 

  55. Ding, C.C., Gao, Y.T., Yu, X., Liu, F.Y., Wu, X.H.: Three-wave resonant interactions: dark-bright-bright mixed N- and high-order solitons, breathers, and their structures. Wave. Random Complex (2022). https://doi.org/10.1080/17455030.2021.1976437

    Article  Google Scholar 

  56. Yang, D.Y., Tian, B., Qu, Q.X., Du, X.X., Hu, C.C., Jiang, Y., Shan, W.R.: Lax pair, solitons, breathers and modulation instability of a three-component coupled derivative nonlinear Schrödinger system for a plasma. Eur. Phys. J. Plus 137, 189 (2022)

    Google Scholar 

  57. Chen, S.S., Tian, B., Qu, Q.X., Li, H., Sun, Y., Du, X.X.: Alfvén solitons and generalized Darboux transformation for a variable-coefficient derivative nonlinear Schrödinger equation in an inhomogeneous plasma. Chaos Solitons Fract. 148, 111029 (2021)

    MATH  Google Scholar 

  58. Liu, F.Y., Gao, Y.T.: Lie group analysis for a higher-order Boussinesq-Burgers system. Appl. Math. Lett. 132, 108094 (2022)

    MathSciNet  MATH  Google Scholar 

  59. Cheng, C.D., Tian, B., Zhang, C.R., Zhao, X.: Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg-de Vries equation in a fluid. Nonlinear Dyn. 105, 2525 (2021)

    Google Scholar 

  60. Gao, X.T., Tian, B., Feng, C.H.: In oceanography, acoustics and hydrodynamics: investigations on an extended coupled (2+1)-dimensional Burgers system. Chin. J. Phys. 77, 2818 (2022)

    MathSciNet  Google Scholar 

  61. Li, L.Q., Gao, Y.T., Yu, X., Jia, T.T., Hu, L., Zhang, C.Y.: Bilinear forms, bilinear Bäcklund transformation, soliton and breather interactions of a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the surface waves in a strait or large channel. Chin. J. Phys. 77, 915 (2022)

    Google Scholar 

  62. Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: Bilinear auto-Bäcklund transformation, breather-wave and periodic-wave solutions for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Eur. Phys. J. Plus 136, 1159 (2021)

    Google Scholar 

  63. Ma, Y.X., Tian, B., Qu, Q.X., Tian, H.Y., Liu, S.H.: Bilinear Bäcklund transformation, breather- and travelling-wave solutions for a (2+1)-dimensional extended Kadomtsev-Petviashvili II equation in fluid mechanics. Mod. Phys. Lett. B 35, 2150315 (2021)

    Google Scholar 

  64. Shen, Y., Tian, B.: Bilinear auto-Bäcklund transformations and soliton solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves. Appl. Math. Lett. 122, 107301 (2021)

    MATH  Google Scholar 

  65. Clarkson, P., Kruskal, M.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201 (1989)

    MathSciNet  MATH  Google Scholar 

  66. Gao, X.T., Tian, B.: Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system. Appl. Math. Lett. 128, 107858 (2022)

    MathSciNet  MATH  Google Scholar 

  67. Gao, X.Y., Guo, Y.J., Shan, W.R.: Similarity reductions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in nonlinear optics, fluid mechanics and plasma physics. Appl. Comput. Math. 20, 421 (2021)

    MathSciNet  MATH  Google Scholar 

  68. Gao, X.Y., Guo, Y.J., Shan, W.R.: Taking into consideration an extended coupled (2+1)-dimensional Burgers system in oceanography, acoustics and hydrodynamics. Chaos Solitons Fract. 161, 112293 (2022)

    MathSciNet  Google Scholar 

  69. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: Considering the shallow water of a wide channel or an open sea through a generalized (2+1)-dimensional dispersive long-wave system. Qual. Theory Dyn. Syst. (2022). https://doi.org/10.1007/s12346-022-00617-7

    Article  MathSciNet  MATH  Google Scholar 

  70. Gao, X.Y., Guo, Y.J., Shan, W.R.: Similarity reductions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid dynamics. Chin. J. Phys. 77, 2707 (2022)

    MathSciNet  Google Scholar 

  71. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: Comment on “Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system’’. Chaos Solitons Fract. 151, 111222 (2021)

    MATH  Google Scholar 

  72. Gao, X.Y., Guo, Y.J., Shan, W.R.: In nonlinear optics, fluid mechanics, plasma physics or atmospheric science: symbolic computation on a generalized variable-coefficient Korteweg-de Vries equation. Acta. Math. Sin.-English Ser. (2022). https://doi.org/10.1007/s10114-022-9778-5

  73. Gao, X.Y., Guo, Y.J., Shan, W.R.: Oceanic long-gravity-water-wave investigations on a variable-coefficient nonlinear dispersive-wave system. Wave. Random Complex (2022). https://doi.org/10.1080/17455030.2022.2039419

    Article  Google Scholar 

  74. Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Acad, San Diego (1997)

    MATH  Google Scholar 

  75. Rizvi, S.T., Seadawy, A.R., Farah, N., Ahmad, S.: Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation. Chaos Solitons Fract. 159, 112128 (2022)

    Google Scholar 

  76. Tao, G., Manafian, J., Ilhan, O.A., Zia, S.M., Agamalieva, L.: Abundant soliton wave solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles by bilinear analysis. Mod. Phys. Lett. B 36, 2150565 (2022)

    MathSciNet  Google Scholar 

  77. Nikhoghossian, A.G.: Conservation Laws in Time-Dependent Radiative Transfer Problems. Astrophysics 65, 81 (2022)

    Google Scholar 

  78. Tassi, E.: Poisson brackets and truncations in nonlinear reduced fluid models for plasmas. Phys. D 437, 133338 (2022)

    MathSciNet  MATH  Google Scholar 

  79. Younas, U., Ren, J., Baber, M.Z., Yasin, M.W., Shahzad, T.: Ion-acoustic wave structures in the fluid ions modeled by higher dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.05.005

  80. Zhou, T.Y., Tian, B., Zhang, C.R., Liu, S.H.: Auto-Bäcklund transformations, bilinear forms, multiple-soliton, quasi-soliton and hybrid solutions of a (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in an electron-positron plasma. Eur. Phys. J. Plus (2022). https://doi.org/10.1140/epjp/s13360-022-02950-x

Download references

Acknowledgements

We express our sincere thanks to the Editors and Advisors/Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11871116 and 11772017, and Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11. X. Y. Gao also thanks the National Scholarship for Doctoral Students of China.

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Xin-Yi Gao, Yong-Jiang Guo & Wen-Rui Shan

Authors
  1. Xin-Yi Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong-Jiang Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wen-Rui Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Xin-Yi Gao, Yong-Jiang Guo or Wen-Rui Shan.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The Hirota bilinear operators \(D_x\), \(D_y\), \(D_z\) and \(D_t\) have been defined as [10]

$$\begin{aligned}&D_{x}^{m_1}D_{y}^{m_2}D_{z}^{m_3}D_{t}^{m_4} G(x,y,z,t)\cdot F(x,y,z,t)= \left( \frac{\partial }{\partial x}-\frac{\partial }{\partial \tilde{x}}\right) ^{m_1}\,\left( \frac{\partial }{\partial y}-\frac{\partial }{\partial \tilde{y}}\right) ^{m_2}\, \nonumber \\&\quad \left( \frac{\partial }{\partial z}-\frac{\partial }{\partial \tilde{z}}\right) ^{m_3}\, \left( \frac{\partial }{\partial t}-\frac{\partial }{\partial \tilde{t}}\right) ^{m_4}\,G(x,y,z,t)\,F(\tilde{x},\tilde{y},\tilde{z},\tilde{t}) \bigg |_{\tilde{x}=x,\,\tilde{y}=y,\,\tilde{z}=z,\,\tilde{t}=t}, \nonumber \\ \end{aligned}$$
(A.1)

with \(\tilde{x}\), \(\tilde{y}\), \(\tilde{z}\) and \(\tilde{t}\) indicating four formal variables, G(x, y, z, t) denoting a \(C^{\infty }\) function of x, y, z and t, \(F(\tilde{x},\tilde{y},\tilde{z},\tilde{t})\) representing a \(C^{\infty }\) function of \(\tilde{x}\), \(\tilde{y}\), \(\tilde{z}\) and \(\tilde{t}\), while \(m_1\), \(m_2\), \(m_3\) and \(m_4\) implying four non-negative integers [10].

Recent applications of the Hirota bilinear operators include the ones to certain Bose-Einstein condensates with the dipole-dipole attractions and repulsions [75], liquids with the gas bubbles [76], time-dependent radiative transfer problems [77], nonlinear reduced fluid models for some plasmas [78] and ion-acoustic wave structures with the effects of magnetic fields in plasma physics [79]. Other recent vigorous references include, e.g., Refs. [22, 26, 27, 43, 80].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, XY., Guo, YJ. & Shan, WR. Bilinear Auto-Bäcklund Transformations and Similarity Reductions for a (3+1)-dimensional Generalized Yu-Toda-Sasa-Fukuyama System in Fluid Mechanics and Lattice Dynamics. Qual. Theory Dyn. Syst. 21, 95 (2022). https://doi.org/10.1007/s12346-022-00622-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00622-w

Keywords

Mathematics Subject Classification

Search

Navigation