Painlevé-Type Auto-Bäcklund Transformations, Periodic and Soliton Solutions of a Damped Variable-Coefficient Fifth-Order Modified Korteweg-de Vries Equation for the Surface Waves in a Strait or Large Channel

• Shu-Peng Feng ^[1] ; Bo Tian ^[1] ; Hao-Dong Liu ^[1]
  1. [1] Beijing University of Posts and Telecommunications

     Beijing University of Posts and Telecommunications

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Modified Korteweg-de Vries equations are used to describe certain phenomena in nonlinear optics, fluid dynamics, plasma physics, ocean physics and gas dynamics. In this paper, we investigate a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the small-amplitude surface waves in a strait or large channel of slowly-varying depth, width and non-vanishing vorticity. Via the truncated Painlevé expansion, Painlevé-type auto-Bäcklund transformations are obtained.

    Based on the Painlevé-type auto-Bäcklund transformations, periodic solutions and one-soliton solutions are derived. The soliton-like solutions are constructed via the modified Kudryashov method. We graphically show the kink-type solitons of the soliton solutions. Graphic analysis shows that the shapes, characteristic lines and velocities of the solitons are related to α1(t) and β(t), while the soliton backgrounds and amplitudes just depend on γ (t), in which α1(t), β(t) and γ (t) are the dispersive, dissipative and line-damping coefficients of the equation we investigated, respectively, where t is the temporal variable.

• Referencias bibliográficas
  • 1. Onorato, M., Janssen, P.A.: On the non-uniqueness of the kernel of the Zakharov equation in intermediate and shallow water: the connection...
  • 2. Kim, I.C., Kaihatu, J.M.: A simplified consistent nonlinear mild-slope equation model for random waves propagation and dissipation. Coast....
  • 3. Churilov, S.M.: Traveling internal waves in a two-layer shallow medium with variable bathymetry and current. Phys. Fluids 35, 026602 (2023)
  • 4. Fang, H., Tang, L., Lin, P.: Bragg scattering of nonlinear surface waves by sinusoidal sandbars. J. Fluid Mech. 979, A13 (2024)
  • 5. Gao, X.Y.: Symbolic computation on a (2+1)-dimensional generalized nonlinear evolution system in fluid dynamics, plasma physics, nonlinear...
  • 6. Gao, X.Y.: Auto-Bäcklund transformation with the solitons and similarity reductions for a generalized nonlinear shallow water wave equation....
  • 7. Gao, X.Y.: In an ocean or a river: Bilinear auto-Bäcklund transformations and similarity reductions on an extended time-dependent (3+1)-dimensional...
  • 8. Akinyemi, L., Manukure, S., Houwe, A., Abbagari, S.: A study of (2+1)-dimensional variable coefficients equation: Its oceanic solitons...
  • 9. Wazwaz, A.M., Alhejaili, W., El-Tantawy, S.A.: On the Painlevé integrability and nonlinear structures to a (3+1)-dimensional Boussinesq-type...
  • 10. Kudryavtsev, A.G., Myagkov, N.N.: On the superposition of solutions of the (3+1) dimensional Charney-Obukhov equation for the ocean....
  • 11. Singh, S., Sakkaravarthi, K., Murugesan, K.: Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional...
  • 12. Singh, S., Sakkaravarthi, K., Manikandan, K., Sakthivel, R.: Superposed nonlinear waves and transitions in a (3+1)-dimensional variable-coefficient...
  • 13. Singh, S., Sakkaravarthi, K., Murugesan, K.: Localized nonlinear waves on spatio-temporally controllable backgrounds for a (3+1)-dimensional...
  • 14. Jisha, C.R., Jang, B.: Novel insights into the exact solutions of the modified (3+1) dimensional fractional KS equation with variable...
  • 15. Yuan, F., Ghanbari, B.: A study of interaction soliton solutions for the (2+1)-dimensional HirotaSatsuma-Ito equation. Nonlinear Dyn....
  • 16. Hashemi, M.S.: A variable coefficient third degree generalized Abel equation method for solving stochastic Schrödinger-Hirota model. Chaos...
  • 17. Liu, F.Y., Xu, S.Y., Triki, H., Choudhuri, A., Zhou, Q.: Spatiotemporal modulated solitons in a quasione-dimensional spin-1 Bose-Einstein...
  • 18. Liu, F.Y., Triki, H., Zhou, Q.: Optical nondegenerate solitons in a birefringent fiber with a 35 degree elliptical angle. Opt. Express...
  • 19. Liu, F.Y., Triki, H., Zhou, Q.: Oscillatory nondegenerate solitons in spin-orbit coupled spin-1/2 BoseEinstein condensates with weak Raman...
  • 20. Ding, C.C., Zhou, Q., Malomed, B.A.: Ultra-high-amplitude Peregrine solitons induced by helicoidal spin-orbit coupling. Phys. Rev. Res....
  • 21. Ding, C.C., Zhu, L.W., Triki, H., Zhou, Q.: Four-wave mixing induced general localized waves for a coupled generalized nonlinear Schrödinger...
  • 22. Ding, C.C., Zhou, Q., Triki, H., Hu, Z.H.: Interaction dynamics of optical dark bound solitons for a defocusing Lakshmanan-Porsezian-Daniel...
  • 23. Zhou, T.Y., Tian, B., Shen, Y., Cheng, C.D.: Lie symmetry analysis, optimal system, symmetry reductions and analytic solutions for a (2+1)-dimensional...
  • 24. Wu, X.H., Gao, Y.T.: N-soliton asymptotic analysis on the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma. Appl. Math. Lett....
  • 25. Shen, Y., Tian, B., Zhou, T.Y., Cheng, C.D.: Localized waves of the higher-order nonlinear SchrödingerMaxwell-Bloch system with the sextic...
  • 26. Liu, J.G., Yang, X.J.: Symmetry group analysis of several coupled fractional partial differential equations. Chaos Solitons Fract. 173,...
  • 27. Acharya, N.P., Basnet, S., Khanal, R.: Nonlinear dust-ion acoustic solitary waves for collisional electronegative dusty plasma in the...
  • 28. Nawaz, H., Jahangir, R., Masood, W., Siddiq, M.: Cubic nonlinearity driven dust ion acoustic solitons with superthermal two-temperature...
  • 29. Wang, C., Wang, J., Liu, Q., Zhao, S., Li, Z., Kaidi, S., Hu, H.B., Chen, X.P., Du, P.: Dynamics and “falling deep” mechanism of submerged...
  • 30. Mohapatra, S.C., Fonseca, R.B., Guedes Soares, C.: Comparison of analytical and numerical simulations of long nonlinear internal solitary...
  • 31. Ma, G., Li, K., Sun, H.: Modeling and simulation of traffic flow based on memory effect and driver characteristics. Chin. J. Phys. 81,...
  • 32. Peng, G., Jia, T., Kuang, H., Tan, H.: Energy consumption in a new lattice hydrodynamic model based on the delayed effect of collaborative...
  • 33. Yue, J., Zhao, Z.L.: Some new lump molecules and hybrid molecular states of a (3+1)-dimensional generalized variable coefficient Kadomtsev-Petviashvili...
  • 34. Feng, C.H., Tian, B., Yang, D.Y., Gao, X.T.: Bilinear form, bilinear Bäcklund transformations, breather and periodic-wave solutions for...
  • 35. Liu, J.G., Wazwaz, A.M., Zhu, W.H.: Solitary and lump waves interaction in variable-coefficient nonlinear evolution equation by a modified...
  • 36. Akinyemi, L., Manukure, S., Houwe, A., Abbagari, S.: A study of (2+1)-dimensional variable coefficients equation: Its oceanic solitons...
  • 37. Yin, T., Pang, J.: Variable coefficient (2+1)D KP equation for Rossby waves and its dynamical analysis. Nonlinear Dyn. 112, 3725–3736...
  • 38. Yoku¸s, A., Duran, S., Kaya, D.: An expansion method for generating travelling wave solutions for the (2+1)-dimensional Bogoyavlensky-Konopelchenko...
  • 39. Song, N., Shang, H.J., Zhang, Y.F., Ma, W.X.: Localized wave solutions to a variable-coefficient coupled Hirota equation in inhomogeneous...
  • 40. Ali, M.R., Ma, W.X., Sadat, R.: Lie symmetry analysis and invariant solutions for (2+1) dimensional Bogoyavlensky-Konopelchenko equation...
  • 41. Zhao, Y., Tian, B.: Hybrid-wave solutions for a (2+1)-dimensional variable-coefficient KadomtsevPetviashvili equation in fluid mechanics...
  • 42. Wang, D., Gao, Y.T., Yu, X., Deng, G.F., Liu, F.Y.: Painlevé analysis, Bäcklund transformation, Lax pair, periodic-and travelling-wave...
  • 43. Liu, F.Y., Gao, Y.T.: Painlevé analysis, auto-Bäcklund transformations, bilinear forms and soliton solutions for a (2+1)-dimensional...
  • 44. Zhang, S., Zhou, H.: Riemann-Hilbert method and soliton dynamics for a mixed spectral complex mKdV equation with time-varying coefficients....
  • 45. Gao, X.Y.: In plasma physics and fluid dynamics: Symbolic computation on a (2+1)-dimensional variable-coefficient Sawada-Kotera system....
  • 46. Mohanty, S.K., Kumar, S., Dev, A.N., Deka, M.K., Churikov, D.V., Kravchenko, O.V.: An efficient technique of G’ G-expansion method for...
  • 47. Liu, X.Z., Yu, J., Lou, Z.M., Qian, X.M.: A nonlocal variable coefficient modified KdV equation derived from a two-layer fluid system...
  • 48. Hashmi, T., Jahangir, R., Masood, W., Alotaibi, B.M., Ismaeel, S.M., El-Tantawy, S.A.: Head-on collision of ion-acoustic (modified) Korteweg-de...
  • 49. Nawaz, H., Jahangir, R., Masood, W., Siddiq, M.: Cubic nonlinearity driven dust ion acoustic solitons with superthermal two-temperature...
  • 50. Mallet, A.: Nonlinear dynamics of large-amplitude, small-scale Alfvén waves. Phys. Plasmas 30, 122103 (2023)
  • 51. Ishaq, M., Xu, H.: Nonlinear dispersive Alfvén waves interaction in magnetized plasma. Phys. Fluids 31, 082105 (2019)
  • 52. Ito, M.: An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders. J. Phys. Soc. Jpn. 49, 771–778 (1980)
  • 53. Shen, S., Pan, Z.: A note on the Jacobi elliptic function expansion method. Phys. Lett. A 308, 143–148 (2003)
  • 54. Wang, X., Zhang, J., Wang, L.: Conservation laws, periodic and rational solutions for an extended modified Korteweg-de Vries equation....
  • 55. Zhang, S., Zhang, L.: Bilinearization and new multi-soliton solutions of mKdV hierarchy with timedependent coefficients. Open Phys. 14,...
  • 56. Wu, Q.L., Zhang, H.Q., Hang, C.: Breather, soliton-breather interaction and double-pole solutions of the fifth-order modified KdV equation....
  • 57. Wazwaz, A.M.: Two new integrable modified KdV equations, of third-and fifth-order, with variable coefficients: multiple real and multiple...
  • 58. 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...
  • 59. Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 24, 522–526 (1983)
  • 60. Zhang, Y.P., Wang, J.Y., Wei, G.M., Liu, R.P.: The Painlevé property, Bäcklund transformation, Lax pair and new analytic solutions of...
  • 61. Chen, Y.Q., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Tian, H.Y., Wang, M.: Reduction and analytic solutions of a variable-coefficient Korteweg-de...
  • 62. Yu, X., Gao, Y.T., Sun, Z.Y., Liu, Y.: Wronskian solutions and integrability for a genralized variablecoefficient forced Korteweg-de Vries...
  • 63. Liu, H.D., Tian, B., Feng, S.P., Chen, Y.Q., Zhou, T.Y.: Integrability, bilinearization, Bäcklund transformations and solutions for a...
  • 64. Liu, H.D., Tian, B., Cheng, C.D., Zhou, T.Y., Gao, X.T.: Painlevé analysis, Bilinear forms, Bäcklund transformations and solitons for...
  • 65. Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17,...
  • 66. Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fract. 24,...
  • 67. Ryabov, P.N., Sinelshchikov, D.I., Kochanov, M.B.: Application of the Kudryashov method for finding exact solutions of the high order...
  • 68. Ntiamoah, D., Ofori-Atta, W., Akinyemi, L.: The higher-order modified Korteweg-de Vries equation: its soliton, breather and approximate...
  • 69. Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992)