Bilinearization and Soliton Solutions for Certain Nonlocal Extended Complex Modified Korteweg-de Vries Equations

• Hao-Dong Liu ^[1] ; Bo Tian ^[1] ; Xiao-Tian Gao ^[1] ; Hong-Wen Shan ^[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. 25, Nº 1, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Nonlocal integrable systems have attracted the increasing attention in recent years, and various scalar nonlocal modified Korteweg-de Vries (mKdV)-type equations have been widely investigated as the important models in mathematical physics and nonlinear science. In this work, we investigate three nonlocal variants of the extended complex mKdV (ecmKdV) equation, corresponding to the reverse space, reverse time, and reverse space-time reductions. Using the improved Hirota method, we derive the bilinear form of the reverse space-time nonlocal ecmKdV equation, and construct the explicit one- and two-soliton solutions for the reverse space, reverse time, and reverse space-time nonlocal ecmKdV equations through certain variable transformations. Furthermore, the asymptotic behavior of the two-soliton solutions for the reverse space-time nonlocal ecmKdV equation is analyzed. Additionally, through certain variable transformations, Lax pairs and conservation laws of those nonlocal ecmKdV equations are constructed under the Ablowitz-Kaup-Newell-Segur procedure.

• Referencias bibliográficas
  • 1. Hollm, M., Seifried, R.: A computational method for the fluid-structure interaction in nonlinear ocean waves using the nonlinear Schrödinger...
  • 2. Veni, S.S., Rajan, M.S., Tabi, C.B., et al.: Propagation dynamics and modulation instability control in inhomogeneous nonlinear Schrödinger...
  • 3. Copie, F., Biondini, G., Oregero, J., et al.: Experimental observation of the spatio-temporal dynamics of breather gases in a recirculating...
  • 4. Gao, X.Y., Zhao, W.G., Zuo, D.W.: Bilinear forms, bilinear auto-Bäcklund transformations and similarity reductions for a (3+1)-dimensional...
  • 5. Gao, X.Y., Liu, J.G., Wang, G.W.: Inhomogeneity, magnetic auto-Bäcklund transformations and magnetic solitons for a generalized variable-coefficient...
  • 6. Liu, H.D., Tian, B., Chen, Y.Q., et al.: N-soliton, Hth-order breather, hybrid and multi-pole solutions for a generalized variable-coefficient...
  • 7. Wang, J.Y., Yang, Y.Q.: Data-driven solutions and parameter estimations for the integrable generalized fifth-order nonlinear Schrödinger...
  • 8. Liu, H.D., Tian, B., Cheng, C.D., et al.: Damped variable-coefficient fifth-order modified Korteweg-de Vries equation in fluid mechanics:...
  • 9. Yang, B., Yang, J.K.: Transformations between nonlocal and local integrable equations. Stud. Appl. Math. 140, 178 (2017)
  • 10. Rabie, W.B., Abbas, W., Ramadan, M.E., et al.: Modulation instability analysis and deriving soliton solutions of new nonlocal Lakshmanan-Porserzian-Daniel...
  • 11. Qin, Q., Li, L., Yu, F.J.: Discrete bright soliton solutions and modulation instability for the nonlocal semi-discrete nonlinear Schrödinger...
  • 12. Hou, L.H., Cheng, L., Yang, Y., et al.: Investigation of forward and inverse problems in a nonlocal reverse-time nonlinear Schrödinger...
  • 13. Lei, L., Tian, S.F., Zhang, X.F.: Riemann-Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal...
  • 14. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)
  • 15. Yang, J.K.: General N-soliton solutions and their dynamics in several nonlocal nonlinear Schrödinger equations. Phys. Lett. A 383, 328...
  • 16. Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity...
  • 17. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7 (2017)
  • 18. Liu, H.D., Tian, B.: Bilinearization, solitons and modulation instability for a variable-coefficient nonlocal Hirota equation. Appl. Math....
  • 19. Lou, S.Y., Huang, F.: Alice-Bob physics: Coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)
  • 20. Wang, X.,Wang, L., Du, Z., et al.: General soliton solutions for the complex reverse space-time nonlocal mKdV equation on a finite background....
  • 21. Ji, J.L., Zhu, Z.N.: On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions. Commun....
  • 22. Song, C.Q., Liu, D.Y., Ma, L.Y.: Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg-de Vries...
  • 23. Tang, X.Y., Liang, Z.F., Hao, X.Z.: Nonlinear waves of a nonlocal modified KdV equation in the atmospheric and oceanic dynamical system....
  • 24. Zhang, F., Hu, Y.R., Xin, X.P., et al.: Darboux transformation, soliton solutions of the variable coefficient nonlocal modified Korteweg-de...
  • 25. Ma, W.X.: Type (−λ, −λ∗) reduced nonlocal integrable mKdV equations and their soliton solutions. Appl. Math. Lett. 131, 108074 (2022)
  • 26. Ma, W.X.: Riemann-Hilbert problems and soliton solutions of type(λ∗, −λ∗) reduced nonlocal integrable mKdV hierarchies. Mathematics 10,...
  • 27. Ma, W.X.: Inverse scattering and soliton solutions of nonlocal complex reverse-space time mKdV equations. J. Geo. Phys. 157, 103845 (2020)
  • 28. Li, Yan, Hu, B.B., Zhang, L., et al.: The exact solutions for the nonlocal Kundu-NLS equation by the inverse scattering transform. Chaos...
  • 29. Feng, B.F., Luo, X.D., Ablowitz, M.J., et al.: General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero...
  • 30. Gai, L.T., Ma, W.X., Jin, L.B.: Hirota condition analysis of N-solitons and their derived waves for a nonlocal (2+1)-dimensional modified...
  • 31. Gürses, N., Pekcan, A.: Nonlocal nonlinear Schrödinger equations and their soliton solutions. J. Math. Phys. 59, 051501 (2018)
  • 32. Gürses,M., Pekcan, A.: Nonlocal modified KdV equations and their soliton solutions by HirotaMethod. Commun. Nonlinear Sci. Numer. Simulat....
  • 33. Zuo, D.W., Zhang, G.F.: Exact solutions of the nonlocal Hirota equations. Appl. Math. Lett. 93, 66 (2019)
  • 34. Zhang, W.X., Liu, Y.Q.: Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV)...
  • 35. Li, L., Duan, C.N., Yu, F.J.: An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de...
  • 36. Bai, Y.S., Zheng, L.N., Ma, W.X., et al.: Hirota bilinear approach to multi-component nonlocal nonlinear Schrödinger equations. Mathematics...
  • 37. Yu, F.J., Fan, R.: Nonstandard bilinearization and interaction phenomenon for PT-symmetric coupled nonlocal nonlinear Schrödinger equations....
  • 38. Kudryashov, N.A., Sinelshchikov, D.I.: A note on the Lie symmetry analysis and exact solutions for the extended mKdV equation. Acta Appl....
  • 39. Liu, H., Li, J.: Lie symmetry analysis and exact solutions for the extended mKdV equation. Acta Appl. Math. 109, 1107 (2010)
  • 40. Liu, H.D., Tian, B., Gao, X.T., et al.: Bilinearization and soliton solutions for certain symmetric nonlocal extended complex modifed...
  • 41. Liu, H.D., Tian, B., Gao, X.T., et al.: Construction, Lax integrability, bilinearization and multi-soliton solutions for a defocusing/focusing...
  • 42. Li, Z.Q., Tian, S.F.: A hierarchy of nonlocal nonlinear evolution equations and ∂¯-dressing method. Appl. Math. Lett. 120, 107254 (2021)
  • 43. Liu, H.D., Tian, B., Gao, X.T., et al.: Solitonic properties for a nonlocal extended complex modifed Korteweg-de Vries equation with the...
  • 44. Hirota, R.: The direct method in soliton theory. Cambridge Univ. Press, New York (2004)
  • 45. Ablowitz, M.J., Kaup, D.J., Newell, A.C., et al.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125 (1973)
  • 46. Ablowitz, M.J., Kaup, D.J., Newell, A.C., et al.: The inverse scattering transform-fourier analysis for nonlinear problems. Stud. Appl....