Skip to main content
SpringerLink
Log in

Characteristic of integrability of nonautonomous KP-modified KP equation and its qualitative studies: soliton, shock, periodic waves, breather, positons and soliton interactions

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This article demonstrates the characteristic of integrability of the nonautonomous KP-mKP equation through Painlevé analysis, bilinear Bäcklund and lax pairs. The nonautonomous KP-mKP equation is converted into the Bell polynomial from which bilinear Bäcklund is constructed and lax pair of the said equation is generated. Further, multi-solitons, smooth positon, breather, and their interaction are fabricated using Hirota’s bilinear approach. Besides, a qualitative analysis of the nonautonomous KP-mKP equation using bifurcation theory is carried out. The deformation of the periodic to quasiperiodic orbit signifying instability of the said system due to damping is observed. Additionally, the external periodic force perturbs the nonautonomous system’s low- and high-energy orbits, resulting in a chaotic structure via the path of intermittency, implying the presence of turbulent flow.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Data availability

Not applicable.

Code Availability

Mathematica codes are available.

References

  1. Jordan, P.M., Puri, A.: A note on traveling wave solutions for a class of nonlinear viscoelastic media. Phys. Lett. A 335(2–3), 150–156 (2005)

    Article  Google Scholar 

  2. Zhou, J.K.: Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press, Wuhan (1986)

    Google Scholar 

  3. Raut, S., Roy, S., Saha, S., Das, A.N.: Studies on the dust-ion-acoustic solitary wave in planar and non-planar super-thermal plasmas with trapped electrons. Plasma Phys. Rep. 48(6), 627–637 (2022)

    Article  Google Scholar 

  4. Ray, A.K., Bhattacharjee, J.K.: Standing and travelling waves in the shallow-water circular hydraulic jump. Phys. Lett. A 371(3), 241–248 (2007)

    Article  Google Scholar 

  5. Yan, Z.: Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos Solitons Fractals 16(5), 759–766 (2003)

    Article  MathSciNet  Google Scholar 

  6. Wang, R.R., Wang, Y.Y., Dai, C.Q.: Influence of higher-order nonlinear effects on optical solitons of the complex Swift–Hohenberg model in the mode-locked fiber laser. Opt. Laser Technol. 152, 108103 (2022)

    Article  Google Scholar 

  7. Whitham, G.B.: Linear and Nonlinear Waves, vol. 1974, p. 651. Wiley-Interscience, Hoboken (1974)

    Google Scholar 

  8. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. In: Doklady Akademii Nauk, vol. 192, no. 4, pp. 753–756. Russian Academy of Sciences (1970)

  9. Gu, Y., Meng, F.: Searching for analytical solutions of the (2+1)-dimensional KP equation by two different systematic methods. Complexity 2019, 1 (2019)

    Google Scholar 

  10. Sarkar, T., Roy, S., Raut, S., Mali, P.C.: Studies on the dust acoustic shock, solitary, and periodic waves in an unmagnetized viscous dusty plasma with two-temperature ions. Braz. J. Phys. 53, 12 (2023)

    Article  Google Scholar 

  11. Ahmad, S., Khan, S.A., Hadi, F.: Damped Kadomtsev–Petviashvili equation for weakly dissipative solitons in dense relativistic degenerate plasmas. Commun. Theor. Phys. 68(6), 783 (2017)

    Article  MathSciNet  Google Scholar 

  12. Roy, A., Raut, S., Barman, R.: Studies on the effect of dust-ion collision on dust-ion acoustic solitary waves in a magnetized dusty plasma in the framework of damped KP equation and modified damped KP equation. Plasma Phys. Rep. 48(4), 367–383 (2022)

    Article  Google Scholar 

  13. Xue, J.K.: Kadomtsev–Petviashvili (KP) Burgers equation in a dusty plasmas with non-adiabatic dust charge fluctuation. Eur. Phys. J. D At. Mol. Opt. Plasma Phys. 26(2), 211–214 (2003)

    MathSciNet  Google Scholar 

  14. Kaur, M., Saini, N.S., Singla, S.: KP Burgers equation in beam plasma with non-Maxwellian electrons. Bull. Am. Phys. Soc. 96(9) (2022)

  15. Dorranian, D., Sabetkar, A.: Dust acoustic solitary waves in a dusty plasma with two kinds of nonthermal ions at different temperatures. Phys. Plasmas 19(1), 013702 (2012)

    Article  Google Scholar 

  16. Wazwaz, A.M.: Multi-front waves for extended form of modified Kadomtsev–Petviashvili equation. Appl. Math. Mech. 32, 875–880 (2011)

    Article  MathSciNet  Google Scholar 

  17. Xu, T., Zhang, H.Q., Zhang, Y.X., Li, J., Feng, Q., Tian, B.: Two types of generalized integrable decompositions and new solitary-wave solutions for the modified Kadomtsev–Petviashvili equation with symbolic computation. J. Math. Phys. 49(1), 013501 (2008)

    Article  MathSciNet  Google Scholar 

  18. Hao, H.H., Zhang, D.J.: Soliton resonances for the modified Kadomtsev–Petviashvili equations in uniform and non-uniform media. Mod. Phys. Lett. B 24(03), 277–288 (2010)

    Article  MathSciNet  Google Scholar 

  19. Sun, Z.Y., Gao, Y.T., Yu, X., Meng, X.H., Liu, Y.: Inelastic interactions of the multiple-front waves for the modified Kadomtsev–Petviashvili equation in fluid dyna plasma physics and electrodynamics. Wave Motion 46, 511–521 (2009)

    Article  MathSciNet  Google Scholar 

  20. Tsuji, H., Oikawa, M.: Two-dimensional interaction of solitary waves in a modified Kadomtsev–Petviashvili equation. J. Phys. Soc. Jpn. 73(11), 3034–3043 (2004)

    Article  Google Scholar 

  21. Wazwaz, A.M.: Solitons and singular solitons for the Gardner-KP equation. Appl. Math. Comput. 204(1), 162–169 (2008)

    MathSciNet  Google Scholar 

  22. Liu, H., Yan, F.: Bifurcation and exact travelling wave solutions for Gardner-KP equation. Appl. Math. Comput. 228, 384–394 (2014)

    MathSciNet  Google Scholar 

  23. Seadawy, A.R., Iqbal, M., Lu, D.: Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg-de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A 544, 123560 (2020)

    Article  Google Scholar 

  24. Roy, S., Raut, S., Kairi, R.R.: Nonlinear analysis of the ion-acoustic solitary and shock wave solutions for non-extensive dusty plasma in the framework of modified Korteweg-de Vries-Burgers equation. Pramana 96(2), 1–13 (2022)

    Article  Google Scholar 

  25. Roy, S., Kairi, R.R., Raut, S.: Cylindrical and spherical dust-ion-acoustic shock solitary waves by Korteweg-de Vries-Burgers equation. Braz. J. Phys. 51(6), 1651–1660 (2021)

    Article  Google Scholar 

  26. Aljahdaly, N.H., El-Tantawy, S.A.: Novel anlytical solution to the damped Kawahara equation and its application for modeling the dissipative nonlinear structures in a fluid medium. J. Ocean Eng. Sci. 7(5), 492–497 (2022)

    Article  Google Scholar 

  27. Raut, S., Barman, R., Sarkar, T.: Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach. Wave Motion 119, 103125 (2023)

    Article  MathSciNet  Google Scholar 

  28. Sen, A., Tiwari, S., Mishra, S., Kaw, P.: Nonlinear wave excitations by orbiting charged space debris objects. Adv. Space Res. 56(3), 429–435 (2015)

    Article  Google Scholar 

  29. Aslanov, V.S., Yudintsev, V.V.: Dynamics, analytical solutions and choice of parameters for towed space debris with flexible appendages. Adv. Space Res. 55(2), 660–667 (2015)

    Article  Google Scholar 

  30. Chadha, N.M., Tomar, S., Raut, S.: Parametric analysis of dust ion acoustic waves in superthermal plasmas through non-autonomous KdV framework. Commun. Nonlinear Sci. Numer. Simul. 123, 107269 (2023)

    Article  MathSciNet  Google Scholar 

  31. Raut, S., Roy, S., Kairi, R.R., Chatterjee, P.: Approximate analytical solutions of generalized Zakharov–Kuznetsov and generalized modified Zakharov–Kuznetsov equations. Int. J. Appl. Comput. Math. 7, 1–25 (2021)

    Article  MathSciNet  Google Scholar 

  32. Roy, S., Raut, S., Kairi, R.R., Chatterjee, P.: Integrability and the multi-soliton interactions of non-autonomous Zakharov–Kuznetsov equation. Eur. Phys. J. Plus 137(5), 1–14 (2022)

    Article  Google Scholar 

  33. Roy, A., Mondal, K.K., Chatterjee, P., Raut, S.: Influence of external periodic force on ion acoustic waves in a magnetized dusty plasma through forced KP equation and modified forced KP equation. Braz. J. Phys. 52(3), 65 (2022)

    Article  Google Scholar 

  34. Roy, S., Raut, S., Kairi, R.R., Chatterjee, P.: Bilinear Bäcklund, Lax pairs, breather waves, lump waves and soliton interaction of (2+1)-dimensional non-autonomous Kadomtsev–Petviashvili equation. Nonlinear Dyn. 111(6), 5721–5741 (2023)

    Article  Google Scholar 

  35. Zhou, Q., Triki, H., Xu, J., Zeng, Z., Liu, W., Biswas, A.: Perturbation of chirped localized waves in a dual-power law nonlinear medium. Chaos Solitons Fractals 160, 112198 (2022)

    Article  MathSciNet  Google Scholar 

  36. Raut, S., Ma, W.X., Barman, R., Roy, S.: A non-autonomous Gardner equation and its integrability: solitons, positons and breathers. Chaos Solitons Fractals 176, 114089 (2023)

    Article  MathSciNet  Google Scholar 

  37. Lan, Z.Z.: Semirational rogue waves of the three coupled higher-order nonlinear Schrödinger equations. Appl. Math. Lett. 147, 108845 (2024)

    Article  Google Scholar 

  38. El-Tantawy, S.A., Salas, A.H., Alyousef, H.A., Alharthi, M.R.: Novel approximations to a nonplanar nonlinear Schrödinger equation and modeling nonplanar rogue waves/breathers in a complex plasma. Chaos Solitons Fractals 163, 112612 (2022)

    Article  Google Scholar 

  39. Dubard, P., Gaillard, P., Klein, C., Matveev, V.B.: On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Spec. Top. 185(1), 247–258 (2010)

    Article  Google Scholar 

  40. Xing, Q., Wang, L., Mihalache, D., Porsezian, K., He, J.: Construction of rational solutions of the real modified Korteweg-de Vries equation from its periodic solutions. Chaos Interdiscip. J. Nonlinear Sci. 27(5), 053102 (2017)

    Article  MathSciNet  Google Scholar 

  41. Wu, Q.L., Zhang, H.Q., Hang, C.: Breather, soliton-breather interaction and double-pole solutions of the fifth-order modified KdV equation. Appl. Math. Lett. 120, 107256 (2021)

    Article  MathSciNet  Google Scholar 

  42. Ma, W.X.: Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schrödinger equations. Proc. Am. Math. Soc. 149(1), 251–263 (2021)

    Article  Google Scholar 

  43. Hirota, R.: The Direct Method in Soliton Theory, p. 155. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  44. Chen, Y.X., Xiao, X.: Vector soliton pairs for a coupled nonautonomous NLS model with partially nonlocal coupled nonlinearities under the external potentials. Nonlinear Dyn. 109(3), 2003–2012 (2022)

    Article  Google Scholar 

  45. Wang, X., Geng, X.G.: N-soliton solution and soliton resonances for the (2+1)-dimensional inhomogeneous Gardner equation. Commun. Theor. Phys. 68(2), 155 (2017)

    Article  MathSciNet  Google Scholar 

  46. Luo, X.Y., Chen, Y.: Darboux transformation and N-soliton solution for extended form of modified Kadomtsev–Petviashvili equation with variable-coefficient. Commun. Theor. Phys. 66(2), 179 (2016)

    Article  MathSciNet  Google Scholar 

  47. Ullah, N., Asjad, M.I., Ur Rehman, H., Akgül, A.: Construction of optical solitons of Radhakrishnan–Kundu–Lakshmanan equation in birefringent fibers. Nonlinear Eng. 11(1), 80–91 (2022)

    Article  Google Scholar 

  48. Fang, Y., Wu, G.Z., Wang, Y.Y., Dai, C.Q.: Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN. Nonlinear Dyn. 105(1), 603–616 (2021)

    Article  Google Scholar 

  49. Wen, X.K., Jiang, J.H., Liu, W., Dai, C.Q.: Abundant vector soliton prediction and model parameter discovery of the coupled mixed derivative nonlinear Schrödinger equation. Nonlinear Dyn. 111, 13343–13355 (2023)

    Article  Google Scholar 

  50. Fang, Y., Han, H.B., Bo, W.B., Liu, W., Wang, B.H., Wang, Y.Y., Dai, C.Q.: Deep neural network for modeling soliton dynamics in the mode-locked laser. Opt. Lett. 48(3), 779–782 (2023)

    Article  Google Scholar 

  51. Guan, X., Liu, W., Zhou, Q., Biswas, A.: Darboux transformation and analytic solutions for a generalized super-NLS-mKdV equation. Nonlinear Dyn. 98, 1491–1500 (2019)

    Article  Google Scholar 

  52. Geng, K.L., Zhu, B.W., Cao, Q.H., Dai, C.Q., Wang, Y.Y.: Nondegenerate soliton dynamics of nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 111(17), 16483–16496 (2023)

    Article  Google Scholar 

  53. Chen, H.Y., Zhu, H.P.: Higher-dimensional vector two-component solitons of a nonautonomous partially nonlocal coupled NLS model in a linear and harmonic potential. Nonlinear Dyn. 111(1), 581–590 (2023)

    Article  Google Scholar 

  54. Chen, Y.X.: Two-component excitation governance of giant wave clusters with the partially nonlocal nonlinearity. Nonlinear Eng. 12(1), 20220319 (2023)

    Article  Google Scholar 

  55. Chen, Y.X.: Vector peregrine composites on the periodic background in spin-orbit coupled Spin-1 Bose–Einstein condensates. Chaos Solitons Fractals 169, 113251 (2023)

    Article  MathSciNet  Google Scholar 

  56. Ma, W.X.: AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials. Appl. Math. Lett. 145, 108775 (2023)

    Article  MathSciNet  Google Scholar 

  57. Zhao, X.H.: Multi-solitons and integrability for a (2+1)-dimensional variable coefficients Date–Jimbo–Kashiwara–Miwa equation. Appl. Math. Lett. 149, 108895 (2023)

    Article  MathSciNet  Google Scholar 

  58. Guang-Mei, W., Yi-Tian, G., Tao, X., Xiang-Hua, M., Chun-Yi, Z.: Painlevé property and new analytic solutions for a variable-coefficient Kadomtsev–Petviashvili equation with symbolic computation. Chin. Phys. Lett. 25(5), 1599 (2008)

    Article  Google Scholar 

  59. Li, X.N., Wei, G.M., Liang, Y.Q.: Painlevé analysis and new analytic solutions for variable-coefficient Kadomtsev–Petviashvili equation with symbolic computation. Appl. Math. Comput. 216(12), 3568–3577 (2010)

    MathSciNet  Google Scholar 

  60. Palit, A., Roy, A., Raut, S.: Qualitative studies of the influence of damping and external periodic force on ion-acoustic waves in a magnetized dusty plasma through modified ZK equation. Braz. J. Phys. 52(4), 1–21 (2022)

    Article  Google Scholar 

  61. Ablowitz, M.J., Ramani, A., Segur, H.: A connection between nonlinear evolution equations and ordinary differential equations of P-type. II. J. Math. Phys. 21(5), 1006–1015 (1980)

    Article  MathSciNet  Google Scholar 

  62. Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 24, 522–526 (1983)

    Article  MathSciNet  Google Scholar 

  63. Jimbo, M., Kruskal, M.D., Miwa, T.: The Painlevé test for the self-dual Yang–Mills equations. Phys. Lett. A 92(2), 59–60 (1982)

    Article  MathSciNet  Google Scholar 

  64. Lambert, F., Springael, J.: Soliton equations and simple combinatorics. Acta Appl. Math. 102, 147–178 (2008)

    Article  MathSciNet  Google Scholar 

  65. Hu, X., Chen, Y.: A direct procedure on the integrability of nonisospectral and variable-coefficient MKdV equation. J. Nonlinear Math. Phys. 19(01), 1250002 (2012)

    MathSciNet  Google Scholar 

  66. Ma, W.X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72(1), 41–56 (2013)

    Article  MathSciNet  Google Scholar 

  67. Wazwaz, A.M.: Solitons and singular solitons for the Gardner-KP equation. Appl. Math. Comput. 204(1), 162–169 (2008)

    MathSciNet  Google Scholar 

  68. Wang, Y.Y., Su, C.Q., Liu, X.Q., Li, J.G.: Nonautonomous solitons for an extended forced Korteweg-de Vries equation with variable coefficients in the fluid or plasma. Waves Random Complex Media 28(3), 411–425 (2018)

    Article  MathSciNet  Google Scholar 

  69. Xing, Q., Wu, Z., Mihalache, D., He, J.: Smooth positon solutions of the focusing modified Korteweg-de Vries equation. Nonlinear Dyn. 89, 2299–2310 (2017)

    Article  MathSciNet  Google Scholar 

  70. Xing, Q., Wang, L., Mihalache, D., Porsezian, K., He, J.: Construction of rational solutions of the real modified Korteweg-de Vries equation from its periodic solutions. Chaos Interdiscip. J. Nonlinear Sci. 27(5), 053102 (2017)

    Article  MathSciNet  Google Scholar 

  71. Schiebold, C.: Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation. Nonlinearity 30(7), 2930 (2017)

    Article  MathSciNet  Google Scholar 

  72. Zhang, Y., Tao, X., Yao, T., He, J.: The regularity of the multiple higher-order poles solitons of the NLS equation. Stud. Appl. Math. 145(4), 812–827 (2020)

    Article  MathSciNet  Google Scholar 

  73. Bilman, D., Buckingham, R.: Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation. J. Nonlinear Sci. 29, 2185–2229 (2019)

    Article  MathSciNet  Google Scholar 

  74. Wang, L., He, J., Xu, H., Wang, J., Porsezian, K.: Generation of higher-order rogue waves from multibreathers by double degeneracy in an optical fiber. Phys. Rev. E 95(4), 042217 (2017)

    Article  MathSciNet  Google Scholar 

  75. Zhang, Z., Yang, X., Li, B.: Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation. Nonlinear Dyn. 100, 1551–1557 (2020)

    Article  Google Scholar 

  76. Manafian, J., Ilhan, O.A., Alizadeh, A.A.: Periodic wave solutions and stability analysis for the KP-BBM equation with abundant novel interaction solutions. Phys. Scr. 95(6), 065203 (2020)

    Article  Google Scholar 

  77. Ahmed, S., Ashraf, R., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Althobaiti, A., El-Shehawi, A.M.: Lump, multi-wave, kinky breathers, interactional solutions and stability analysis for general (2+1)-rth dispersionless Dym equation. Results Phys. 25, 104160 (2021)

  78. Yokus, A., Isah, M.A.: Stability analysis and solutions of (2+1)-Kadomtsev–Petviashvili equation by homoclinic technique based on Hirota bilinear form. Nonlinear Dyn. 109, 3029–3040 (2022)

  79. Schneider, T.M., Eckhardt, B., Yorke, J.A.: Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99(3), 034502 (2007)

    Article  Google Scholar 

  80. Chen, M., Sun, M., Bao, H., Hu, Y., Bao, B.: Flux-charge analysis of two-memristor-based Chua’s circuit: dimensionality decreasing model for detecting extreme multistability. IEEE Trans. Ind. Electron. 67(3), 2197–2206 (2019)

    Article  Google Scholar 

  81. Li, H., Li, K., Chen, M., Bao, B.: Coexisting infinite orbits in an area-preserving Lozi map. Entropy 22(10), 1119 (2020)

    Article  MathSciNet  Google Scholar 

  82. Abdul Rahim, M.F., Natiq, H., Fataf, N.A.A., Banerjee, S.: Dynamics of a new hyperchaotic system and multistability. Eur. Phys. J. Plus 134, 1–9 (2019)

    Article  Google Scholar 

Download references

Funding

Mr. Tanay Sarkar (JRF) sincerely appreciates the Fellowship granted by University Grants Commission (UGC) [No.1155/(CSIR-UGC NET DEC.2017)], India.

Author information

Authors and Affiliations

  1. Department of Mathematics, Mathabhanga College, Cooch Behar, West Bengal, 736164, India

    Santanu Raut

  2. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    Tanay Sarkar

  3. Department of Mathematics, Cooch Behar Panchanan Barma University, Cooch Behar, West Bengal, 736101, India

    Subrata Roy

  4. Department of Mathematics, Surya Sen Mahavidyalaya, Siliguri, 734004, India

    Aniruddha Palit

Authors
  1. Santanu Raut
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tanay Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Subrata Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Aniruddha Palit
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Santanu Raut contributed to writing—original draft preparation and writing—review and editing. Tanay Sarkar was involved in the methodology and writing—review and editing. Subrata Roy contributed to the software, visualization, and investigation. Aniruddha Palit contributed to software, visualization, and conceptualization.

Corresponding author

Correspondence to Aniruddha Palit.

Ethics declarations

Conflict of interest

There is no conflict of interest between the authors.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raut, S., Sarkar, T., Roy, S. et al. Characteristic of integrability of nonautonomous KP-modified KP equation and its qualitative studies: soliton, shock, periodic waves, breather, positons and soliton interactions. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09378-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11071-024-09378-1

Keywords

Search

Navigation