[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Optimal Classifications, New Doubly Periodic, Multiple Soliton Solutions, Waves Dynamics and Conserved Quantities of a New Integrable (1+1)-D Boussinesq Equation with Dissipative Phase in Fluid Dynamics

Journal of Applied Nonlinear Dynamics 15(4) (2026) 847--883 | DOI:10.5890/JAND.2026.12.005

Oke Davies Adeyemo

Material Science, Innovation and Modelling Research Focus Area, Department of Mathematical Sciences,

North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, Republic of South Africa

Download Full Text PDF

 

Abstract

The theory of natural evolution and computations has contributed to many advancements in science and engineering. Consequently, this article explores the analytical examinations of a new integrable (1+1)-dimensional Boussinesq equation recently formulated in the literature which is applicable in optics, fluid dynamics, ocean science and other nonlinear sciences and engineering. Lie group theory in differential equations is then applied to identify point symmetries within the model, enabling the derivation of nonlinear ordinary differential equations through symmetry reductions. In addition, direct integration is invoked to secure some soliton solutions, such as bright, periodic, and singular. More importantly, the Jacobi elliptic function approach is further engaged to secure abundant general exact travelling wave solutions in the structures of periodic as well as singular soliton solutions to the model. This technique enables the attainment of various exact soliton solutions, including topological and non-topological soliton solutions (both complex and non-complex). Additionally, general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model, are also secured. In the same vein, a power series approach is utilized to solve part of the difficult nonlinear ordinary differential equations obtained. Besides, by taking some essential limits of part of the solutions, one obtains various soliton results, including triangular solutions of the understudy model. These are in the form of hyperbolic and trigonometric functions, achieved with regards to the secant, tangent, and cotangent functions. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the understudy integrable Boussinesq equation. Conclusively, the study further produces conserved quantities of note, such as energy, mass, and momentum, which are secured through the use of Ibragimov's theorem, as well as the multiplier approach.

References

  1. [1]  Adeyemo, O.D., Khalique, C.M., Gasimov, Y.S., and Villecco, F. (2023), Variational and non-variational approaches with Lie algebra of a generalized (3+1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation in engineering and physics, Alexandria Engineering Journal, 63, 17-43.
  2. [2]  Adeyemo, O.D., Motsepa, T., and Khalique, C.M. (2022), A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences, Alexandria Engineering Journal, 61, 185-194.
  3. [3]  Khalique, C.M. and Adeyemo, O.D. (2020), A study of (3+1)-dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation via Lie symmetry approach, Results in Physics, 18, 103197.
  4. [4]  Du, X.X., Tian, B., Qu, Q.X., Yuan, Y.Q., and Zhao, X.H. (2020), Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos, Solitons & Fractals, 134, 109709.
  5. [5]  Zhang, C.R., Tian, B., Qu, Q.X., Liu, L., and Tian, H.Y. (2020), Vector bright solitons and their interactions of the couple Fokas–Lenells system in a birefringent optical fiber, Zeitschrift für Angewandte Mathematik und Physik, 71, 1-19.
  6. [6]  Gao, X.Y., Guo, Y.J., and Shan, W.R. (2020), Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto-and non-auto-Bäcklund transformations, Applied Mathematics Letters, 104, 106170.
  7. [7]  Adeyemo, O.D., Zhang, L., and Khalique, C.M. (2022), Optimal solutions of Lie subalgebra, dynamical system, travelling wave solutions and conserved currents of (3+1)-dimensional generalized Zakharov–Kuznetsov equation type I, European Physical Journal Plus, 137, 954. https://doi.org/10.1140/epjp/s13360-022-03100-z
  8. [8]  Adeyemo, O.D. and Khalique, C.M. (2023), Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electromagnetism and complex media, Alexandria Engineering Journal, 73, 751-769.
  9. [9]  Adeyemo, O.D. and Khalique, C.M. (2023), An optimal system of Lie subalgebras and group-invariant solutions with conserved currents of a (3+1)-D fifth-order nonlinear model with applications in electrical electronics, chemical engineering and pharmacy, Journal of Nonlinear Mathematical Physics, 30, 843-916. https://doi.org/10.1007/s44198-022-00101-5
  10. [10]  Al Khawajaa, U., Eleuchb, H., and Bahloulid, H. (2019), Analytical analysis of soliton propagation in microcavity wires, Results in Physics, 12, 471-474.
  11. [11]  Adeyemo, O.D., Zhang, L., and Khalique, C.M. (2022), Bifurcation theory, Lie group-invariant solutions of subalgebras and conservation laws of a generalized (2+1)-dimensional BK equation Type II in plasma physics and fluid mechanics, Mathematics, 10, 2391.
  12. [12]  Wazwaz, A.M. (2017), Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation, Open Engineering, 7, 169-174.
  13. [13]  Adeyemo, O.D. and Khalique, C.M. (2023), Lie group theory, stability analysis with dispersion property, new soliton solutions and conserved quantities of 3D generalized nonlinear wave equation in liquid containing gas bubbles with applications in mechanics of fluids, biomedical sciences and cell biology, Communications in Nonlinear Science and Numerical Simulation, 123, 107261.
  14. [14]  Ablowitz, M.J. and Clarkson, P.A. (1991), Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, UK.
  15. [15]  Adeyemo, O.D. (2024), Applications of cnoidal and snoidal wave solutions via an optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering, Journal of Ocean Engineering and Science, 9, 126-153. https://doi.org/10.1016/j.joes.2022.04.012
  16. [16]  Ovsiannikov, L.V. (1982), Group analysis of differential equations, Academic Press, New York, USA.
  17. [17]  Olver, P.J. (1993), Applications of Lie groups to differential equations, second ed., Springer-Verlag, Berlin, Germany.
  18. [18]  Kudryashov, N.A. and Loguinova, N.B. (2008), Extended simplest equation method for nonlinear differential equations, Applied Mathematics and Computation, 205, 396-402.
  19. [19]  Vitanov, N.K. (2010), Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity, Communications in Nonlinear Science and Numerical Simulation, 15, 2050-2060.
  20. [20]  Chen, Y. and Yan, Z. (2005), New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method, Chaos, Solitons & Fractals, 26, 399-406.
  21. [21]  Li, L., Duan, C., and Yu, F. (2019), An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (mKdV) equation, Physics Letters A, 383, 1578-1582.
  22. [22]  Wang, M., Li, X., and Zhang, J. (2005), The (G'/G)- expansion method and travelling wave solutions for linear evolution equations in mathematical physics, Physics Letters A, 24, 1257-1268.
  23. [23]  Feng, L., Tian, S., Zhang, T., and Zhou, J. (2017), Lie symmetries, conservation laws and analytical solutions for two-component integrable equations, Chinese Journal of Physics, 55, 996-1010.
  24. [24]  Zhang, Y., Ye, R., and Ma, W.X. (2020), Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations, Mathematical Methods in the Applied Sciences, 43, 613-627.
  25. [25]  Weiss, J., Tabor, M., and Carnevale, G. (1985), The Painl{e}vé property and a partial differential equations with an essential singularity, Physics Letters A, 109, 205-208.
  26. [26]  Zhang, L. and Khalique, C.M. (2018), Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete and Continuous Dynamical Systems - Series S, 11, 777-790.
  27. [27]  Chun, C. and Sakthivel, R. (2010), Homotopy perturbation technique for solving two point boundary value problems-comparison with other methods, Computer Physics Communications, 181, 1021-1024.
  28. [28]  Darvishi, M.T. and Najafi, M. (2011), A modification of extended homoclinic test approach to solve the (3+1)-dimensional potential-YTSF equation, Chinese Physics Letters, 28, 040202.
  29. [29]  Wazwaz, A.M. (2007), Traveling wave solution to (2+1)-dimensional nonlinear evolution equations, Journal of Natural Sciences and Mathematics, 1, 1-13.
  30. [30]  Wazwaz, A.M. (2002), Partial differential equations, CRC Press, Boca Raton, Florida, USA.
  31. [31]  Zhou, Y., Wang, M., and Wang, Y. (2003), Periodic wave solutions to a coupled KdV equations with variable coefficients, Physics Letters A, 308, 31-36.
  32. [32]  Gu, C.H. (1990), Soliton theory and its application, Zhejiang Science and Technology Press, Zhejiang, China.
  33. [33]  Salas, A.H. and Gomez, C.A. (2010), Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation, Mathematical Problems in Engineering, 2010, 2010.
  34. [34]  Zeng, X. and Wang, D.S. (2009), A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation, Applied Mathematics and Computation, 212, 296-304.
  35. [35]  Date, M., Jimbo, M., Kashiwara, M., and Miwa, T. (1981), Operator apporach of the Kadomtsev-Petviashvili equation - transformation groups for soliton equations III, Journal of the Physical Society of Japan, 50, 3806-3812.
  36. [36]  Kuo, C.K. and Ma, W.X. (2020), An effective approach to constructing novel KP-like equations, Waves in Random and Complex Media, 32, 629-640.
  37. [37]  Ma, W.X. and Fan, E. (2011), Linear superposition principle applying to Hirota bilinear equations, Computers & Mathematics with Applications, 61, 950-959.
  38. [38]  Wazwaz, A.M. (2012), Multiple-soliton solutions for a (3+1)-dimensional generalized KP equation, Communications in Nonlinear Science and Numerical Simulation, 17, 491-495.
  39. [39]  Ma, W.X. (2015), Lump solutions to the Kadomtsev-Petviashvili equation, Physics Letters A, 379, 1975-1978.
  40. [40]  Zhao, Z. and Han, B. (2017), Lump solutions of a (3+1)-dimensional B-type KP equation and its dimensionally reduced equations, Analysis and Mathematical Physics, 9, 119-130.
  41. [41]  Khalique, C.M., Adeyemo, O.D., and Mohapi, I. (2024), Exact solutions and conservation laws of a new fourth-order nonlinear (3+1)-dimensional Kadomtsev-Petviashvili-like equation, Applied Mathematics & Information Sciences, 18, 1-25.
  42. [42]  Boussinesq, J.V. (1877), Essai sur la théorie des eaux courantes, Mémoires Présentés par divers savants à l'Académie des Sciences de l'Institut National de France, XXIII, 55-108.
  43. [43]  Wazwaz, A.M. and Kaur, L. (2019), New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions, Nonlinear Dynamics, 97, 83-94.
  44. [44]  McKean, H.P. (1978), Boussinesq's equation as a Hamiltonian system, Advances in Mathematics Supplementary Studies, 3, 217-226.
  45. [45]  McKean, H.P. (1981), Boussinesq's equation on the circle, Communications on Pure and Applied Mathematics, 34, 599-691.
  46. [46]  Wazwaz, A.M. (2012), Two kinds of multiple wave solutions for the potential YTSF equation and a potential YTSF-type equation, Journal of Applied Nonlinear Dynamics, 1, 51-58.
  47. [47]  Leblond, H. and Mihalache, D. (2013), Models of few optical cycle solitons beyond the slowly varying envelope approximation, Physics Reports, 523, 61-126.
  48. [48]  Su, T. (2017), Explicit solutions for a modified 2+1-dimensional coupled Burgers equation by using Darboux transformation, Applied Mathematics Letters, 69, 15-21.
  49. [49]  Xu, S.L., Zhou, Q., Zhao, D., Beli{c}, M.R., and Zhao, Y. (2020), Spatiotemporal solitons in cold Rydberg atomic gases with Bessel optical lattices, Applied Mathematics Letters, 106, 106230.
  50. [50]  Baleanu, D. and Fernandez, A. (2019), On fractional operators and their classifications, Mathematics, 7, 830.
  51. [51]  Adel, M., Baleanu, D., Sadiya, U., Arefin, M.A., and Uddin, M.H. (2022), Inelastic soliton wave solutions with different geometrical structures to fractional order nonlinear evolution equations, Results in Physics, 38, 105661.
  52. [52]  Alhejaili, W., Tiofack, C.G.L., Mohamadou, A., and El-Tantawy, S.A. (2023), Multidimensional dust-acoustic rogue waves in electron-depleted complex magnetoplasmas, Physics of Fluids, 35, 063102.
  53. [53]  Mouhammadoul, B.B., Alrowaily, A.W., Ismaeel, S., and El-Tantawy, S.A. (2023), On the super positron-acoustic rogue waves in q-nonextensive magnetoplasmas, Physics of Fluids, 35, 054109.
  54. [54]  Khuri, S.A. (2005), Soliton and periodic solutions for higher order wave equations of KdV type (I), Chaos, Solitons & Fractals, 26, 25-32.
  55. [55]  Khuri, S.A. (2008), Exact solutions for a class of nonlinear evolution equations: a unified ansätze approach, Chaos, Solitons & Fractals, 36, 1181-1188.
  56. [56]  Wazwaz, A.M. (2013), A variety of distinct kinds of multiple soliton solutions for a (3+1)‐dimensional nonlinear evolution equation, Mathematical Methods in the Applied Sciences, 36, 349-357.
  57. [57]  Wazwaz, A.M. (2018), Multiple complex and multiple real soliton solutions for the integrable sine-Gordon equation, Optik, 172, 622-627.
  58. [58]  Clarkson, P.A. and Kruskal, M.D. (1989), New similarity reductions of the Boussinesq equation, Journal of Mathematical Physics, 30, 2201-2213.
  59. [59]  Cao, Y., He, D., and Mihalache, D. (2018), Families of exact solutions of a new extended (2+1)-dimensional Boussinesq equation, Nonlinear Dynamics, 91, 2593-2605.
  60. [60]  Hirota, R. and Ito, M. (1983), Resonance of solitons in one dimension, Journal of the Physical Society of Japan, 52, 744-748.
  61. [61]  Yin, Y.H., Ma, W.X., Liu, J.G., and Lü, X. (2018), Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction, Computers & Mathematics with Applications, 76, 1275-1283.
  62. [62]  Gao, L.N., Zi, Y.Y., Yin, Y.H., Ma, W.X., and Lü, X. (2017), Bäcklund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation, Nonlinear Dynamics, 89, 2233-2240.
  63. [63]  Lü, X. and Ma, W.X. (2016), Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation, Nonlinear Dynamics, 85, 1217-1222.
  64. [64]  Lü, X., Chen, S.T., and Ma, W.X. (2016), Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation, Nonlinear Dynamics, 86, 523-534.
  65. [65]  Lü, X., Ma, W.X., Yu, J., and Khalique, C.M. (2016), Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation, Communications in Nonlinear Science and Numerical Simulation, 31, 40-46.
  66. [66]  Clarkson, P.A. and Dowie, E. (2017), Rational solutions of the Boussinesq equation and applications to rogue waves, Transactions of Mathematics and its Applications, 1.
  67. [67]  Wazwaz, A.M. and Kaur, L. (2019), New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions, Nonlinear Dynamics, 97, 83-94.
  68. [68]  Lawden, D.F. (1989), Elliptic functions and applications, Springer-Verlag, 8.
  69. [69]  Chen, Y. and Yan, Z. (2006), The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos, Solitons & Fractals, 29, 948-964.
  70. [70]  Adeyemo, O.D. and Khalique, C.M. (2023), An optimal system of Lie subalgebras and group-invariant solutions with conserved currents of a (3+1)-D fifth-order nonlinear model with applications in electrical electronics, chemical engineering and pharmacy, Journal of Nonlinear Mathematical Physics, 30, 843-916. https://doi.org/10.1007/s44198-022-00101-5
  71. [71]  Huai-Tang, C. and Hong-Qing, Z. (2004), New double periodic and multiple soliton solutions of the generalized (2+ 1)-dimensional Boussinesq equation, Chaos, Solitons & Fractals, 20, 765-769.
  72. [72]  He, J.H. (2012), Comment on variational iteration method for fractional calculus using he's polynomials, Abstract and Applied Analysis, 2012.
  73. [73]  Zhou, Y., Ma, W., and Li, Z. (1996), Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216, 67-75.
  74. [74]  Adeyemo, O.D., Zhang, L., and Khalique, C.M. (2022), Optimal solutions of Lie subalgebra, dynamical system, travelling wave solutions and conserved currents of (3+1)-dimensional generalized Zakharov-Kuznetsov equation type I, European Physical Journal Plus, 137, 954. https://doi.org/10.1140/epjp/s13360-022-03100-z
  75. [75]  Adeyemo, O.D. and Khalique, C.M. (2024), Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics, Journal of Taibah University for Science, 18, 2331984.
  76. [76]  Khalique, C.M. and Adeyemo, O.D. (2020), Closed-form solutions and conserved vectors of a generalized (3+1)-dimensional breaking soliton equation of engineering and nonlinear science, Mathematics, 8, 1692.
  77. [77]  Ibragimov, N.H. (2007), A new conservation theorem, Journal of Mathematical Analysis and Applications, 333, 311-328.
  78. [78]  Raza, N., Gandarias, M.L., and Basendwah, G.A. (2024), Symmetry reductions and conservation laws of a modified-mixed KdV equation: exploring new interaction solutions, AIMS Mathematics, 9, 10289-10303.
  79. [79]  Adeyemo, O.D. and Khalique, C.M. (2024), Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering, Open Physics, 22, 20230155.

Copyright © 2011-2026 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates