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In the Shallow Water: Auto-Bäcklund, Hetero-Bäcklund and Scaling Transformations via a (2+1)-Dimensional Generalized Broer-Kaup System

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Abstract

These days, watching the shallow water waves, people think about the nonlinear Broer-type models, e.g., a (2+1)-dimensional generalized Broer-Kaup system modeling, e.g., certain nonlinear long waves in the shallow water. For that system, with reference to, e.g., the wave height and wave horizontal velocity, this paper avails of symbolic computation to obtain (A) an auto-Bäcklund transformation with some solitons; (B) a group of the scaling transformations and (C) a group of the hetero-Bäcklund transformations, to a known linear partial differential equation, from that system. Results rely on the coefficients in that system

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Notes

  1. More fluid-dynamics investigations could be found, e.g., in Refs. [17,18,19,20,21,22,23,24,25,26].

  2. When \(\mathcal{A}=\mathcal{B}=\mathcal{C}=0\), \(\eta =\chi \), \(\tau =-\frac{1}{2} T\) and \(\chi =X\), System (2) can be reduced to System (1).

  3. Such a Bell-polynomial deduction could be seen, e.g., in Refs. [86,87,88].

References

  1. Rabie, W.B., Khalil, T.A., Badra, N., Ahmed, H.M., Mirzazadeh, M., Hashemi, M.S.: Soliton solutions and other solutions to the (4+1)-dimensional Davey-Stewartson-Kadomtsev-Petviashvili equation using modified extended mapping method. Qual. Theory Dyn. Syst. 23, 87 (2024)

    MathSciNet  Google Scholar 

  2. Agrawal, D., Abbas, S.: Existence of periodic solutions for a class of dynamic equations with multiple time varying delays on time scales. Qual. Theory Dyn. Syst. 23, 32 (2024)

    MathSciNet  Google Scholar 

  3. Gao, X.Y., Guo, Y.J., Shan, W.R.: On the oceanic/laky shallow-water dynamics through a Boussinesq-Burgers system. Qual. Theory Dyn. Syst. 23, 57 (2024)

    MathSciNet  Google Scholar 

  4. Feng, C.H., Tian, B., Yang, D.Y., Gao, X.T.: Bilinear form, bilinear Bäcklund transformations, breather and periodic-wave solutions for a (2+1)-dimensional shallow water equation with the time-dependent coefficients. Qual. Theory Dyn. Syst. 22, 147 (2023)

    Google Scholar 

  5. Shen, Y., Tian, B., Yang, D.Y., Zhou, T.Y.: Studies on a three-field lattice system: \(N\)-fold Darboux transformation, conservation laws and analytic solutions. Qual. Theory Dyn. Syst. 22, 74 (2023)

    MathSciNet  Google Scholar 

  6. Zhou, T.Y., Tian, B., Chen, Y.Q.: Elastic two-kink, breather, multiple periodic, hybrid and half-/local-periodic kink solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves. Qual. Theory Dyn. Syst. 22, 34 (2023)

    MathSciNet  Google Scholar 

  7. 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. 21, 104 (2022)

    MathSciNet  Google Scholar 

  8. Gao, X.Y., Guo, Y.J., Shan, W.R.: Theoretical investigations on a variable-coefficient generalized forced-perturbed Korteweg-de Vries-Burgers model for a dilated artery, blood vessel or circulatory system with experimental support. Commun. Theor. Phys. 75, 115006 (2023)

    MathSciNet  Google Scholar 

  9. Kumar, A., Hayatdavoodi, M.: Effect of currents on nonlinear waves in shallow water. Coast. Eng. 181, 104278 (2023)

    Google Scholar 

  10. Cheng, C.D., Tian, B., Shen, Y., Zhou, T.Y.: Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3+1)-dimensional extended shallow water wave equation. Phys. Fluids 35, 087123 (2023)

    Google Scholar 

  11. Gao, X.Y., Guo, Y.J., Shan, W.R.: Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: Hetero-Bäcklund transformations, bilinear forms and \(M\) solitons. Chaos Solitons Fract. 162, 112486 (2022)

    Google Scholar 

  12. 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. Heat Fluid Flow 32, 2282 (2022)

  13. Kassem, M.M., Rashed, A.S.: \(N\)-solitons and cuspon waves solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt equations via hidden symmetries of Lie optimal system. Chin. J. Phys. 57, 90 (2019)

    MathSciNet  Google Scholar 

  14. Yamgoué, S.B., Deffo, G.R., Pelap, F.B.: A new rational sine-Gordon expansion method and its application to nonlinear wave equations arising in mathematical physics. Eur. Phys. J. Plus 134, 380 (2019)

    Google Scholar 

  15. Nabelek, P.V., Zakharov, V.E.: Solutions to the Kaup-Broer system and its (2+1) dimensional integrable generalization via the dressing method. Phys. D 409, 132478 (2020)

    MathSciNet  Google Scholar 

  16. Gao, X.Y.: Auto-Bäcklund transformation with the solitons and similarity reductions for a generalized nonlinear shallow water wave equation. Qual. Theory Dyn. Syst. (2024) in press, https://doi.org/10.1007/s12346-024-01034-8

  17. Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: Pfaffian solutions and nonlinear waves of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics. Phys. Fluids 35, 025103 (2023)

    Google Scholar 

  18. Gao, X.Y.: Considering the wave processes in oceanography, acoustics and hydrodynamics by means of an extended coupled (2+1)-dimensional Burgers system. Chin. J. Phys. 86, 572 (2023). https://doi.org/10.1016/j.cjph.2023.10.051

    Article  MathSciNet  Google Scholar 

  19. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C.: \(N\)-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics. Nonlinear Dyn. 111, 16339 (2023)

    Google Scholar 

  20. 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 generalized nonlinear evolution system in a fluid or a plasma. Chin. J. Phys. 84, 343 (2023)

    MathSciNet  Google Scholar 

  21. Cheng, C.D., Tian, B., Zhou, T.Y., Shen, Y.: Wronskian solutions and Pfaffianization for a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma. Phys. Fluids 35, 037101 (2023)

    Google Scholar 

  22. Feng, C.H., Tian, B., Yang, D.Y., Gao, X.T.: Lump and hybrid solutions for a (3+1)-dimensional Boussinesq-type equation for the gravity waves over a water surface. Chin. J. Phys. 83, 515 (2023)

    MathSciNet  Google Scholar 

  23. Gao, X.Y.: Two-layer-liquid and lattice considerations through a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system. Appl. Math. Lett. 152, 109018 (2024). https://doi.org/10.1016/j.aml.2024.109018

    Article  MathSciNet  Google Scholar 

  24. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: Extended (2+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics: Solitons, breathers, lumps and interactions. Eur. Phys. J. Plus 138, 305 (2023)

    Google Scholar 

  25. Cheng, C.D., Tian, B., Ma, Y.X., Zhou, T.Y., Shen, Y.: Pfaffian, breather and hybrid solutions for a (2+1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. Phys. Fluids 34, 115132 (2022)

    Google Scholar 

  26. Chen, Y., Lü, X.: Wronskian solutions and linear superposition of rational solutions to B-type Kadomtsev-Petviashvili equation. Phys. Fluids 35, 106613 (2023)

    Google Scholar 

  27. Zhao, Z.L., Han, B.: On optimal system, exact solutions and conservation laws of the Broer-Kaup system. Eur. Phys. J. Plus 130, 223 (2015)

    Google Scholar 

  28. Cao, X.Q., Guo, Y.N., Hou, S.H., Zhang, C.Z., Peng, K.C.: Variational principles for two kinds of coupled nonlinear equations in shallow water. Symmetry-Basel 12, 850 (2020)

    Google Scholar 

  29. Zheng, C.L., Fei, J.X.: Complex wave excitations in generalized Broer-Kaup system. Commun. Theor. Phys. 48, 657 (2007)

    Google Scholar 

  30. Yomba, E., Peng, Y.Z.: Fission, fusion and annihilation in the interaction of localized structures for the (2+1)-dimensional generalized Broer-Kaup system. Chaos Solitons Fract. 28, 650 (2006)

    MathSciNet  Google Scholar 

  31. Lü, Z. S.: Infinite generation of soliton-like solutions for the (2+1)-dimensional generalized Broer-Kaup system. Nonlinear Anal.-Theor. 67, 2283 (2007)

  32. Huang, D.J., Zhang, H.Q.: New explicit exact solutions to (2+1)-dimensional generalized Broer-Kaup system. Commun. Theor. Phys. 43, 397 (2005)

    MathSciNet  Google Scholar 

  33. Ma, Z.Y.: Special conditional similarity reduction solutions for two nonlinear partial differential equations. Commun. Theor. Phys. 48, 199 (2007)

    MathSciNet  Google Scholar 

  34. Li, J.M., Ding, W., Tang, X.Y.: Symmetry and similarity solutions of a (2+1)-dimensional generalized Broer-Kaup system. Commun. Theor. Phys. 47, 1058 (2007)

    MathSciNet  Google Scholar 

  35. Bai, C.L., Bai, C.J., Zhao, H.: A new class of (2+1)-dimensional combined structures with completely elastic and non-elastic interactive properties. Z. Naturforsch. A 61, 53 (2006)

    Google Scholar 

  36. Lu, D.C., Hong, B.J.: New exact solutions for the (2+1)-dimensional generalized Broer-Kaup system. Appl. Math. Comput. 199, 572 (2008)

    MathSciNet  Google Scholar 

  37. Ma, S.H., Fang, J.P., Zhu, H.P.: Special soliton structures and the phenomena of fission and annihilation of solitons for the (2+1)-dimensional Broer-Kaup system with variable coefficients. Acta Phys. Sin. 56, 6777 (2007)

    MathSciNet  Google Scholar 

  38. Dai, C.Q., Cen, X., Wu, S.S.: Exotic localized structures based on a variable separation solution of the (2+1)-dimensional higher-order Broer-Kaup system. Nonlinear Anal. 10, 259 (2009)

    MathSciNet  Google Scholar 

  39. Wen, X.Y.: Construction of new exact rational form non-travelling wave solutions to the (2+1)-dimensional generalized Broer-Kaup system. Appl. Math. Comput. 217, 1367 (2010)

    MathSciNet  Google Scholar 

  40. Wang, H., Tian, Y.H.: Non-Lie symmetry groups and new exact solutions of a (2+1)-dimensional generalized Broer-Kaup system. Commun. Nonlinear Sci. Numer. Simul. 16, 3933 (2011)

    MathSciNet  Google Scholar 

  41. Ma, Z.Y.: Approximate soliton solutions for a (2+1)-dimensional Broer-Kaup system by He’s methods. Comput. Math. Appl. 58, 2410 (2009)

    MathSciNet  Google Scholar 

  42. Huang, D.J., Zhang, H.Q.: Variable-coefficient projective Riccati equation method and its application to a new (2+1)-dimensional simplified generalized Broer-Kaup system. Chaos Solitons Fract. 23, 601 (2005)

    MathSciNet  Google Scholar 

  43. Liu, Q., Zhu, J.M., Hong, B.H.: A modified variable-coefficient projective Riccati equation method and its application to (2+1)-dimensional simplified generalized Broer-Kaup system. Chaos Solitons Fract. 37, 1383 (2008)

    MathSciNet  Google Scholar 

  44. Ma, W.X., Zhou, Z.X.: Coupled integrable systems associated with a polynomial spectral problem and their Virasoro symmetry algebras. Prog. Theor. Phys. 96, 449 (1996)

    MathSciNet  Google Scholar 

  45. Ma, W.X.: Four-component integrable hierarchies and their Hamiltonian structures. Commun. Nonlinear Sci. Numer. Simul. 126, 107460 (2023)

    MathSciNet  Google Scholar 

  46. Ma, W.X.: AKNS type reduced integrable hierarchies with Hamiltonian formulations. Rom. J. Phys. 68, 116 (2023)

    Google Scholar 

  47. Zhang, S.L., Wu, B., Lou, S.Y.: Painlevé analysis and special solutions of generalized Broer-Kaup equations. Phys. Lett. A 300, 40 (2002)

    MathSciNet  Google Scholar 

  48. Li, X.N., Wei, G.M., Liu, Y.P., Liang, Y.Q., Meng, X.H.: Painlevé analysis and new analytic solutions for (1+1)-dimensional higher-order Broer-Kaup system with symbolic computation. Int. J. Mod. Phys. B 28, 1450067 (2014)

    Google Scholar 

  49. Alquran, M., Al-deiakeh, R.: Lie-Bäcklund symmetry generators and a variety of novel periodic-soliton solutions to the complex-mode of modified Korteweg-de Vries equation. Qual. Theory Dyn. Syst. 23, 95 (2024)

    Google Scholar 

  50. Yasmin, H., Alshehry, A.S., Ganie, A.H., Mahnashi, A.M., Shah, R.: Perturbed Gerdjikov-Ivanov equation: Soliton solutions via Bäcklund transformation. Optik 298, 171576 (2024)

    Google Scholar 

  51. Carillo, S., Chichurin, A., Filipuk, G., Zullo, F.: Schwarzian derivative, Painlevé XXV-Ermakov equation, and Bäcklund transformations. Math. Nachr. 297, 83 (2024)

    MathSciNet  Google Scholar 

  52. Rahioui, M., Kinani, E.H., Ouhadan, A.: Nonlocal residual symmetries, \(N\)-th Bäcklund transformations and exact interaction solutions for a generalized Broer-Kaup-Kupershmidt system. Z. Angew. Math. Phys. 75, 37 (2024)

    Google Scholar 

  53. Singh, S., Saha Ray, S.: Bilinear representation, bilinear Bäcklund transformation, Lax pair and analytical solutions for the fourth-order potential Ito equation describing water waves via Bell polynomials. J. Math. Anal. Appl. 530, 127695 (2024)

    Google Scholar 

  54. Mann, N., Rani, S., Kumar, S., Kumar, R.: Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation. Math. Comput. Simulat. 219, 473 (2024)

    MathSciNet  Google Scholar 

  55. Niwas, M., Kumar, S.: Multi-peakons, lumps, and other solitons solutions for the (2+1)-dimensional generalized Benjamin-Ono equation: An inverse-expansion method and real-world applications. Nonlinear Dyn. 111, 22499 (2023)

    Google Scholar 

  56. Kumar, S., Dhiman, S.K.: Exploring cone-shaped solitons, breather, and lump-forms solutions using the Lie symmetry method and unified approach to a coupled breaking soliton model. Phys. Scr. 99, 025243 (2024)

    Google Scholar 

  57. Kumar, S., Mohan, B.: A novel analysis of Cole-Hopf transformations in different dimensions, solitons, and rogue waves for a (2+1)-dimensional shallow water wave equation of ion-acoustic waves in plasmas. Phys. Fluids 35, 127128 (2023)

    Google Scholar 

  58. Kumar, S., Hamid, I., Abdou, M.A.: Dynamic frameworks of optical soliton solutions and soliton-like formations to Schrödinger-Hirota equation with parabolic law non-linearity using a highly efficient approach. Opt. Quant. Electron. 55, 1261 (2023)

    Google Scholar 

  59. Kumar, S., Niwas, M.: Analyzing multi-peak and lump solutions of the variable-coefficient Boiti-Leon-Manna-Pempinelli equation: a comparative study of the Lie classical method and unified method with applications. Nonlinear Dyn. 111, 22457 (2023)

    Google Scholar 

  60. Kumar, S., Mann, N., Kharbanda, H., Inc, M.: Dynamical behavior of analytical soliton solutions, bifurcation analysis, and quasi-periodic solution to the (2+1)-dimensional Konopelchenko-Dubrovsky (KD) system. Anal. Math. Phys. 13, 40 (2023)

    MathSciNet  Google Scholar 

  61. Rani, S., Kumar, S., Mann, N.: On the dynamics of optical soliton solutions, modulation stability, and various wave structures of a (2+1)-dimensional complex modified Korteweg-de-Vries equation using two integration mathematical methods. Opt. Quant. Electron. 55, 731 (2023)

    Google Scholar 

  62. Hamid, I., Kumar, S.: Symbolic computation and novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach. Opt. Quant. Electron. 55, 755 (2023)

    Google Scholar 

  63. Abdou, M.A., Ouahid, L., Kumar, S.: Plenteous specific analytical solutions for new extended deoxyribonucleic acid (DNA) model arising in mathematical biology. Mod. Phys. Lett. B 37, 2350173 (2023)

    MathSciNet  Google Scholar 

  64. Kumar, S., Ma, W.X., Dhiman, S.K., Chauhan, A.: Lie group analysis with the optimal system, generalized invariant solutions, and an enormous variety of different wave profiles for the higher-dimensional modified dispersive water wave system of equations. Eur. Phys. J. Plus 138, 434 (2023)

    Google Scholar 

  65. Wu, X.H., Gao, Y.T.: Generalized Darboux transformation and solitons for the Ablowitz-Ladik equation in an electrical lattice. Appl. Math. Lett. 137, 108476 (2023)

    MathSciNet  Google Scholar 

  66. Shen, Y., Tian, B., Zhou, T.Y., Cheng, C.D.: Multi-pole solitons in an inhomogeneous multi-component nonlinear optical medium. Chaos Solitons Fract. 171, 113497 (2023)

    MathSciNet  Google Scholar 

  67. Zhou, T.Y., Tian, B., Shen, Y., Gao, X.T.: Bilinear form, bilinear auto-Bäcklund transformation, soliton and half-periodic kink solutions on the non-zero background of a (3+1)-dimensional time-dependent-coefficient Boiti-Leon-Manna-Pempinelli equation. Wave Motion 121, 103180 (2023)

    Google Scholar 

  68. Wu, X.H., Gao, Y.T., Yu, X., Li, L.Q., Ding, C.C.: Vector breathers, rogue and breather-rogue waves for a coupled mixed derivative nonlinear Schrödinger system in an optical fiber. Nonlinear Dyn. 111, 5641 (2023)

    Google Scholar 

  69. Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: \(N\)-soliton, \(M\)th-order breather, \(H\)th-order lump, and hybrid solutions of an extended (3+1)-dimensional Kadomtsev-Petviashvili equation. Nonlinear Dyn. 111, 10407 (2023)

    Google Scholar 

  70. Chen, S.S., Tian, B., Tian, H.Y., Hu, C.C.: Riemann-Hilbert approach, dark solitons and double-pole solutions for Lakshmanan-Porsezian-Daniel equation in an optical fiber, a ferromagnetic spin or a protein. Z. Angew. Math. Mech. e202200417 (2024). https://doi.org/10.1002/zamm.202200417

  71. Wu, X.H., Gao, Y.T., Yu, X., Liu, F.Y.: Generalized Darboux transformation and solitons for a Kraenkel-Manna-Merle system in a ferromagnetic saturator. Nonliner Dyn. 111, 14421 (2023)

    Google Scholar 

  72. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: N-fold Darboux transformation and solitonic interactions for the Kraenkel-Manna-Merle system in a saturated ferromagnetic material. Nonlinear Dyn. 111, 2641 (2023)

    Google Scholar 

  73. Shen, Y., Tian, B., Yang D.Y., Zhou, T.Y.: Hybrid relativistic and modified Toda lattice-type system: equivalent form, \(N\)-fold Darboux transformation and analytic solutions. Eur. Phys. J. Plus 138, 744 (2023)

  74. Gao, X.T., Tian, B.: Similarity reductions on a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system describing certain electromagnetic waves in a thin film. Int. J. Theor. Phys. 63, 99 (2024)

    MathSciNet  Google Scholar 

  75. Yin, Y.H., Lü, X., Jiang, R., Jia, B., Gao, Z.: Kinetic analysis and numerical tests of an adaptive car-following model for real-time traffic in ITS. Phys. A 635, 129494 (2024)

    MathSciNet  Google Scholar 

  76. Peng, X., Zhao, Y.W., Lü, X.: Data-driven solitons and parameter discovery to the (2+1)-dimensional NLSE in optical fiber communications. Nonlinear Dyn. 112, 1291 (2024)

    Google Scholar 

  77. Chen, S.J., Yin, Y.H., Lü, X.: Elastic collision between one lump wave and multiple stripe waves of nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simul. 130, 107205 (2024)

    MathSciNet  Google Scholar 

  78. Cao, F., Lü, X., Zhou, Y.X., Cheng, X.Y.: Modified SEIAR infectious disease model for Omicron variants spread dynamics. Nonlinear Dyn. 111, 14597 (2023)

    Google Scholar 

  79. Zhou, T.Y., Tian, B., Shen, Y., Gao, X.T.: Auto-Bäcklund transformations and soliton solutions on the nonzero background for a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in a fluid. Nonlinear Dyn. 111, 8647 (2023)

    Google Scholar 

  80. Gao, X.Y.: Oceanic shallow-water investigations on a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system. Phys. Fluids 35, 127106 (2023). https://doi.org/10.1063/5.0170506

  81. Zhou, T.Y., Tian, B.: Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber. Appl. Math. Lett. 133, 108280 (2022)

    Google Scholar 

  82. Gao, X.Y., Guo, Y.J., Shan, W.R.: Ultra-short optical pulses in a birefringent fiber via a generalized coupled Hirota system with the singular manifold and symbolic computation. Appl. Math. Lett. 140, 108546 (2023)

    MathSciNet  Google Scholar 

  83. Ma, W.X.: Lump waves in a spatial symmetric nonlinear dispersive wave model in (2+1)-dimensions. Mathematics 11, 4664 (2023)

    Google Scholar 

  84. Gao, X.Y., Guo, Y.J., Shan, W.R.: Bilinear forms through the binary Bell polynomials, \(N\) solitons and Bäcklund transformations of the Boussinesq-Burgers system for the shallow water. Commun. Theor. Phys. 72, 095002 (2020)

    Google Scholar 

  85. Gao, X.Y., Guo, Y.J., Shan, W.R., Zhou, T.Y., Wang, M., Yang, D.Y.: In the atmosphere and oceanic fluids: Scaling transformations, bilinear forms, Bäcklund transformations and solitons for a generalized variable-coefficient Korteweg-de Vries-modified Korteweg-de Vries equation. China Ocean Eng. 35, 518 (2021)

    Google Scholar 

  86. Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258 (1934)

  87. Lambert, F., Loris, I., Springael, J., Willox, R.: On a direct bilinearization method: Kaup’s higher-order water wave equation as a modified nonlocal Boussinesq equation. J. Phys. A 27, 5325 (1994)

    MathSciNet  Google Scholar 

  88. Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys: Conf. Ser. 411, 012021 (2013)

    Google Scholar 

  89. “Partial differential equation”, https://encyclopedia.thefreedictionary.com/Partial+differential+equation (2024)

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Gao, XY. In the Shallow Water: Auto-Bäcklund, Hetero-Bäcklund and Scaling Transformations via a (2+1)-Dimensional Generalized Broer-Kaup System. Qual. Theory Dyn. Syst. 23, 184 (2024). https://doi.org/10.1007/s12346-024-01025-9

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