Abstract
In this work, a generalized (\(2+1\))-dimensional variable-coefficient shallow water wave equation is investigated, which describes the interaction between Riemann waves propagating along the y-axis and long waves propagating along the x-axis in a fluid. Based on the Hirota bilinear form and three-wave method, the breather wave solution is obtained. The interaction solutions between lump wave and periodic wave are presented. The interaction solutions between lump wave and solitary waves are also studied. None of these obtained solutions have been found in other literature.
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References
Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 1975–8 (2015)
Zhang, R.F., Li, M.C., Al-Mosharea, E., Zheng, F.C., Bilige, S.: Rogue waves, classical lump solutions and generalized lump solutions for Sawada–Kotera-like equation. Int. J. Mod. Phys. B 36(5), 2250044 (2022)
Rao, J., Kanna, T., Dumitru, M., He, J.S.: Resonant collision of lumps with homoclinic orbits in the two-dimensional multi-component long-wave-short-wave resonance interaction systems. Physica D 439, 133281 (2022)
Chen, S.J., Lü, X.: Lump and lump-multi-kink solutions in the (\(3+1\))-dimensions. Commun. Nonlinear Sci. 109, 106103 (2022)
Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: Shallow-water-wave studies on a (\(2+1\))-dimensional Hirota–Satsuma–Ito system: X-type soliton, resonant Y-type soliton and hybrid solutions. Chaos Soliton Fract. 157, 111861 (2022)
Behzad, G.: Employing Hirota’s bilinear form to find novel lump waves solutions to an important nonlinear model in fluid mechanics. Results Phys. 29, 104689 (2021)
Liu, J.G., Zhu, W.H., Osman, M.S., Ma, W.H.: An explicit plethora of different classes of interactive lump solutions for an extension form of 3D-Jimbo-Miwa model. Eur. Phys. J. Plus 135(5), 412 (2020)
Akinyemi, L., Morazara, E.: Integrability, multi-solitons, breathers, lumps and wave interactions for generalized extended Kadomtsev–Petviashvili equation. Nonlinear Dyn. 111, 4683–4707 (2023)
Liu, J.G., Eslami, M., Rezazadeh, H., Mirzazadeh, M.: Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation. Nonlinear Dyn. 95(2), 1027–33 (2019)
Liu, J.G., Ye, Q.: Stripe solitons and lump solutions for a generalized Kadomtsev–Petviashvili equation with variable coefficients in fluid mechanics. Nonlinear Dyn. 96, 23–9 (2019)
Zeng, S., Liu, Y., Chen, X., Zhang, W.X.: Various breathers, Lumps, line solitons and their interaction solutions for the (\(2+1\))-dimensional variable-coefficient Sawad–Kotera equation. Results Phys. 42, 105992 (2022)
Yan, X., Liu, J., Xin, X.: Soliton solutions and lump-type solutions to the (\(2+1\))-dimensional Kadomtsev–Petviashvili equation with variable coefficient. Phys. Lett. A 457, 128574 (2023)
Hu, Z., Wang, F., Zhao, Y., Lan, Z., Li, M.: Nonautonomous lump waves of a (\(3+1\))-dimensional Kudryashov–Sinelshchikov equation with variable coefficients in bubbly liquids. Nonlinear Dyn. 104(4), 4367–78 (2021)
Liu, J.G., Xiong, W.P.: Multi-wave, breather wave and lump solutions of the Boiti–Leon–Manna–Pempinelli equation with variable coefficients. Results Phys. 19, 103532 (2020)
Li, Q., Shan, W., Wang, P., Cui, H.: Breather, lump and N-soliton wave solutions of the (\(2+1\))-dimensional coupled nonlinear partial differential equation with variable coefficients. Commun. Nonlinear Sci. 106, 106098 (2022)
Lan, Z.Z., Gao, Y.T., Yang, J.W., et al.: Solitons, Bäcklund transformation, lax pair, and infinitely many conservation law for a (\(2+1\))-dimensional generalised variable-coefficient shallow water wave equation. Z Naturforsch A. 71, 69 (2016)
Yun, H.W., Yong, C.: Integrability of an extended (\(2+1\))-dimensional shallow water wave equation with Bell polynomials. Chin. Phys. B 22, 050509 (2013)
Akinyemi, L.: Shallow ocean soliton and localized waves in extended (\(2+1\))-dimensional nonlinear evolution equations. Phys. Lett. A 463(5), 128668 (2023)
Dikwa, J., Houwe, A., Abbagari, S., Akinyemi, L., Inc, M.: Modulated waves patterns in the photovoltaic photorefractive crystal. Opt. Quant. Electron. 54, 842 (2022)
Zhao, Y.H., Mathanaranjan, T., Rezazadeh, H., Akinyemi, L., Inc, M.: New solitary wave solutions and stability analysis for the generalized -dimensional nonlinear wave equation in liquid with gas bubbles. Results Phys. 43, 106083 (2022)
Abbagari, S., Houwe, A., Akinyemi, L., Inc, M., Bouetou, T.: Discrete modulation instability and localized modes in chiral molecular chains with first- and third-neighbor interactions. Phys. Scr. 98, 025210 (2023)
Funding
Project supported by National Natural Science Foundation of China (Grant No. 12161048), Doctoral Research Foundation of Jiangxi University of Chinese Medicine (Grant No. 2021WBZR007) and Jiangxi University of Chinese Medicine Science and Technology Innovation Team Development Program (Grant No. CXTD22015).
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Liu, JG., Zhu, WH. & Wu, YK. New breather wave and interaction solutions of the generalized (\(2+1\))-dimensional variable-coefficient shallow water wave equation. Nonlinear Dyn 111, 16441–16447 (2023). https://doi.org/10.1007/s11071-023-08710-5
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DOI: https://doi.org/10.1007/s11071-023-08710-5