Existence of Classical Solutions for Broer-Kaup Equations
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Authors: D. BOURENI, S. G. GEORGIEV, A. KHELOUFI AND K. MEBARKI
DOI: 10.46793/KgJMat2501.125B
Abstract:
In this paper we investigate the Cauchy problem for one dimensional Broer-Kaup equations for existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.
Keywords:
Broer-Kaup equations, classical solution, fixed point, initial value problem.
References:
[1] C. L. Chen and S. Y. Lou, CTE solvability and exact solution to the Broer-Kaup system, Chin. Phys. Lett. 30(11) (2013), Article ID 110202, 4 pages. https://doi.org/10.1088/0256-307X/30/11/110202
[2] L. J. F. Broer, Approximate equations for long water waves, Appl. Sci. Res. 31 (1975), 377–395. https://doi.org/10.1007/BF00418048
[3] X. Q. Cao, S. C. Hou, Y. N. Guo, C. Z. Zhang and K. C. Peng, Variational Principle for (2+1)-dimensional Broer-Kaup equations with fractal derivatives, Fractals 28(7) (2020), Article ID 2050107, 13 pages. https://doi.org/10.1142/S0218348X20501078
[4] G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Differential Equations Math. Phys. 36 (2005), 81–134.
[5] S. Djebali and K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contractions, Topol. Methods Nonlinear Anal. 54(2A) (2019), 613–640. https://doi.org/10.12775/TMNA.2019.055
[6] M. F. El-Sabbagh and S. I. El-Ganaini, The He’s variational principle to the Broer-Kaup (BK) and Whitham Broer-Kaup (WBK) systems, International Mathematical Forum 7(43) (2012), 2131–2141.
[7] S. Guo, Y. Zhou and C. Zhao, The improved ()-expansion method and its applications to the Broer-Kaup equations and approximate long water wave equations, Appl. Math. Comput. 216(7) (2010), 1965–1971. https://doi.org/10.1016/j.amc.2010.03.026
[8] B. Jiang and Q.-S. Bi, Peaked periodic wave solutions to the Broer-Kaup equation, Commun. Theor. Phys. 67 (2017), 22–26. https://doi.org/10.1088/0253-6102/67/1/22
[9] D. J. Kaup, A higher order water wave equation and method for solving it, Progress of Theoretical Physics 54 (1975), 396–408. https://doi.org/10.1143/PTP.54.396
[10] S. Kumar, K. Singh and R. K. Gupta, Painlevé analysis, Lie symmetries and exact solutions for (2 + 1)-dimensional variable coefficients Broer-Kaup equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1529–1541. https://doi.org/10.1016/j.cnsns.2011.09.003
[11] B. A. Kupershmidt, Mathematics of dispersive water waves, Commn. Math. Phys. 99(1) (1985), 51–73. https://doi.org/10.1007/BF01466593
[12] S. Y. Lou and X. B. Hu, Broer-Kaup systems from Darboux transformation related symmetry constraints of Kadomtsev-Petviashvili equation, Commun. Theor. Phys. 29(1) (1998), 145–148. https://doi.org/10.1088/0253-6102/29/1/145
[13] Q. Meng, W. Li and B. He, Smooth and peaked solitary wave solutions of the Broer-Kaup system using the approach of dynamical system, Commun. Theor. Phys. 62(3) (2014), 308–314. https://doi.org/10.1088/0253-6102/62/3/03
[14] A. Neirameh, Solitary wave solutions of the BK equation and ALWW system by using the first integral method, Comput. Methods Differ. Equ. 1(2) (2013), 146–156.
[15] A. Polyanin and A. Manzhirov, Handbook of Integral Equations, CRC Press, London, 1998.
[16] J. Satsuma, K. Kajiwara, J. Matsukidaira and J. Hietarinta, Solutions of the Broer-Kaup system through its trilinear form, J. Phys. Soc. Jpn. 61(9) (1992), 3096–3102. https://doi.org/10.1143/JPSJ.61.3096
[17] S. Georgiev, A. Kheloufi and K. Mebarki, Classical solutions for the Korteweg-De Vries equation, New Trends Nonlinear Anal. Appl. (2022) (to appear).
[18] A. K. Svinin, Differential constraints for the Kaup-Broer system as a reduction of the 1D Toda lattice, Inverse Probl. 17(4) (2001), 1061–1066. https://doi.org/10.1088/0266-5611/17/4/332
[19] M. Wang, J. Zhang and X. Li, Application of the ()-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations, Appl. Math. Comput. 206 (2008), 321–326. https://doi.org/10.1016/j.amc.2008.08.045
[20] H. Wang, H. Chen, Z. Ouyang and F. Li, Lie symmetry analysis and conservation law for the equation arising from higher order Broer-Kaup equation, Open Math. 17 (2019), 1045–1054. https://doi.org/10.1515/math-2019-0080
[21] Z. Wen, New exact explicit nonlinear wave solutions for the Broer-Kaup equation, J. Appl. Math. 2014 (2014), Article ID 984791, 7 pages. https://doi.org/10.1155/2014/984791
[22] S. Xie, X. Hong and J. Lu, The bifurcation travelling waves of a generalized Broer-Kaup equation, J. Appl. Anal. Comput. 11(1) (2021), 210–226. https://doi.org/10.11948/20190268
[23] X.-P. Xin, Interaction solutions for (1+1)-dimensional higher-order Broer-Kaup system, Commun. Theor. Phys. 66 (2016), 479–482. https://doi.org/10.1088/0253-6102/66/5/479
[24] H.-M. Yin, B. Tian, J. Chai and X.-Y. Wu, Stochastic soliton solutions for the (2+1)-dimensional stochastic Broer-Kaup equations in a fluid or plasma, Appl. Math. Lett. 82 (2018), 126–131. https://doi.org/10.1016/j.aml.2017.12.005
[25] Q. Zeng and Y. Li, Application of the sub-ode method for the Broer-Kaup equation, Advanced Materials Research 760–762 (2013), 1655–1660. https://doi.org/10.4028/www.scientific.net/AMR.760-762.1655
[26] C.-L. Zheng and J.-X. Fei, Complex wave excitations in generalized Broer-Kaup system, Commun. Theor. Phys. 48(4) (2007), 657–661. https://doi.org/10.1088/0253-6102/48/4/018
[27] Zh. J. Zhou and Z. B. Li, A Darboux transformation and new exact solutions for Broer-Kaup system, Acta Phys. Sin. 52(2) (2003), 262–266. https://doi.org/10.7498/aps.52.262
[28] S. Zhu and J. Song, Residual symmetries, nth Bäcklund transformation and interaction solutions for (2 + 1)-dimensional generalized Broer-Kaup equations, Appl. Math. Lett. 83 (2018), 33–39. https://doi.org/10.1016/j.aml.2018.03.021
[29] J. Zhu and X. Wang, Broer-Kaup system revisit: Inelastic interaction and blowup solutions, J. Math. Anal. Appl. 496 (2021), Article ID 124794, 15 pages. https://doi.org/10.1016/j.jmaa.2020.124794