×

Studies of wave interaction of high-order Korteweg-de Vries equation by means of the homotopy strategy and neural network prediction. (English) Zbl 07412676

Summary: The shallow water wave interaction governed by the fifth-order Korteweg-de Vries (KdV) equation is investigated. The novel periodic wave interaction solutions are captured, which seems to be overlooked previously. The Fourier spectrum is employed to analyze the energy transfer process between two traveling waves. The rhodonea curve is given to present the rationality and irrationality of wavenumbers. Several neural networks are used to predict wave interaction solutions. And their performance is verified by several evaluation indexes.

MSC:

81-XX Quantum theory
82-XX Statistical mechanics, structure of matter
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liu, X.; Bai, C., Exact solutions of some fifth-order nonlinear equations, Appl. Math. J. Chin. Univ. Ser. B, 15, 1, 28-32 (2000) · Zbl 0954.35143
[2] Lax, P. D., Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21, 467-490 (1968) · Zbl 0162.41103
[3] Caudrey, P. J.; Dodd, R. K.; Gibbon, J. D., A new hierarchy of Korteweg-De Vries equations, Proc. R. Soc. Lond. A, 351, 407-422 (1976) · Zbl 0346.35024
[4] Conte, R.; Mussette, M., Link between solitary waves and projective Riccati equations, J. Phys. A, Math. Gen., 25, 5609-5623 (1992) · Zbl 0782.35065
[5] Li, Z. B.; Pan, S. Q., Exact solitary wave and soliton solutions of the generalized fifth order KdV equation, Acta Phys. Sin., 50, 3, 402-404 (2001), (in Chinese) · Zbl 1202.35235
[6] Li, Z. B.; Pan, S. Q., Exact solitary wave solution for nonlinear wave equations using symbolic computation, Acta Math. Sci., 17, 1, 81-89 (1997) · Zbl 0893.65066
[7] Kaya, D., An explicit and numerical solutions of some fifth-order KdV equation by decomposition method, Appl. Math. Comput., 144, 353-363 (2003) · Zbl 1024.65096
[8] Wazwaz, A. M., Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput., 182, 283-300 (2006) · Zbl 1107.65092
[9] Chun, C., Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method, Phys. Lett. A, 372, 2760-2766 (2008) · Zbl 1220.35148
[10] Bilige, S.; Chaolu, T., An extended simplest equation method and its application to several forms of the fifth-order KdV equation, Appl. Math. Comput., 216, 3146-3153 (2010) · Zbl 1195.35257
[11] Seadawy, A. R., New exact solutions for the KdV equation with higher order nonlinearity by using the variational method, Comput. Math. Appl., 62, 3741-3755 (2011) · Zbl 1236.35156
[12] Wang, G.; Kara, A. H.; Fakhar, K.; Vega-Guzman, J.; Biswase, A., Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation, Chaos Solitons Fractals, 86, 8-15 (2016) · Zbl 1360.35232
[13] Lü, J. Q.; Bilige, S.; Chaolu, T., The study of lump solution and interaction phenomenon to (2+1)-dimensional generalized fifth-order KdV equation, Nonlinear Dyn., 91, 1669-1676 (2018)
[14] Liu, N., Soliton and breather solutions for a fifth-order modified KdV equation with a nonzero background, Appl. Math. Lett., 104, Article 106256 pp. (2020) · Zbl 1441.35212
[15] Liao, S. J., On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves, Commun. Nonlinear Sci. Numer. Simul., 16, 1274-1303 (2011) · Zbl 1221.76046
[16] Xu, D. L.; Lin, Z. L.; Liao, S. J.; Stiassnie, M., On the steady-state fully resonant progressive waves in water of finite depth, J. Fluid Mech., 710, 379-418 (2012) · Zbl 1275.76052
[17] Liu, Z.; Xu, D. L.; Li, J.; Peng, T.; Alsaedi, A.; Liao, S. J., On the existence of steady-state resonant waves in experiments, J. Fluid Mech., 763, 1-23 (2015)
[18] Akgöbek, Ö.; Yakut, E., Efficiency measurement in Turkish manufacturing sector using Data Envelopment Analysis (DEA) and Artificial Neural Networks (ANN), J. Econ. Financ. Stud., 2, 35-45 (2014)
[19] Lapedes, A. S.; Farber, R. M., How neural nets work, (Neural Information Processing Systems. Neural Information Processing Systems, NIPS (1987)), 442-456
[20] Sharda, R.; Patil, R. B., Connectionist approach to time series prediction: an empirical test, J. Intell. Manuf., 3, 317-323 (1992)
[21] Zhou, H.; Zheng, P., Time series forecasting based on hierarchical genetic algorithm and BP neural network, J. Syst. Simul., 19, 5055-5058 (2007)
[22] Awad, M.; Pomares, H.; Rojas, I.; Salameh, O.; Hanmdon, M., Prediction of time series using RBF neural networks: a new approach of clustering, Int. Arab J. Inf. Technol., 6, 138-144 (2009)
[23] Phillips, O. M., On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions, J. Fluid Mech., 9, 193-217 (1960) · Zbl 0094.41101
[24] Ishaq, M.; Xu, H.; Sun, Q., Interactions of multiple three-dimensional nonlinear high frequency magnetosonic waves in magnetized plasma, Phys. Fluids, 32, Article 077109 pp. (2020)
[25] Liu, Z.; Liao, S. J., Steady-state resonance of multiple wave interactions in deep water, J. Fluid Mech., 742, 664-700 (2014)
[26] Liu, Z.; Xu, D. L.; Liao, S. J., Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups, Phys. Fluids, 29, Article 127104 pp. (2017)
[27] Liu, Z.; Xu, D. L.; Liao, S. J., Finite amplitude steady-state wave groups with multiple near resonances in deep water, J. Fluid Mech., 835, 624-653 (2018) · Zbl 1421.76026
[28] Yang, X. Y.; Dias, F.; Liao, S. J., On the steady-state resonant acoustic-gravity waves, J. Fluid Mech., 849, 111-135 (2018) · Zbl 1415.76087
[29] Sun, F. C.; Cai, J. X., N-soliton solutions to the (2+1)-dimensional variable-coefficient breaking soliton equation and its application, J. North China Univ. Technol., 24, 49-54 (2012)
[30] Maïna, I.; Tabi, C. B.; Ekobena, H. P.F.; Mohamadou, A.; Kofané, T. C., Discrete impulses in ephaptically coupled nerve fibers, Chaos, 25, Article 043118 pp. (2015) · Zbl 1374.92024
[31] Etémé, A. S.; Tabi, C. B.; Mohamadou, A., Synchronized nonlinear patterns in electrically coupled Hindmarsh-Rose neural networks with long-range diffusive interactions, Chaos Solitons Fractals, 104, 813-826 (2017) · Zbl 1380.92007
[32] Etémé, A. S.; Tabi, C. B.; Mohamadou, A.; Kofané, T. C., Elimination of spiral waves in a two-dimensional Hindmarsh-Rose neural network under long-range interaction effect and frequency excitation, Physica A, 533, Article 122037 pp. (2019) · Zbl 07570034
[33] Abdalla, E.; Maroufi, B.; Melgar, B. C.; Sedra, M. B., Information transport by sine-Gordon solitons in microtubules, Physica A, 301, 169-173 (2001) · Zbl 0991.92004
[34] Penrose, R., Shadows of the Mind (1994), Oxford University Press
[35] Qiao, B.; Song, K., Information soliton, J. Mod. Phys., 4, 923-929 (2013)
[36] Hirota, R., Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn., 33, 1456-1458 (1972)
[37] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Method for solving the sine-Gordon equation, Phys. Rev. Lett., 30, 25, 1262-1264 (1973)
[38] Abraham, A., Artificial neural network, (Sydenham, Peter H.; Thorn, Richard, Handbook of Measuring System Design (2005), John Wiley & Sons, Ltd.), 901-908
[39] Graupe, D., Principles of Artificial Neural Networks (2013), World Scientific: World Scientific Singapore · Zbl 1273.68001
[40] Dayhoff, J. E.; DeLeo, J. M., Artificial neural networks: opening the black box, Cancer, 91, 8, 1615-1635 (2001)
[41] Zhang, G. Q.; Patuwo, B. E.; Hu, M. Y., Forecasting with artificial neural networks: the state of the art, Int. J. Forecast., 14, 35-62 (1998)
[42] Gui, X. C., Realization of BP networks and their application on MATLAB, J. Zhanjiang Norm. Coll., 25, 3, 79-83 (2004)
[43] Specht, D. F., A general regression neural network, IEEE Trans. Neural Netw., 2, 6, 568-576 (1991)
[44] Broomhead, D. S.; Lowe, D., Multivariable functional interpolation and adaptive network, Complex Syst., 2, 321-355 (1988) · Zbl 0657.68085
[45] Rumelhart, D. E.; Hinton, G. E.; Williams, R. J., Learning representations by back-propagating errors, Nature, 323, 533-536 (1986) · Zbl 1369.68284
[46] Lyu, J.; Zhang, J., BP neural network prediction model for suicide attempt among Chinese rural residents, J. Affect. Disord., 246, 465-473 (2019)
[47] Alilou, V. K.; Yaghmaee, F., Application of GRNN neural network in non-texture image inpainting and restoration, Pattern Recognit. Lett., 62, 24-31 (2015)
[48] Li, C.; Bovik, A. C.; Wu, X., Blind image quality assessment using a general regression neural network, IEEE Trans. Neural Netw., 22, 05, 793-799 (2011)
[49] Sun, Y.; Zhang, H. Z.; Chang, Y. N., Application of GRNN in time series prediction for deformation of surrounding rocks in soft rock roadway, (2011 Fourth International Conference on Intelligent Computation Technology and Automation (2011)), 63-66
[50] Moody, J.; Darken, C. J., Fast learning in networks of locally-tuned processing units, Neural Comput., 1, 281-294 (1989)
[51] Wedding, D. K.; Cios, K. J., Time series forecasting by combining RBF networks, certainty factors, and the Box-Jenkins model, Neurocomputing, 10, 149-168 (1996) · Zbl 0850.68058
[52] Fritzke, B., Fast learning with incremental RBF networks, Neural Process. Lett., 1, 1, 2-5 (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.