Different Groups of Variational Principles for Whitham-Broer-‎Kaup Equations in Shallow Water

Document Type : Research Paper


1 College of Meteorology and Oceanography, National University of Defense Technology, Changsha, 410073, China

2 College of Computer, National University of Defense Technology, Changsha, 410073, China


Because the variational theory is the theoretical basis for many kinds of analytical or numerical methods, it is an essential but difficult task to seek explicit functional formulations whose extrema are sought by the nonlinear and complex models. By the semi-inverse method and designing trial-Lagrange functional skillfully, two different groups of variational principles are constructed for the Whitham-Broer-Kaup equations, which can model a lot of nonlinear shallow-water waves. Furthermore, by a combination of different variational formulations, new families of variational principles are established. The obtained variational principles provide conservation laws in an energy form and are proved correct by minimizing the functionals with the calculus of variations. All variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities, and might find lots of applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be extended to more other nonlinear equations.


Main Subjects

[1] Ablowitz, M. J., Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scatting, Cambridge University Press, Cambridge, 1991.
[2] Gu, C. H.,  et al., Soliton Theory and Its Application, Zhejiang Science and Technology Publishing House, Hangzhou, 1990.
[3] He, J. H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16(2), 2012, 331-334.
[4] Wang, M. L., et al., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216, 1996, 67-75.
[5] Liu, S. S., Fu, Z. T., Expansion method about the Jacobi elliptic function and its applications to nonlinear wave equations, Acta Physica Sinica, 50, 2001, 2068-2073.
[6] Ma, H. C., Exact solutions of nonlinear fractional partial differential equations by fractional sub-equation method, Thermal Science, 19, 2015, 1239-1244.
[7] Li, Z. B., Exact Solutions of Time-fractional Heat Conduction Equation by the Fractional Complex Transform, Thermal Science, 16, 2012, 335-338.
[8] He, J. H., Exp-function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation, 14, 2013, 363-366.
[9] He, J. H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 2006, 1141-1199.
[10] Guner, O., Bekir, A., Exp-function method for nonlinear fractional differential equations, Nonlinear Science Letters A, 8, 2017, 41-49.
[11] He, J. H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57(8), 2019, 1932-1934.
[12] He, C. H., Shen, Y., Ji, F.Y., He, J. H., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28(1), 2020, 2050011.
[13] He, J. H., Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Engineering Journal, 2020, http://doi.org/10.1016/j.asej.2020.01.016
[14] Wu, Y., Variational approach to higher-order water-wave equations, Chaos, Solitons & Fractals, 32, 2007, 195-203.
[15] Gazzola, F., Wang, Y., Pavani, R.: Variational formulation of the Melan equation, Mathematical Methods in the Applied Sciences, 41(3), 2018, 943-951.
[16] Baleanu, D., A modified fractional variational iteration method for solving nonlinear gas dynamic and coupled KdV equations involving local fractional operator, Thermal Science, 22, 2018, S165-S175.
[17] Durgun, D. D., Fractional variational iteration method for time-fractional nonlinear functional partial differential equation having proportional delays, Thermal Science, 22, 2018, S33-S46.
[18] He, J. H., Liu, F. J., Local Fractional Variational Iteration Method for Fractal Heat Transfer in Silk Cocoon Hierarchy, Nonlinear Science Letters A, 4(1), 2013, 15-20.
[19] Yang, X. J., Baleanu, D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 17(2), 2013, 625-628.
[20] Mir, S. H., Mustafa, I., Ali, A., Analytical treatment of the couple stress fluid-filled thin elastic tubes, Optik, 145, 2017, 336-345.
[21] Maysaa, M. A. Q., Esma, A., Mustafa, I., Optical solitons in multiple-core couplers with the nearest neighbors linear coupling, Optik, 142, 2017, 343-353.
[22] Bulent, K., Mustafa, I., The First Integral Method for the time fractional Kaup-Boussinesq System with time dependent coefficient, Applied Mathematics and Computation, 254, 2015, 70-74.
[23] Mustafa, I., The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, Journal of Mathematical Analysis and Applications, 345, 2008, 476-484.
[24] He, J. H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons & Fractals, 19, 2004, 847-851
[25] He, J. H., A modified Li-He’s variational principle for plasma, International Journal of Numerical Methods for Heat and Fluid Flow, 2019, DOI: 10.1108/HFF-06-2019-0523.
[26] He, J. H., Generalized equilibrium equations for shell derived from a generalized variational principle, Applied Mathematics Letters, 64, 2017, 94-100.
[27] He, J. H., Sun, C., A variational principle for a thin film equation, Journal of Mathematical Chemistry, 57, 2019, 2075-2081.
[28] He, J. H., Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, Journal of Applied and Computational Mechanics, 6(4), 2020, 735-740.
[29] He, J. H., Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mechanica, 231, 2020, 899-906.
[30] He, J. H., A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals, 2019, https://doi.org/10.1142/S0218348X20500243
[31] Cao, X. Q., et al., Variational principles for two kinds of extended Korteweg-de Vries equations, Chinese Physics B, 20(9) , 2011, 94-102.
[32] Cao X. Q.,et al., Generalized variational principles for Boussinesq equation systems, Acta Physica Sinica, 60(8), 2011, 105-113.
[33] He, J. H., Ain, Q. T., New promises and future challenges of fractal calculus: from two-scale Thermodynamics to fractal variational principle, Thermal Science, 24(2A), 2020, 659-681.
[34] Wang, Y., et al., A variational formulation for anisotropic wave traveling in a porous medium, Fractals, 27, 2019, 1950047.
[35] Wang, K. L., He, C. H., A remark on Wang's fractal variational principle, Fractals, 27, 2019, 1950132.
[36] El-Kalaawy, O.H., New Variational principle-exact solutions and conservation laws for modified ion-acoustic shock waves and double layers with electron degenerate in plasma, Physics of Plasmas, 24, 2017, 032308.
[37] El-Kalaawy, O.H., Variational principle, conservation laws and exact solutions for dust ion acoustic shock waves modeling modified Burger equation, Computers and Mathematics with Applications, 72, 2016, 1013-1041.
[38] Whitham, G. B., Variational methods and applications to water wave, Proceedings of the Royal Society of London. Series A, 299, 1967, 6-25
[39] Broer, L. J., Approximate equations for long water waves, Applied Science Research, 31(5), 1975, 377-395.
[40] Kaup, D. J., A higher-order water wave equation and method for solving it, Progress of Theoretical Physics, 54(2), 1975, 396-408
[41] Kaupershmidt, B. A., Mathematics of dispersive water waves, Communications in Mathematical Physics, 99(1), 1985, 51-73.
[42] Fan, E. G., Zhang, H. Q., Backlund transformation and exact solutions for Whitham-Broer-Kaup Equations in shallow water, Applied Mathematics and Mechanics, 19(8), 1998, 713-716.
[43] Wang, Y., et al., A Fractional Whitham-Broer-Kaup Equation and its Possible Application to Tsunami Prevention, Thermal Science, 21(4), 2017, 1847-1855.
[44] Wang, L., Chen, X., Approximate Analytical Solutions of Time Fractional Whitham-Broer-Kaup Equations by a Residual Power Series Method, Entropy, 17, 2015, 6519-6533.