New groups of solutions to the Whitham-Broer-Kaup equation
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2683-7
New groups of solutions to the Whitham-Broer-Kaup equation∗ Yaji WANG1 , Hang XU1 , Q. SUN2,† 1. State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; 2. Australian Research Council Centre of Excellence for Nanoscale BioPhotonics, School of Science, RMIT University, Melbourne, VIC 3001, Australia (Received Jul. 28, 2020 / Revised Sept. 9, 2020)
Abstract The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature. Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions. Key words Whitham-Broer-Kaup equation, travelling wave, bifurcation analysis, projective Riccati-equation method Chinese Library Classification O53 2010 Mathematics Subject Classification
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34DXX, 35Q53
Introduction
Study of the dynamics and behavior of the tsunami waves is not only of scientific importance but also of significant contribution to the practices in our daily life. A tsunami wave is often characterized as a shallow-water wave since its wavelength is usually far longer than the water depth. Many mathematical models have been developed to describe the shallow water waves in which the typical models include the Korteweg-de Vries (KdV) equation[1] , the Boussinesq equation[2] , the Degasperis-Procesi equation[3] , the Benjamin-Bona-Mahony (BBM) equation[4] , the Kadomtsev-Petviashvili (K-P) equation[5] , and the Whitham-Broer-Kaup (WBK) model[6] . Particularly, the WBK model is usually used to describe the tsunami wave dynamics under gravity, which is formulated based on the assumption that the fluid is incompressible and irrotational. Different analytical methods have been developed and used to obtain the solutions of the WBK model and the soliton solutions of nonlinear wave equations, such as the improved ∗ Citation: WANG, Y. J., XU, H., and SUN, Q. New groups of solutions to the Whitham-Broer-Kaup equation. Applied Mathematics and Mechanics (English Edition) (2020) https://doi.org/10.1007/s10483-020-2683-7 † Corresponding author, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11872241), the Discovery Early Career Researcher Award (No. DE150100169), and the Centre of Excellence Grant funded by the Australian Research Council (No. CE140100003) ©The Author(s) 2020
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Yaji WANG, Hang XU, and Q. SUN
Riccati equations method[6–10] , the first integral method[11] , the Backlund transformation method[12] , the optimal homotopy asymptotic method[13] , the hyperbolic function method[14] , the homogeneous balance method[15] , the pa
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