Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation |
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| Affiliation: | 1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi''an, Shanxi 710072, PR China;2. Department of Mathematics, Xuchang University, Xuchang, Henan 461000, PR China;1. Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning, Guangxi, 530003, PR China;2. Department of Mathematics, Tianjin University of Technology, Tianjin, 300384, PR China;3. Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada;1. Central European Institute of Technology, CEITEC VUT, Brno University of Technology, Technická 10, CZ–616 69, Brno, Czech Republic;2. Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, CZ–616 69, Brno, Czech Republic;3. Faculty of Engineering Science and Technology, Norwegian University of Science and Technology, Rich. Birkelandsvei 1A, 7491, Trondheim, Norway;1. Department of Theoretical Mechanics, Tomsk State University, Tomsk, Russia;2. Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, Tomsk, Russia;3. Department of Applied Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania;4. Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey;5. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia |
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| Abstract: | The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation (B(m,n) equation), utt−uxx−a(un)xx+b(um)xxxx=0, is studied by using bifurcation theory of dynamical system. As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given. |
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