Exact solution of Boussinesq equations for propagation of nonlinear waves
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Exact solution of Boussinesq equations for propagation of nonlinear waves O. V. Kaptsova , D. O. Kaptsov Institute of Computational Modeling, SB RAS, Krasnoyarsk, Russia Received: 22 May 2020 / Accepted: 28 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we consider two Boussinesq models that describe propagation of small-amplitude long water waves. Exact solutions of the classical Boussinesq equation that represent the interaction of wave packets and waves on solitons are found. We use the Hirota representation and computer algebra methods. Moreover, we find various solutions for one of the variants of the Boussinesq system. In particular, these solutions can be interpreted as the fusion and decay of solitary waves, as well as the interaction of more complex structures.
1 Introduction The study of water waves is of great practical and theoretical interest. There are currently a number of mathematical models describing nonlinear shallow water waves [1–5]. These include the Boussinesq-type approximations. The one-dimensional model [6,7] is given by a system of equations of the form u t + uu x + ηx + c1 u t x x + c2 ηx x x = 0, ηt + u x + (uη)x + c3 u x x x + c4 ηx x x = 0,
(1)
where u is the fluid velocity, η is the elevation of the free surface, c1 , . . . , c4 are constants. Recall that more than a hundred years ago, Boussinesq derived the famous equation which, by a scaling transformation, is reduced to the form ηtt = ηx x + 3(η2 )x x + ηx x x x .
(2)
The study of the system (1) was carried out by analytical and numerical methods. In particular, Hirota and Satsuma [8] found N -soliton solutions of the system u t + uu x + ηx = 0, ηt + (u + uη)x + u x x x = 0.
(3)
O. V. Kaptsov: This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2020-1631). a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:723
An infinite family of rational solutions of the system were obtained in [9,10]. Using the Darboux transformation, Zhang et al. [11] derived some solutions describing the elastic-fusioncoupled interaction. Furthermore, Hirota [12] found N -soliton solutions of the Boussinesq equation (2). In addition, in [13,14], solutions of this equation that describe wave packets and waves on solitons were obtained. The structure of the paper is the following: In the second section, we construct solutions of the Boussinesq equation (2) that describe the elastic interaction of wave packets with each other, as well as with solitons. We find solutions of the system (3) in the third section. For this purpose, the Hirota–Satsuma ansatz [9] is substituted into (3), and as a result, a large system of nonlinear algebraic equations is obtained. The solut
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