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Introduction
Boussinesq equation [1, 2–3]
1.1
is an important model equation in fluid dynamics, which is used to describe the propagation of waves in nonlinear and dispersive media. It is suitable for the seepage problem of porous underground strata, and is widely used to describe many physical phenomena such as shallow water waves, plasma, and nonlinear lattices [4, 5–6]. If , it is called the “bad" Boussinesq equation, which was proposed by Boussinesq [1] in 1872 to describe the problem of shallow water waves. When , it is called the “good" Boussinesq equation, which can describe the nonlinear vibration of strings and can also be used to describe electromagnetic waves in nonlinear dielectric materials. In [4, 5], a Boussinesq system suitable for two-way propagation of surface water waves is proposed, where the main aspect of propagation is the balance between the nonlinear effect of convection and the linear effect of frequency dispersion. In addition, it is proved that the linear well-posed first-order modified model is actually locally nonlinear well-posed. [6] attempted to explain the famous Stokes paradox and showed that the motion of small-amplitude shallow water long waves under gravity can be described by the Boussinesq equation.Due to its wide application in various physical scenarios, people have carried out a lot of research on the Boussinesq equation. Zakharov [7] used the inverse scattering method to study the solvability of the initial value problem of equation (1.1). Weiss [8, 9–10] studied the Painlevé property, Bäclund transformation, Lax pair, et al. of equation (1.1). Manoranjan [11] obtained the soliton solutions of equation (1.1). Yu et al. [12] gave the multi-soliton solution of equation (1.1) by using the homogeneous balance principle. In [13], the variational form of Boussinesq wave equation is established by using the semi-inverse method, and the soliton solutions of the equation (1.1) in the form of sech-function and exp-function are obtained. Jafari et al. [14] used Riccati equation as the simplest equation, and constructed the exact solution of equation (1.1) by using the simplest equation method and Lie symmetry method. Guo et al. [15] applied the consistent tanh expansion (CTE) method to the (2+1)-dimensional Boussinesq equation (1.1), which describes the propagation of ultrashort pulses in a quadratic nonlinear medium, and clearly gave the interaction...