Application of the bifurcation method to the modified Boussinesq equation

In this paper, we investigate the modified Boussinesq equation utt − uxx − εuxxxx − 3(u)xx + 3(uux)x = 0. Firstly, we give a property of the solutions of the equation, that is, if 1 + u(x, t) is a solution, so is 1 − u(x, t). Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.


Introduction
In recent years, nonlinear phenomena have been studied in all fields of science and engineering, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and so on.Many nonlinear evolution equations play an important role in the analysis of these phenomena.
The bad and good Boussinesq equations [13] are as follows and Email: lishaoyongok@sohu.com 2 S. Li which were introduced by the French scientist Joseph Boussinesq  to describe the 1870s model equation for the propagation of long waves on the surface of water with a small amplitude.Equation (1.1) is used to describe the two-dimensional flow of shallow-water waves having small amplitudes.There is a dense connection to the so-called Fermi-Pasta-Ulam (FPU) problem.The existence of Lax pair, Bäcklund transformation and some solitontype solutions is known [13,21].Equation (1.2) describes the two-dimensional irrotational flow of an inviscid liquid in a uniform rectangular channel.There are known results due to local well-posedness, global existence and blow-up of some solutions [3].The bad and good Boussinesq equations have been studied by using the VIM, HPM, ADM, Exp-function method, and F-expansion method [1,2,7,15].Dai et al. [4] studied the explicit homoclinic orbits solutions for the bad Boussinesq equation with periodic boundary condition and even constraint, and periodic soliton solutions for the good Boussinesq equation with even constraint.G. Forozani and M. Ghorveei Nosrat [5], by adding a nonlinear term of the form 3(u 2 u x ) x to the bad and good Boussinesq equations, studied the following modified bad and good Boussinesq equations and They obtained variant solutions such as kink, anti-kink, compacton and periodic solutions for these equations by using the standard tanh, the extended tanh method and a mathematical method based on the reduction of order.
In the present paper, combining the modified bad and good Boussinesq equations, we consider the following modified Boussinesq equation where ε is a nonzero constant.When ε = 1, equation (1.5) reduces to the modified bad Boussinesq equation (1.3).When ε = −1, equation (1.5) reduces to the modified good Boussinesq equation (1.4).In order to search for the traveling wave solutions of equation (1.5), here we study equation (1.5) by using the bifurcation method mentioned above.Firstly, we give a property of the solutions of equation (1.5), that is, if 1 + u(x, t) is a solution, so is 1 − u(x, t).Secondly, we obtain some explicit expressions of solutions for equation (1.5), which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions.After checking over these solutions carefully, we find that some solutions are, in fact, exactly the same as those solutions given in [5].To our knowledge, many other solutions are new.This paper is organized as follows.In Section 2, we state our main results which are included in three propositions.In Section 3, we give the theoretical derivations for the propositions respectively.A brief conclusion is given in Section 4.

Main results
In this section we list our main results.To relate conveniently, for given constant wave speed c, let ξ = x − ct, (2.1) Using the notations above, our main results are stated in Proposition 2.1 (the property of the solutions of equation (1.5)) and Propositions 2.2, 2.3 (the exact explicit expressions of solutions for equation (1.5)).

Proposition 2.1. There exists a property of the solutions of equation
Proposition 2.2.When ε > 0, equation (1.5) has the following exact solutions.
(1) Let g denote the integral constant in equation (3.3).If g = 0, we obtain two kink-shaped solutions ) and four periodic blow-up solutions ) (2) If 0 < g < g 0 , we obtain two solitary wave solutions ) and two periodic blow-up solutions where α depends on g, it is such that ) ) .
(3 • ) If g = g 0 , we obtain two solitary wave solutions

The theoretic derivations for main results
In this section, we will give the derivations for our main results.Firstly we derive Proposition 2.1, the property of the solutions of equation (1.5).If 1 + u(x, t) is a solution of equation (1.5), that is 1 + u(x, t) satisfies equation (1.5), then we have On the other hand, substituting 1 − u(x, t) into the left side of equation (1.5), we have Secondly we derive Propositions 2.2 and 2.3, the explicit expressions of solutions for equation (1.5).We look for the traveling wave solutions of equation (1.5) in the form of where ξ was given in (2.1).Substituting (3.2) into equation (1.5) and integrating twice with respect to ξ, we get where g and g * are two integral constants.In order to use the bifurcation method of dynamical systems, we consider the case g * = 0. Letting y = φ ′ , we obtain the following planar system which has the first integral where h is an integral constant.Now we consider the phase portraits of system (3.4).Set and

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It is easy to obtain the two extreme points of f 0 (φ) as follows 2).Let (φ * , 0) be one of the singular points of system (3.4).Then the characteristic values of the linearized system of system (3.4) at the singular point (φ * , 0) are Figure 3.1: The phase portraits of system (3.4) when ε > 0.
According to the qualitative theory of dynamical systems, we therefore know the following From the analysis above, we obtain the phase portraits of system (3.4) in Figure 3.1 (when ε > 0) and Figure 3.2 (when ε < 0).Now we will obtain the explicit expressions of solutions for equation (1.5) when ε > 0.
(ii) Secondly, from the phase portrait, we note that there are two special orbits Γ l 2 and Γ r 2 , which have the same Hamiltonian as that of the center point (0, 0).On the (φ, y)-plane the expressions of the two orbits are given as Substituting (3.6) into dφ/dξ = y and integrating them along the two orbits Γ l 2 and Γ r 2 , it follows that Computing the integrals above, we have (ii) Secondly, from the phase portrait, we note that there are two special orbits Γ l 4 and Γ r 4 , which have the same Hamiltonian as that of the center point (φ 2 , 0).On the (φ, y)-plane the expressions of these orbits are given as where Substituting (3.12) into dφ/dξ = y and integrating them along the orbits Γ l 4 and Γ r 4 , it follows that Computing the integrals above, we have , where η 2 is given in (2.11).Noting that u(x, t) = 1 + φ(ξ) with ξ = x − ct, we get two periodic blow-up solutions u 7± (x, t) as (2.9).
(3) If g = g 0 , from the phase portrait, we see that there are two orbits Γ l 5 and Γ r 5 , which have the same Hamiltonian with the degenerate saddle point (φ 4 , 0).On the (φ, y)-plane the expressions of these orbits are given as where Computing the integrals above, we have