BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE ZAKHAROV-RUBENCHIK EQUATION

. The bounded traveling wave solutions of the Zakharov-Rubenchik equation are investigated by using the method of dynamical system theorems in this paper. After suitable transformations we ﬁnd that the traveling wave equations of the Zakharov-Rubenchik equation are fully determined by a second- order singular ordinary diﬀerential equation (ODE) with three real coeﬃcients which can be arbitrary constants. We derive abundant exact bounded periodic and solitary wave solutions of the Zakharov-Rubenchik equation via studying the bifurcations and exact solutions of the derived ODE.


1.
Introduction. In order to model the nonlinear interaction of high-frequency and low-frequency waves in an arbitrary medium, Zakharov [25] proposed a equation given by iB t + B = Bρ, ρ tt − ρ = (|B| 2 ).
Here B represents the slowly varying envelope of high-frequency electric field and ρ the low-frequency variation of density of ions. This equation is the so-called Zakharov equation and some generalized forms of this equation have been investigated in some literatures [1,5,9,21,22,23,24]. In 2007, Wang et al [24] investigated the exact and numerical solutions of a generalized Zakharov equation with certain initial conditions by using the modified Adomian decomposition method (mADM). Later on, Javidi and Golbabai [9] succeed in finding some exact and numerical solutions of the generalized Zakharov equation by the well-known variational iteration method. In 2009, Guo and Zhang [5] studied the existence and uniqueness of smooth solution for a generalized Zakharov equation, and proved the global existence of smooth solution in one spatial dimension at the same time. The homotopy analysis method (HAM) is applied in [1] to obtain approximations to the analytic solution of a generalized form of the Zakharov equation.
The global well-posedness of the one-dimensional Zakharov-Rubenchik equation has been considered in [21]. As a result, the existence and orbital stability of solitary wave solutions to equations (1) have been proved. [22] proved that the point-wised converge of the solutions B ε of equation (1) to the solution B of the nonlinear Schrödinger equation by making the adiabatic limit θ tend to 0. In 2005, Ponce and Saut [23] studied the well-posedness of a multi-dimensional version of system (1). The global well-posedness of one dimensional Zakharov-Rubenchik was applied [15] to study the norm growth of solutions corresponding to the Schrödinger equation term.
In this paper, we aim to investigate the bounded traveling wave solutions of equations (1). Amounts of methods have been proposed to explore the exact solutions of nonlinear PDEs which model a majority of real-world physical phenomena [4,10,16,18,29], for instance, the inverse scattering method, Backlund transformation method, Darboux transformation method, Hirota bilinear method, invariant subspace method, symmetry analysis approach, dynamical system approach and some special function etc (refer to [7,8,11,19,30,31] and their references). The dynamical system approach has been well applied to study the dynamics of some nonlinear partial differential equations [3,12,17,26,32] since the corresponding traveling wave systems of PDEs are dynamical systems which are always involving with some arbitrary constants such as wave speed, constant of integration and parameters of original systems. We will use dynamical system approach [6,13,14,20,27,28] to study the bifurcation and exact bounded solutions of the assistant second-order ODE to investigate the bounded traveling wave solutions of the the one-dimensional Zakharov-Rubenchik equation.
This paper is organized as follows. In Section 2, by rescaling and using the traveling wave frame, we reduce the partial differential equations (PDEs) (1) to ODEs at first. By integrations and further calculations, we find that all traveling wave solutions of (1) are determined by a second-order ODE with three real coefficients which can be arbitrary constants. Therefore, it is necessary to study the bifurcations of this ODE which has singularity to derive all the bounded traveling wave solutions of system (1). The phase portraits in each bifurcation set are plotted. In Section 3, we show the various bounded traveling wave solutions of equations (1) and their qualitative properties via the analysis of phase portraits derived in previous section. Some exact solitary wave solutions are obtained by integration along the bounded homoclinic orbits.

2.
Reduction of the one-dimensional Zakharov-Rubenchik equation. Letting ρ = ψ 1 + ψ 2 and u = √ β(ψ 1 − ψ 2 ), system (1) becomes Denote β ), then system (2) can be rewritten as To investigate the traveling wave solutions to (2) and (3), we perform the traveling wave transformations B(x, t) = e −iλt e if (ξ) ϕ(ξ), ψ 1 (x, t) = φ 1 (ξ), ψ 2 (x, t) = φ 2 (ξ) and ξ = x − ct, where λ and c are two real numbers, f , ϕ, φ 1 and φ 2 are real functions of ξ, then system (2.2) reduces to the following nonlinear ODEs: where denotes the derivative with respect to ξ. In what follows, we will investigate the solutions of system (4) in order to study the traveling wave solutions of system (3). Integrating the second and third equation of system (4) once respectively, we have where g 1 and g 2 are constants of integration. Comparing the real parts and imaginary parts of the first equation of (4), respectively, yields and We see from (6) that which is a separable first-order differential equation of the functions ϕ and f . For the case when c ω − 2f = 0, separating the dependent variables ϕ and f first and then integrating (8) once gives where g 3 is a constant of integration. It follows from (9) that where g = ± 1 2 e −2g3 . Note that g defined previously seems to be a nonzero constant, but it can be extended to an arbitrary real number including zero, which can be known from the fact that ϕ = 0 is a solution of equation (8). Obviously, for the case when ϕ = 0, (10) can be rewritten as Substituting (5) and (11) into (7) yields where a = 1 Equation (12) is a second-order ODE, which is equivalent to the following planar dynamical system ϕ = y, By letting dξ = ϕ 3 dη, system (13) becomes φ = ϕ 3 y, where · denotes the derivative with respect to η. It is easy to see that system (14) is a Hamiltonian system with Hamiltonian Clearly, (14) has the same phase portraits as (13) except the singular line ϕ = 0 for the case when b = 0. Let φ = ϕ 2 , then (14) becomes φ = 2yφ 2 , Note that the function φ in system (16) is a nonnegative function, so we only need to study the bifurcations of (13) by the analysis of (16) on the right half phase plane.
However, for the case when c Substituting (5) and (17) into (7) leads to It is easy to see that equation (18) is just a special case of equation (12) with b = 0, so the coefficient b involving in all above equations (12)- (16) is in the range [0, +∞).
3.1. Bifurcation analysis of system (13) with b = 0. Let M (ϕ e , 0) be the coefficient matrix of the linearized system of (13) with b = 0 at the equilibrium point (ϕ e , 0) and J(ϕ e , 0) be its Jacobian determinant. Then we have J(ϕ e , 0) = −(a + 3dϕ 2 e ). By the theory of planar dynamic system, we know that for an equilibrium point of a planar integrable systems, if J < 0 then the equilibrium point is a saddle point; If J > 0 and T race(M (φ e ), 0) = 0 then it is a center; If J = 0 and Poincaré index of the equilibrium point is 0 then it is a cusp. Let f (ϕ) = aϕ + dϕ 3 , we find that (ϕ e , 0) is a saddle point if f (ϕ e ) > 0 and is a center if f (ϕ e ) < 0, and is of cuspidal type if f (ϕ e ) = 0, where f (ϕ e ) = 0. By careful calculations, we have the following statements.
Equation f (ϕ) = 0 has three roots and thus system (13) has three equilibrium points (ϕ e± , 0) and (0, There are two homoclinic orbits connecting the saddle point and three families of periodic orbits, among which two families of periodic orbits surround the centers respectively and one family of periodic orbits enclose the two homoclinic orbits (see Fig. 1(a)).
System (13) has three equilibrium points (ϕ e± , 0) and (0, 0) similarly. However, (ϕ e± , 0) are saddle points since f (ϕ e± ) > 0 and (0, 0) is a center since f (0) < 0. There are two heteroclinic orbits connecting the two saddle points which construct the boundary of a family of periodic orbits surrounding the center (see Fig. 1 Hence, there are a family of periodic orbits surrounding the center (see Fig. 1(c)).
Equation f (ϕ) = 0 has only one root ϕ = 0 and thus system (13) with b = 0 has unique equilibrium point (0, 0) which is a saddle point for d ≥ 0 & a > 0 and a cusp for a = 0 & d = 0. Therefore, system (13) with b = 0 has no bounded orbits in this case.
3.2. Bifurcation analysis of system (13) with b > 0. We now analyze the bifurcation of system (13) via studying system (16) by the theory of planar dynamic system. Let f 1 (φ) = dφ 3 + aφ 2 + b, then the positive roots of f 1 (φ) determine the equilibrium points of system (16). By direct calculations, we firstly get the following results for system (16).
Equation f 1 (φ) = 0 has no positive root, so (16) has no equilibrium point on the right-half phase plane.
For the case when b = − 4a 3 27d 2 , equation f 1 (φ) = 0 has one positive root φ 2 , and the corresponding equilibrium point (φ 2 , 0) is a cusp. For the case when b > − 4a 3 27d 2 , equation f 1 (φ) = 0 has no positive root. Therefore, we know that system (16) has no bounded orbits on the right-half plane for the case when 0) is a center since f 1 (φ e ) < 0. Therefore, system (16) has a family of periodic orbits surrounding the center on its right-half phase plane.
Recalling that φ = ϕ 2 , so we can get the phase portrait in each bifurcation set of system (13) via the bifurcation analysis for system (16) on the right-half phase plane. Here we just focus on the bounded orbits of system (13), so we only consider the bounded orbits of system (16). We see from the analysis above that system (16) has bounded orbits if and only if the parametersa, b and d satisfy the conditions: In what follows, we consider the bounded orbits of system (13) for which the parameters satisfy one of the two conditions.
(B) If a, b and d satisfy condition (2), namely, b > 0, d < 0 & a ≥ 0 or d ≤ 0 & a < 0, then system (13) has two centers (± √ φ e , 0). There are two family of periodic orbits surrounding the two centers respectively (see Fig. 2(B)). In this section, we apply the results obtained in the previous sections to study the existence and qualitative properties of bounded solutions and explore some exact traveling wave solutions of the one-dimensional Zakharov-Rubenchik equation (1).

Bounded solutions determined by
It follows from Section 3.1 that system (13) has bounded orbits if and only if the parameters a and d satisfies: For case (I), that is, d < 0 and a > 0, H(ϕ, y) = h ≤ − a 2 4d . h = 0 corresponds to the two homoclinic orbits connecting the saddle point (0, 0); H(ϕ, y) = h with h ∈ (0, − a 2 4d ) determines two periodic orbits surrounding the two centers and inside the homoclinic orbits, respectively. H(ϕ, y) = h with h ∈ (−∞, 0) determines a big periodic orbit which surrounds the two homoclinic orbits.
Solving for y from (19) with h = 0 and then substituting the result in the first equation of (13), one has where ϕ ± = ± − 2a d . Integrating (20) gives where ξ 0 is an arbitrary constant. Clearly, (21) are two solutions of (12) with b = 0 which has a single peak and tends to 0 as ξ approaches infinity.
For the case when d = 0, a < 0 and s 0 ∈ (0, φ e ), Substituting (46) in (36) and then solving for s leads to Therefore, are two solutions of (12) which correspond to the periodic orbits passing through (± √ s 0 , 0), respectively.

5.
Conclusion. In this paper, we obtain traveling wave solutions of the Zakharov-Rubenchik equation with different parameter values. By using traveling wave transformations and a series of calculations, we transformed the equation to a secondorder ODE which have coefficients a, b and d. By using bifurcation and dynamic system theorem, all possible bounded real solutions of the involving planar dynamic systems were studied and then some exact solitary wave solutions were obtained.