Exact periodic wave solutions of the cubic-quintic Zakharov equation and their evolution with Hamilton energy

In this paper, we study the exact periodic wave solutions of the Zakharov equation with cubic and quintic nonlinear terms, and their evolution with the energy of Hamiltonian system corresponding to the amplitudes. Based on the theory of plane dynamical system, we first make a detailed qualitative analysis to the plane dynamical system corresponding to the amplitudes of traveling wave solutions of the studied equation, then by applying the analysis method based on the first integral and several appropriate transformations, all seven families of elliptic function periodic wave solutions of the Zakharov equation are obtained. In addition, by studying the evolution limit of periodic wave solutions with respect to Hamilton energy and using the analysis method based on the first integral, all ten pairs of solitary wave solutions of the studied equations are also given under various parameter conditions. From the evolution analysis to the periodic wave solutions with respect to Hamilton energy, it can be seen that it is the energy H of the Hamiltonian system corresponding to the studied equation taking values in different ranges that makes the traveling wave solution of this equation appear as periodic wave solution or solitary wave solution.


Introduction
The famous Zakharov equation [1] { is a governing equation of plasma physics derived by Zakharov et al when they studied the nonlinear coupling of Langmuir wave and ion acoustic wave in unmagnetized plasma in 1972.
The generalized form [2,3] of equation (1.1) is also a plasma simulation model, in which the cubic term describes the nonlinear self-action in the high-frequency subsystem corresponding to the steady-state self-focusing in plasma physics.Further, make the nonlinear term saturated, then the equation describing the nonlinear medium is the Zakharov equation with cubic and quintic nonlinear terms as follows [4], which is studied in this paper: 4 u, (1.3) where the complex-valued function u(x, t) is the envelope of high-frequency electric field, the real-valued function H(x, t) is the plasma density measured by the equilibrium value, b 1 and b 2 are arbitrary constants.According to Zakharov's study in 1975, due to the correction brought by the nonlinear saturated quintic term |u| 4 u, equation (1.3) depicts the particle motion in the potential composed of the attractive Coulomb potential and the repulsive short-range potential, and the particle can be repulsed to the central position in this optical field.
In the previous literature on the Zakharov equation, the solution-solving and some related problems of the above three types of Zakharov equations were discussed and studied respectively.For equation (1.1), [5,6] proposed an implicit finite difference scheme for calculating the periodic initial value problem, and then gave the solitary wave solutions and the collision solitary wave solutions of equation (1.1); [7] studied the unique local solvability and the smoothing effect of equation (1.1); [8,9] studied the orbital stability of solitary wave solutions and periodic wave solutions of equation (1.1) respectively.For equation (1.2), in [10][11][12], the solitary wave solutions of equation (1.2) were obtained by using the He's semi-inverse method and the He's variational iteration method respectively; [13] obtained the analytic approximate solutions of equation (1.2) by the homotopy analysis method; In [14], the dynamic behaviors of the exact traveling wave solutions of equation (1.2) were studied by using the dynamical system method; [15] studied the orbital stability of periodic wave solutions of equation (1.2).For the Zakharov equation (1.3) with cubic and quintic nonlinear terms studied in this paper, in [16], the bright-dark solitary wave solutions and trigonometric function periodic wave solutions of equation (1.3) were obtained by the Fan's direct algebraic method; In [17], some exact solitary wave solutions of equation (1.3) were obtained by the first integral method, which correspond to the homoclinic orbits in figure 2.1 and the heteroclinic orbits in figure 2.2 of our paper respectively; By bifurcation theory, [18] obtained a family of elliptic function periodic wave solutions, which corresponds to a family of closed orbits located inside the heteroclinic orbits and around the singular point P2 in figure 2.4 of our paper, at the end of section 3 in this paper, the periodic wave solutions obtained in [18] will be compared with the periodic wave solutions we obtained.
Through the above literature review on the exact solutions of Zakharov equation, it can be seen that for the Zakharov equation (1.3) with both cubic and quintic nonlinear terms, the previous literature mainly focuses on the study of its solitary wave solutions, while only individual solutions are obtained for periodic wave solutions.We have not yet found a comprehensive and systematic study on the exact periodic wave solutions of the Zakharov equation (1.3).Considering the fact that periodic motion is the most common and basic motion in nature, and many other motions can be evolved from it, people have always attached great importance to the study of periodic wave solutions of nonlinear systems and related problems [19][20][21][22].In this paper, we make a comprehensive and systematic study on the solution-solving problem of the periodic wave solutions of the Zakharov equation (1.3) with cubic and quintic nonlinear terms, and obtain all exact periodic wave solutions of equation (1.3).In addition, the evolutionary relationship of periodic wave solutions and solitary wave solutions with respect to Hamilton energy is studied.In order to obtain the related results of this study, we apply the theory of planar dynamical system and the analysis method based on the first integral.
The specific arrangement of this paper is as follows: In section 2, we make a qualitative analysis to the dynamical system corresponding to the bounded traveling wave solutions of Zakharov equation (1.3), and obtain the existence condition, number and approximate position of solitary wave solutions and periodic wave solutions of this equation.In section 3, we mainly study the solution-solving of periodic wave solutions of the Zakharov equation (1.3).
Here we apply the analysis method based on the first integral of the corresponding system and use several appropriate transformations to find all seven families of periodic wave solutions of equation (1.3) corresponding to the closed orbits in the global phase diagrams.In section 4, we study the evolution of periodic wave solutions of equation (1.3) with respect to Hamilton energy corresponding to the amplitudes of these solutions.For the periodic wave solutions corresponding to the closed orbits surrounded by or surrounding homoclinic orbits and symmetrical heteroclinic orbits, when the corresponding Hamilton energy H tends to the Hamilton energy corresponding to homoclinic orbits or symmetrical heteroclinic orbits, do the periodic wave solutions corresponding to these closed orbits expand or shrink to the solitary wave solutions corresponding to the homoclinic orbits or symmetric heteroclinic orbits?By studying the evolution limit of periodic wave solutions obtained in section 3 with respect to Hamilton energy, we give five pairs of bell-shaped and kink-shaped solitary wave solutions of equation (1.3).For three kinds of special cases that periodic wave solutions do not tend to solitary wave solutions, we directly use the analysis method based on the first integral to find the remaining five pairs of solitary wave solutions, of which two pairs are algebraic solitary wave solutions [23][24][25] with amplitude as algebraic function.In addition, we also give the three-dimensional numerical simulation diagrams depicting the evolution of the periodic wave solutions tending to the bell-shaped and kink-shaped solitary wave solutions, so as to more intuitively show their evolutionary relationship with respect to Hamilton energy.
It is worth pointing out: (1) For a nonlinear system, the higher-order nonlinear term often means a more complex physical phenomenon, so the research on its related problems is more important.However, in the process of studying nonlinear evolution equations, the higher-order nonlinear term makes the corresponding dynamical system have more complex bounded orbits, which increase the difficulty of solution-solving.Therefore, in the previous literature, only the periodic wave solutions were obtained for the evolution equations with cubic nonlinear term, for the evolution equations with both cubic and quintic nonlinear terms such as the Zakharov equation (1.3) in this paper, although the solitary wave solutions have been obtained, we have not found a literature to solve the exact periodic wave solutions of this kind of equations comprehensively and study the evolutionary relationship of them and solitary wave solutions.In order to overcome the difficulty caused by the cubic and quintic nonlinear terms of the studied equation, we use several skillful transformations to obtain all seven families of periodic wave solutions of Zakharov equation (1.3); (2) In the process of solving the elliptic function periodic wave solutions of Zakharov equation (1.3), we give the specific form of the modulus k contained in each periodic wave solution, which is a function of Hamilton energy h.Through the evolution analysis, it can be seen that it is the energy h of the Hamiltonian system corresponding to the studied equation taking values in different ranges that makes the traveling wave solutions of the equation show different forms, namely, periodic wave solutions and solitary wave solutions.This conclusion does not seem to have been mentioned and discussed in the previous literature.The conclusion is of practical significance, because for a nonlinear system that may have various complex phenomena, it is easier to control and apply the real models corresponding to these equations if the fundamental causes of the complex phenomena are grasped.

Qualitative analysis to bounded traveling wave solutions of Zakharov equation (1.3)
2.1.Dynamical system corresponding to the traveling wave solutions of equation (1.3) Assume that equation (1.3) has bounded traveling wave solutions in the form of where ξ = x − vt.Substitute (2.1) into equation (1.3) and by simplification, we have where 2), we get if we take k = v 2 and substitute (2.5) into (2.3), then after arrangement, it can be seen that a(ξ) satisfy where (2.8) The above derivation shows that if equation (1.3) has the solution in the form of (2.1), k and v satisfy the relation k = v 2 , and the assumption (2.4) holds, then the amplitude a(ξ) of the solution (u(ξ), H(ξ)) of form (2.1) for equation (1.3) must satisfies equation (2.7).Conversely, if the amplitude a(ξ) is the solution of equation (2.7) with second derivative, and H(ξ) satisfies the form (2.5), then the solution in the form of (2.1) for equation (1.3) can be obtained according to the relation k = v 2 .Therefore, in order to study the periodic wave solutions and the solitary wave solutions of equation (1.3), we can start from equation (2.7).So we always assume that the relation k = v 2 holds in the following discussion.Now the qualitative analysis on equation (2.7) is made by using the planar dynamical system theory [26,27].
Let x = a(ξ) and y = a ′ (ξ), then equation (2.7) can be transformed to the following planar dynamical system where Since the number of finite-far singular points for system (2.9) in the (x, y) plane depends on the number of real roots for the algebraic equation f(x) = 0, we investigate the real roots of the algebraic equation x 4 + px 2 + q = 0. Let z = x 2 , then the algebraic equation can be transformed to Under the condition of p 2 − 4q ⩾ 0, equation (2.12) has the following real roots Moreover, it is easy to verify: (1) z 2 ⩽ z 1 ; (2) when p ⩽ 0, there must be z 1 ⩾ 0, and when p > 0, the necessary and sufficient condition of z 1 ⩾ 0 is q ⩽ 0; (3) the necessary condition of z 2 ⩾ 0 is p ⩽ 0, and when p < 0, z 2 ⩾ 0 is equivalent to q ⩾ 0.
Since this paper focuses on the periodic wave solutions of equation (1.3) and their evolutionary relationship with solitary wave solutions, in the following discussion, we always assume that one of the following two conditions holds: S 1 : p < 0, q ⩾ 0, p 2 − 4q ⩾ 0; S 2 : q < 0.
It can be seen that system (2.9) is a Hamiltonian system, thus we have the first integral where H = H(x, y) is also called the Hamilton energy function at (x, y).Assume the singular point of system (2.9) is P i (x i , 0), then the corresponding Jacobi matrix is from the above, we get (2.17) If x i ̸ = 0, then x i satisfies x 4 i + px 2 i + q = 0, and we have det (2.18) 2.2.Analysis to singular points of system (2.9) We first analyze the finite-far singular points of system (2.9).

.22)
With regard to the Hamilton energy at the singular point, the following conclusions can be proved easily: (2) In the case of q < 0 In this situation, the system (2.9) has three singular points Pi (x i , 0) (i = 1, 2, 3), where x2 x When b 2 > 0, det (J (x 1 , 0)) = det (J (x 3 , 0)) < 0, det (J (x 2 , 0)) > 0, then Pi (i = 1, 3) are saddle points, P2 is a center, and the Hamilton energy at these singular points has the following relation: P2 is a saddle point, and the Hamilton energy at these singular points has the following relation: (3) In the case of q = 0, p < 0 In this situation, the system (2.9) has three singular points Pi (x i , 0) (i = 1, 2, 3), where x1 = − √ −p,x 2 = 0,x 3 = √ −p.According to (2.17), we can get Notice that P2 (0, 0) is a higher-order singular point in this situation, the two real roots of the characteristic equation for system (2.9) at P2 are both zero, then now system (2.9) can be expressed as ) , thus, according to the discriminant method of singular point and higher-order singular point in [26,27], we can obtain the following conclusions: When b 2 > 0, P2 is a center, Pi (i = 1, 3) are saddle points, and the Hamilton energy at these singular points has the following relation: When b 2 < 0, P2 is a saddle point, Pi (i = 1, 3) are centers, and the Hamilton energy at these singular points has the following relation: In this situation, the system (2.9) has three singular points Pi (x i , 0) Since det (J (x 1 , 0)) = det (J (x 3 , 0)) = 0 and the Poincaré index of x1 and x3 is zero, we know that P1 and P3 are cusp points.Therefore, for the singular points of system (2.9) in this situation, we can obtain the following conclusions: When b 2 > 0, Pi (i = 1, 3) are cusp points, P2 is a saddle point, and the Hamilton energy at these singular points has the following relation: When b 2 < 0, Pi (i = 1, 3) are cusp points, P2 is a center, and the Hamilton energy at these singular points has the following relation: (5) With regard to the infinite-far singular points of system (2.9).By making the Poincaré transformations u = y x , z = 1 x and v = x y , z = 1 y to system (2.9) respectively, and the analysis shows that the system (2.9)only has one pair of infinite-far singular points A 1 and A 2 on the yaxis.When b 2 < 0, there exists a hyperbolic region around each A i (i = 1, 2), and when b 2 > 0, there exists an elliptic region around each A i (i = 1, 2).In addition, the circle of the Poincaré disk is also an orbit.

Global phase diagrams and analysis conclusions of system (2.9)
According to the above discussion, under different conditions of parameters b 2 , p and q, we can obtain nine global phase diagrams of system (2.9) as follows.
Since the homoclinic orbit of the planar dynamical system (2.9) corresponds to the bellshaped solution of equation (2.7), the saddle-saddle heteroclinic orbit corresponds to the kinkshaped solution of equation (2.7), and the closed orbit corresponds to the periodic solution of equation (2.7).Moreover, the bell-shaped, kink-shaped and periodic solutions of equation (2.7) correspond to the bell-shaped, kink-shaped solitary wave solutions and the periodic wave solutions of form (2.1) for equation (1.3), respectively.Therefore, we can obtain the following theorems from propositions 2.1 and 2.2.
(1) When q > 0, p < −4 √ q 3 , equation (1.3) has two kink-shaped solitary wave solutions and two bell-shaped solitary wave solutions (corresponding to the symmetrical heteroclinic orbits L ± (P 1 , P 5 ) and homoclinic orbits L ± (P 3 , P 3 ) in figure 2.1), there are also three families of periodic wave solutions (corresponding to a family of closed orbits surrounded by the symmetrical heteroclinic orbits L ± (P 1 , P 5 ) and two families of closed orbits surrounded by two homoclinic orbits L ± (P 3 , P 3 ) respectively in figure 2.1); (2) When q > 0, p = −4 √ q 3 , equation (1.3) has four kink-shaped solitary wave solutions and two families of periodic wave solutions (corresponding to the symmetrical heteroclinic orbits L ± (P 1 , P 3 ) and L ± (P 3 , P 5 ) , and two families of closed orbits surrounded by the symmetrical heteroclinic orbits respectively in figure 2 3) has two bell-shaped solitary wave solutions and two families of periodic wave solutions (corresponding to the homoclinic orbits L (P 1 , P 1 ) and L (P 5 , P 5 ), and the closed orbits surrounded by the homoclinic orbits in figure 2.3) ; (4) When 'q < 0' or 'q = 0, p < 0', equation (1.3) has two kink-shaped solitary wave solutions and a family of periodic wave solutions (corresponding to the symmetrical heteroclinic orbits L ± ( P1 , P3 ) and the closed orbits surrounded by the symmetrical heteroclinic orbits in figure 2.4).

Periodic wave solutions of Zakharov equation (1.3)
In this section, according to the global phase diagrams of system (2.9), the periodic solutions of equation (2.7) can be get under various parameter conditions respectively, then we obtain the periodic wave solutions of form (2.1) for equation (1.3).
Since the bounded orbits in the global phase diagrams of dynamical system (2.9) correspond to the bounded solutions of equation (2.7), and the first integral (2.15) is the general expression of orbits in the global phase diagrams of system (2.9), then the bounded solutions of equation (2.7) can be obtained by applying the first integral (2.15).
It can be seen that the system (2.9) is a Hamiltonian system, which satisfies that the Hamilton energy H of points on the same orbit is equal.So, under a given parameter condition and for ∀H ∈ R, the closed orbit C H = {(x, y) ∈ ℜ × ℜ|H(x, y) = H} with energy H in the global phase diagram can be uniquely determined.
With regard to the first integral (2.15), we get where h = 2 b2 H, and H = H(x, y) denotes the Hamilton energy of system (2.9) at (x, y).Since y = x ′ , through separation of variables, the issue that finding the bounded solutions of equation (2.7) can be transformed into solving the following integral ˆdx where and when b 2 > 0, taking F h (x) in the square root of (3.2), when b 2 < 0, taking −F h (x) in the square root of (3.2).It can be easily known that for a given parameter condition and a given

Y Guo et al
Hamilton energy value H, the orbit CH of the phase diagram corresponds to the solution of equation (3.2) determined by taking h = h = 2 b2 H. Based on the above discussion, the periodic wave solutions of form (2.1) for equation (1.3) obtained in this section and the solutions corresponding to the closed orbits in figures 2.1-2.9 are one-to-one correspondence.
3.1.Periodic wave solutions corresponding to the closed orbits around singular points P 2 and P 4 respectively in figures 2.1 and 2.2 x When b 2 > 0, q > 0, p < −4 √ q 3 , from figure 2.1, system (2.9) has two families of closed orbits located inside the homoclinic orbits L ± (P 3 , P 3 ) and around the centers P 2 and P 4 respectively.The Hamilton energy corresponding to the closed orbits satisfies from figure 2.2, system (2.9) has two families of closed orbits located inside the heteroclinic orbits L ± (P 1 , P 3 ) and L ± (P 3 , P 5 ), and around P 2 and P 4 respectively.The Hamilton energy corresponding to the closed orbits satisfies H (x 2 , 0) < H < H(0, 0).Since the Hamilton energy of points on the same periodic orbit is equal, we assume that the Hamilton energy of the points on the periodic orbits around P 2 and P 4 respectively in figures 2.1 and 2.2 is where (η 1 , 0) is the intersection point of the periodic orbit and the x-axis, and h 1 satisfies In this situation, the curve Y = F h1 (x) = x 6 + 3 2 px 4 + 3qx 2 + 3h 1 has six intersection points with the x-axis which are symmetrical about the y-axis, then F h1 (x) can be expressed as where λ 1 , λ 2 and λ 3 are three real roots of equation z 3 + 3 2 pz 2 + 3qz + 3h 1 = 0, and 0 < λ 1 < λ 2 < λ 3 .Substituting the above F h1 (x) into (3.2), and make x 2 = φ, we get Here we take φ ∈ (λ 1 , λ 2 ) and let α 1 = λ1−λ2 λ1−λ3 , then make the transformation substituting the above into (3.5),we have where . By applying the Jacobi elliptic function integral knowledge [28,29], we can get Substituting the above into (3.6) and simplifying it, we have where k 2 = λ3(λ2−λ1) λ2(λ3−λ1) .Based on the transformation a 2 (ξ) = x 2 = φ, we obtain the periodic solutions of equation (2.7) To sum up, we obtain the following theorem: where a ± P1 (ξ) are given by (3.8).λ 1 , λ 2 and λ 3 in modulus k 2 are three real roots of equation z 3 + 3 2 pz 2 + 3qz + 6 b2 H = 0, satisfying 0 < λ 1 < λ 2 < λ 3 ; a + P1 (ξ) and a − P1 (ξ) correspond to the closed orbits surrounded by the homoclinic orbits L + (P 3 , P 3 ) and L − (P 3 , P 3 ) respectively in figure 2.1, or they correspond to the closed orbits surrounded by the heteroclinic orbits L ± (P 3 , P 5 ) and L ± (P 1 , P 3 ) respectively in figure 2.2.

Periodic wave solutions corresponding to the closed orbits around singular points P 2 and P 4 respectively in figure 2.3
When b 2 > 0, q > 0, −4 √ q 3 < p < −2 √ q, from figure 2.3, system (2.9) has two families of closed orbits located inside the homoclinic orbits L (P 1 , P 1 ) and L (P 5 , P 5 ), and around the singular points P 2 and P 4 respectively.The Hamilton energy corresponding to the closed orbits satisfies H (x 2 , 0) < H < H (x 1 , 0) (figure 3.3).
curve family and bounded orbits.
Since the Hamilton energy of points on the same periodic orbit is equal, we assume that the Hamilton energy of the points on the periodic orbits around P 2 and P 4 respectively in figure 2.3 is where (η 3 , 0) is the intersection point of the periodic orbit and the x-axis, and h 3 satisfies In this situation, the curve Y = F h3 (x) = x 6 + 3 2 px 4 + 3qx 2 + 3h 3 has six intersection points with the x-axis which are symmetrical about the y-axis, then F h3 (x) can be expressed as where λ 1 , λ 2 and λ 3 are three real roots of equation z 3 + 3 2 pz 2 + 3qz + 3h 3 = 0, and 0 < λ 1 < λ 2 < λ 3 .Substituting the above F h3 (x) into (3.2), and make x 2 = φ, we get

Periodic wave solutions corresponding to the closed orbits around centers P 1 , P 5 and P 3 respectively in figures 2.6 and 2.7
x When b 2 < 0, q > 0, p ⩽ −4 √ q 3 , from figure 2.6, system (2.9) has two families of closed orbits around the centers P 1 and P 5 respectively.And there is also a family of closed orbits located inside the heteroclinic orbits L (P 2 , P 4 ) and L (P 4 , P 2 ), and around the center P 3 .The Hamilton energy corresponding to the closed orbits satisfies H(0, 0) < H < H (x 2 , 0); y When b 2 < 0, q > 0, −4 √ q, from figure 2.7, system (2.9) has two families of closed orbits around the centers P 1 and P 5 respectively.And there is also a family of closed orbits located inside the heteroclinic orbits L (P 2 , P 4 ) and L (P 4 , P 2 ), and around the center P 3 .

Periodic wave solutions corresponding to the closed orbits when H
Since the Hamilton energy of points on the same periodic orbit is equal, we assume that the Hamilton energy of the points on the corresponding periodic orbits in figures 2.6 and 2.8 is where (η 5 , 0) is the intersection point of the periodic orbit and the x-axis, and h 5 satisfies In this situation, the curve ) has four intersection points with the x-axis which are symmetrical about the y-axis, then −F h 5 (x) can be expressed as where λ 1 , λ 2 and λ 3 are three real roots of equation z 3 + 3 2 pz 2 + 3qz + 3h 5 = 0, and λ 1 ⩽ 0 < λ 2 < λ 3 .Substituting the above −F h 5 (x) into (3.2), and make x 2 = φ, we get
{ Under the condition of b 2 < 0, q > 0, p = −2 √ q, when the Hamilton energy satisfies H(0, 0) < H < H (x 1 , 0), system (2.9) has a family of closed orbits located inside the heteroclinic orbits L , and around the center (0, 0) in figure 2.9; when the Hamilton energy satisfies H > H (x 1 , 0), system (2.9) also has a family of closed orbits surrounding the heteroclinic orbits L A same characteristic of such closed orbits is that there are only two intersection points when the orbit meets the x-axis, and these two points are symmetrical about the y-axis (figure 3.6).Assume that the Hamilton energy of the points on such orbits is where (η 6 , 0) is the intersection point of the periodic orbit and the x-axis, when q > 0, p < −2 √ q, h 6 satisfies In this situation, the curve ) has two intersection points with the x-axis which are symmetrical about the y-axis, then −F h 6 (x) can be expressed as where ∆ = ( 3 2 p + λ ) 2 + 12h 6 λ < 0, λ is the square value of the abscissa at intersection point of −F h 6 (x) and x-axis, and it is also the unique positive real root of equation z 3 + 3 2 pz 2 + 3qz + 3h 6 = 0. Substituting the above −F h 6 (x) into (3.2), and make x 2 = φ, we get ) . ( Now assume that ρ 1 and ρ 2 (here ρ 1 > ρ 2 ) are the two roots of the following linear duality equation with respect to ρ: thus we have 0, and d 2 > 0. Now we make the following transformation to the right side of (3.40): then the right side of (3.40) can be expressed as then the integral in (3.44) can be transformed into ˆdt √ (c where . Now substituting the above into (3.44),we can get ˆdφ By simplification and according to (3.40), we have By applying the Jacobi elliptic function integral knowledge [28,29], we have where k 2 = −c2d1 c1d2−c2d1 .Substituting the above into (3.45) and (3.43), by simplification, we obtain Based on the transformation a 2 (ξ) = x 2 = φ, we obtain the periodic solutions of equation (2.7) To sum up, we obtain the following theorem: Theorem 3.6.Assume b 2 < 0, if one of the following conditions holds: (1) q > 0, p < −2 √ q, the Hamilton energy satisfies H > H (x 2 , 0); (2) q > 0, −4 √ q 3 < p < −2 √ q, the Hamilton energy satisfies H(0, 0) < H < H (x 1 , 0); (3) 'q < 0' or 'q = 0, p < 0', the Hamilton energy satisfies H > H(0, 0); (4) q > 0, p = −2 √ q, the Hamilton energy satisfies H(0, 0) , λ is the unique positive real root of equation z 3 + 3 2 pz 2 + 3qz + 3h 6 = 0. Now, we compare the periodic wave solutions we obtained with the periodic wave solutions of Zakharov equation (1.3) obtained in [18].
The coefficients of equations ( 1) and (2) studied in [18] and the abscissa of intersection points of the periodic orbit and the x-axis (see the left side) have the following relationship with the coefficients of equation (1.3) studied in this paper and the abscissa of intersection points of the periodic orbit and the x-axis (see the right side): From theorem 3.2, when 'b 2 > 0, q < 0' or 'b 2 > 0, q = 0, p < 0', the amplitudes (3.14) of periodic wave solutions (3.15) of equation ( 1.3) are , where k 2 = λ2(λ3−λ1) λ3(λ2−λ1) .Now according to the parameter relationship (3.54) and Jacobi elliptic function relation sn 2 (z) + cn 2 (z) = 1, we have where ) . It can be seen that the above formula is completely consistent with the amplitude of periodic wave solutions (43) obtained in [18].Through the above comparison, we know that the elliptic function periodic wave solutions obtained in [18] correspond to the family of closed orbits located inside the heteroclinic orbits L ( P1 , P3 ) and L ( P3 , P1 ) , and around the center P2 in figure 2.4 of this paper.Therefore, only individual periodic wave solutions of the Zakharov equation (1.3) have been obtained in the previous literature.In this paper, by using the analysis method based on the first integral and various effective transformations, the elliptic function periodic wave solutions corresponding to all closed orbits in the global phase diagrams 2.1-2.9 are obtained, that is, all the periodic wave solutions of the Zakharov equation (1.3) are obtained.In addition, the evolutionary relationship of the periodic wave solutions obtained in this section with respect to the Hamiltonian system energy corresponding to the studied equation will be discussed below.

Relationship between Hamilton energy and periodic wave solutions, solitary wave solutions of Zakharov equation (1.3)
Through the analysis method based on the first integral and various effective transformations, the periodic wave solutions of equation (1.3) under various conditions of parameters b 2 , p and q are obtained in section 3.In this section, we further study the evolution of the periodic wave solutions of equation (1.3) with respect to Hamilton energy.

Evolution of periodic wave solutions for equation (1.3) with respect to the limit of Hamilton energy
In this subsection, we study the evolution of periodic wave solutions when the Hamilton energy H corresponding to the periodic wave solutions of equation (1.3) tends to the Hamilton energy corresponding to the solitary wave solutions.
Label  Considering the length of this paper, we choose the case 'b 2 > 0, q > 0, p < −4 √ q 3 ' as an example to analyze, and the analysis of other cases is similar.Now we discuss in two cases.
(1) In the first case, when the Hamilton energy H satisfies H (x 2 , 0) < H < H(0, 0), equation (1.3) has periodic wave solutions ) (given by (3.9)) whose amplitudes are a ± P1 (ξ) (given by (3.8)), which correspond to the closed orbits centered on P 2 and P 4 in system (2.9).According to figure 2.1 or 3.1(a), when H = H(0, 0) = 0, system (2.9) has two symmetrical homoclinic orbits L ± (P 3 , P 3 ), that is, equation (2.7) has two bell-shaped solutions, then the corresponding equation (1.3) has two bell-shaped solitary wave solutions.Here we discuss the limit of ( b2 H → 0 − , each parameter in (3.7) and (3.8) has the following limit: )) From the above, we know that when H = H(0, 0), the two bell-shaped solitary wave solutions of form (2.1) for equation (1.3) corresponding to the two homoclinic orbits L ± (P 3 , P 3 ) of system (2.9) are where a ± S1 (ξ) are given by (4.1), and the following limits hold: { lim (2) In the second case, when the Hamilton energy H satisfies H(0, 0) < H < H (x 1 , 0), equation (1.3) has periodic wave solutions ) (given by (3.15)) whose amplitudes are a ± P2 (ξ) (given by (3.14)), which correspond to the closed orbits surrounding the homoclinic orbits L ± (P 3 , P 3 ).According to figure 2.1 or 3.1(a), when H = H (x 1 , 0), system (2.9) has two symmetrical heteroclinic orbits L (P 1 , P 5 ) and L (P 5 , P 1 ), that is, equation (2.7) has two kink-shaped solutions, then the corresponding equation (1.3) has two kink-shaped solitary wave solutions.Here we discuss the limit of each parameter in (3.13) and (3.14) has the following limit: From the above, we know that when H = H (x 2 , 0) = H 2 , equation (1.3) has the kink-shaped solitary wave solutions where a ± S4 (ξ) are given by (4.9), which correspond to the heteroclinic orbits L (P 2 , P 4 ) and L (P 4 , P 2 ) in figure 2.6 or 2.7, and the following limits hold: Now we study the limit of Same as getting the limits of the parameters in (4.9), for the amplitudes a ± P5 (ξ) of periodic wave solutions (4.12) From the above, we know that when H = H 2 , equation (1.3) has the bell-shaped solitary wave solutions { where a ± S5 (ξ) are given by (4.12), which correspond to the homoclinic orbits L (P 4 , P 4 ) and L (P 2 , P 2 ) respectively in figure 2.6 or 2.7.

Conclusion
In this paper, by applying the theory of planar dynamical system and the analysis method based on the first integral, we obtain all seven families of elliptic function periodic wave solutions of form (2.1) of Zakharov equation (1.3) with cubic and quintic nonlinear terms.And by studying the evolutionary limit of periodic wave solutions with respect to Hamilton energy and directly using the analysis method based on the first integral, all ten pairs of solitary wave solutions of equation (1.3) under various parameter conditions are given.In addition, through the evolution analysis of Hamilton energy corresponding to the periodic wave solution of Zakharov equation (1.3), it can be seen that it is the energy H of the Hamiltonian system corresponding to the studied equation taking values in different ranges that makes the traveling wave solution of this equation appear as periodic wave solution or solitary wave solution.In this paper, we apply the theory of planar dynamical system and the analysis method based on the first integral, and use multiple more skillful transformations, which not only overcome the computing difficulties caused by the cubic and quintic nonlinear terms, but also correspond the obtained solutions to the orbits in the global phase diagrams.Moreover, we reveal the relationship between the energy of the Hamiltonian system corresponding to the studied equation and its traveling wave solutions.The ideas and methods of this paper can also be applied to study other nonlinear evolution equations.
where H 1 and H 2 are given by (2.21) and (2.22) respectively.