Solitary and Periodic Wave Solutions of Sasa–Satsuma Equation and Their Relationship with Hamilton Energy

In this paper, we study the exact solitary wave solutions and periodic wave solutions of the S-S equation and give the relationships between solutions and the Hamilton energy of their amplitudes. First, on the basis of the theory of dynamical system, we make qualitative analysis on the amplitudes of solutions. )en, by using undetermined hypothesis method, the first integral method, and the appropriate transformation, two bell-shaped solitary wave solutions and six exact periodic wave solutions are obtained. Furthermore, we discuss the evolutionary relationships between these solutions and find that the appearance of these solutions for the S-S equation is essentially determined by the value which the Hamilton energy takes. Finally, we give some diagrams which show the changing process from the periodic wave solutions to the solitary wave solutions when the Hamilton energy changes.


Sasa-Satsuma equation (called S-S equation for short)
iz T + 1 2 z XX +|z| 2 z + iε z XXX + 6|z| 2 z X + 3z |z| 2 X � 0 was proposed by Sasa and Satsuma in studying the integrability of the Schrödinger equation with higher-order nonlinear terms, and it is a typical equation which describes the propagation of short pulses in an optical fiber [1][2][3][4]. e 1-soliton solutions and the N-soliton solutions of equation (1) were also given by Sasa and Satsuma in [1].
Although the S-S equation has been solved by various methods in many literatures, there are still some solitary wave solutions and periodic wave solutions for the S-S equation that have not been solved. In this paper, we will study solutions of the form (2) for the S-S equation (1) and expect to find its new periodic wave solutions or solitary wave solutions. At the same time, we will study the relationships between solutions and Hamilton energy for the S-S equation.
First, by employing the theory and method of planar dynamical system, we make detailed qualitative analysis on the amplitudes of solutions for the S-S equation and give the global phase portraits under different parameters. We also obtain the conditions for the existence of solitary wave solutions and periodic wave solutions. In Section 3, according to the conclusions of the qualitative analysis, we use the undetermined hypothesis method to obtain two bellshaped solitary wave solutions for the S-S equation, where one is new, bounded, and rational [17,18]. In Section 4, by the first integral method and appropriate transformation, we obtain six periodic wave solutions for equation (1), and the amplitudes of two solutions are in the fractional form of the Jacobi elliptic function sn 2 (φ, ξ), which are similar as the amplitudes of the periodic wave solutions found in [13]. However, for the rest four kinds of periodic wave solutions, their amplitudes are composed by dn(φ, k), cn(φ, k), and cos φ, which have not been obtained in the previous literatures. In Section 5, we study the evolutionary relationships between the solitary wave solutions, periodic wave solutions, and the Hamilton energy for the S-S equation and show the reason of the appearance of different kinds of solutions is that the corresponding Hamilton energy h takes different values. When h tends to be H(x 2 , 0), the periodic wave solutions will take the solitary wave solutions as the limit. At last, we will show this evolutionary process in the diagrams.
It is worth to point out that (1) by combining the qualitative analysis with the first integral method, we not only obtain the new bell-shaped solitary wave solutions (unimodal) and periodic wave solutions for equation (1), but also show the mathematical meaning for the solutions by establishing the corresponding relationships between the solutions and the bounded trajectories in the phase portraits.
(2) In the previous literatures [12,[19][20][21] for solving the solitary wave and periodic wave solutions of the nonlinear evolutionary equations, the relationships between solutions and the energy of system are rarely described. In this paper, we not only obtain the solutions of equation (1) but also show the relationships between the amplitudes of the solutions and the Hamilton energy for the corresponding system. en, we have the conclusion that when Hamilton energy h takes different values, equation (1) has different solutions in the form of (2). For a nonlinear complex system, it is important to understand the essential reasons for the various and complex phenomena in this system and only by this way can people really control and apply the system which corresponds to the S-S equation.

Qualitative Analysis
In this section, we will employ the theory and method of planar dynamical system [29,30] to make qualitative analysis on the traveling wave solutions of the form (2) for the S-S equation (1) and give the global phase portraits for the amplitudes of these solutions under different parameters. en, several conclusions for the existence of these solutions can be obtained.
Assuming that the S-S equation (1) has solutions in the form of (2), by substituting (2) into equation (1), we have en, taking the real and imaginary parts of (7) as zero, we can get In (8), we set the coefficients of u, u ″ , and |u| 2 u as zero and obtain 2 Complexity According to (10), we have By substituting (11) into (9), we have en, integrating on both sides of equation (12) with respect to ξ, we can obtain where g 0 is a constant. According to (13), we have where p � ((1 − 12εc)/48ε 2 ) and q � (− g 0 /4ε). From the above discussions, we can deduce that for studying the solutions of equation (1), we can start from equation (14).
Let x � u(ξ) and y � u ′ (ξ), equation (14) is equivalent to the following planar dynamic system: dx dξ � P(x, y), Since system (15) is a Hamiltonian system, it has the following first integral: where h � H(x, y) is the Hamilton energy at the point (x, y) for system (15). On the plane (x, y), the number of finite singularities of system (15) depends on the number of the real roots for equation f(x) � x 3 + px + q � 0. Here, we note the discriminant of f(x) � 0 as and then we can get the following conclusions: (a). If Δ > 0, f(x) � 0 has one real root and two complex roots.
(b). If Δ � 0, f(x) � 0 has three real roots (two of them are equal): (c). If Δ < 0, f(x) � 0 has three different real roots: where cos θ � (3 In this paper, we mainly discuss the bounded traveling wave solutions of equation (1), so we always assume that p < 0 and Δ ≤ 0 in the following. Under this assumption, there exist three finite singular points P i (x i , 0)(i � 1, 2, 3) in system (15), where x i (i � 1, 2, 3) is the real root of f(x) � 0 and satisfies Here we assume x 1 ≤ x 2 ≤ x 3 , and the Jacobi matrix at P i for system (15) can be expressed as follows: where f ′ (x i ) � 3x 2 i + p(i � 1, 2, 3). (1) In the case of q � 0, we have Δ < 0 now, and equation f(x) � 0 has three real roots (15). From (21), we have the determinants of the Jacobi matrix J(x i , 0)(i � 1, 2, 3) as follows: So P 1 and P 3 are centers, and P 2 is a saddle point. (2) In the case of q < 0.
(I) When Δ � 0, the three real roots x 1 � x 2 < 0 < x 3 for the equation f(x) � 0 correspond to two Complexity different finite singular points P 1 (x 1,2 , 0) and P 3 (x 3 , 0) of system (15). From (20), we have en, we obtain the determinants of the Jacobi matrix J(x i , 0) at the singular points P 1,2 and P 3 as follows: So P 1,2 is a cusp and P 3 is a center.
(II) When Δ < 0, there exist three different real roots in the equation f(x) � 0 which satisfy 3 , and the corresponding system (15) has three different finite singular points P i (x i , 0)(i � 1, 2, 3). en, from (21), we get the determinants of the Jacobi matrix J( So, P 1 and P 3 are centers, and P 2 is a saddle point. (3) In the case of q > 0.
(I) When Δ � 0, the three real roots (15), so we get the determinants of the Jacobi matrix So P 1 is a center and P 2,3 is a cusp. (II) When Δ < 0, the three different real roots of the equation 3 , and corresponding system (15) has three different finite singular points P i (x i , 0)(i � 1, 2, 3).
From (21), we get the determinants of the Jacobi matrix J(x i , 0)(i � 1, 2, 3) at P i (x i , 0)(i � 1, 2, 3) as follows: So, P 1 and P 3 are centers and P 2 is a saddle point.
According to the above qualitative analysis, we can obtain five global phase portraits of system (15) under different parameters (see Figures 1(a)-1(e)). en, we can get the following propositions from the above five global phase portraits.  (2) When Δ � 0, there exist one homoclinic orbit and numerous closed orbits in system (15) in the case of q < 0 or q > 0 (see Figures 1(d) and 1(e)).
Because the homoclinic orbits correspond to the bellshaped solutions of equation (14), and the periodic trajectories correspond to the periodic solutions of equation (14); on the basis of Proposition 1, we can obtain the following theorem. Theorem 1. Assume Δ ≤ 0 and p < 0.
(1) When Δ < 0 in the case of q < 0, q � 0, or q > 0, there exist two bell-shaped solitary wave solutions (corresponding to the homoclinic orbits in Figures 1

Solitary Wave Solutions of Equation (1)
In this section, we use the undetermined hypothesis method to obtain two kinds of solitary wave solutions of the form (2) for equation (1). Since equation (1) can be transformed into equation (14), we only need to consider the bounded solutions for equation (14) [31,32]. Here, we assume that equation (14) has solutions with the following forms: 4 Complexity where A, B, D, m are undetermined parameters. Substituting (28) and (29) into (14), we can get the solutions of equation (14), and then, we have the solitary wave solutions of equation (1). So the following theorem can be obtained. (1) has bell-shaped solitary wave solution: where the bounded solutions u ± S1 (ξ) of equation (14) can be expressed as follows: Besides, (31) can be equivalently expressed as (2) When p < 0 and Δ � 0, equation (1) has bell-shaped solutions: where the bounded solutions u ± S2 (ξ) of equation (14) can be expressed as follows: Note 1 (1) u − Si and u + Si (i � 1, 2) represent the solutions with − and + before radical sign, respectively.
(2) When q � 0 and x 0 � 0, u ± s1 (ξ) can be simplified as u ± S1 (ξ), which can be expressed as follows: So when q � 0, equation (1) has bell-shaped solutions: (3) Equation f(x) � x 3 + px + q � 0 has three real roots which satisfy x 1 < x 2 < x 3 , since we set x 0 as the real root for equation f(x) � 0 and it satisfies p + 3x 2 0 < 0, according to the discussions in Section 3, we know that here x 0 is x 2 .
Here, we point out that (1). When q ≠ 0 and Δ < 0, the amplitude u + S1 (ξ) of bellshaped solution z + S1 (ξ) for equation (1) corresponds to the homoclinic trajectory with P 2 as a saddle point and P 3 as a center in the global phase portraits Figures 1(b) and 1(c). e amplitude u − S1 (ξ) of bellshaped solution z − S1 (ξ) of equation (1) corresponds to the homoclinic trajectory with P 2 as a saddle point and P 1 as a center in the global phase portraits Figures 1(b) and 1(c). (2). When q � 0 and Δ < 0, the amplitudes u ± S1 (ξ) of bellshaped solutions z ± S1 (ξ) for equation (1) correspond to the homoclinic trajectories in the global phase portrait- Figure 1

Periodic Wave Solutions of Equation (1)
Similar to the solving process in the last section, we need to solve the periodic wave solutions for equation (14) if we want to get the periodic wave solutions in the form of (2) for equation (1).
Equation (14) corresponds to the planar dynamic system (15), and its first integral (16) is the general expression of the trajectories which correspond to the solutions for equation (14). erefore, we can use the first integral (16) to obtain the periodic wave solutions for equation (14), and then, we can get the periodic wave solutions for the S-S equation (1). In this section, we always assume that p < 0 and Δ ≤ 0 and three real roots From the first integral (16) of system (15), we can obtain By the separate variable method, we can translate the problem of solving equation (14) into the following integral: where Note 2. In eorem 2, by taking the Hamilton energy h � H(x 2 , 0) in (39) and integrating on it, the amplitude u ± S1 (ξ) of the bell-shaped solutions z ± S1 (ξ) for equation (1) can be obtained. Similarly, we can obtain the amplitude u ± S2 (ξ) of the bell-shaped solutions z ± S2 (ξ) for equation (1) in the case of Δ � 0 and q ≠ 0.

Periodic Wave Solutions Corresponding to Periodic Trajectories Surrounded by Symmetric Homogeneous Orbits.
When q � 0 and Δ < 0, there exist symmetric homoclinic orbits in system (15). Since the Hamilton energy on the same periodic trajectory is equal, we can set the Hamiltonian energy on the same periodic trajectory as follows: where (η 1 , 0) is the intersection of the periodic trajectory and the x-axis and satisfies |η 1 | < �� � − p √ . According to (16) Substituting three solutions en, we have the − F h (x) function curve and periodic trajectories when h 1 ∈ (− p 2 , 0) as shown in Figure 2.
en, (39) can be transformed into (42) According to the properties of the elliptic function integral [33,34], the following lemma can be obtained.
; then by integrating on the both sides of (42), we have where en, we can obtain the following theorem.
Here, we point out that the amplitudes u ± P1 (ξ) of the periodic wave solutions z ± P1 (ξ) for equation (1) correspond to the periodic trajectories, which are contained in the homoclinic trajectory centered on P 1 and P 3 in the global phase portrait (Figure 1(a)), respectively. Figures 1(b) and 1(c). When q ≠ 0 and Δ < 0, there exist asymmetric homoclinic orbits in system (15). In global phase portraits Figures 1(b) and 1(c), we set the Hamilton energy at the point (x, y) on the periodic trajectory surrounded by the homoclinic orbits as

Exact Periodic Wave Solutions Corresponding to Closed Trajectories Contained in Asymmetric Homoclinic Orbits in
where H(x, y) is given by (16) and (η 2 , 0) is the intersection of the periodic trajectory and the x-axis. From Figures 1(b) and 1(c), we can know that P i (x i , 0)(i � 1, 2, 3) are three singular points of system (15), and their abscissas x i (i � 1, 2, 3) satisfy x 1 < x 2 < x 3 . It can be verified that when q < 0, its corresponding − F h (x) function curve and periodic trajectories are shown in Figure 3. Set the four different real 2,3,4), and the roots satisfy can be expressed as follows: Since x � u(ξ), y � u ′ (ξ), the bounded solutions of equation (14) satisfy (1) When u ∈ (x 1 ′ , From (39) and (51), we have where . According to the definition of Jacobi elliptic function [33,34], we get en, by substituting (53) , we get the bounded solution of equation (14): (54) From (39) and (55), we have where By substituting (57) , we obtain the bounded solution of equation (14): From the above discussions, we have the following lemma.
en, according to Lemma 2 and equation (11), we can get the following theorem.
Here, we point out that, in eorem 4, u P2 (ξ) corresponds to the periodic trajectory contained in the homoclinic trajectory with P 1 as the center, and u P3 (ξ) corresponds to the periodic trajectory contained in the homoclinic trajectory with P 3 as the center in Figures 1(b)  and 1(c).   (1), which correspond to the periodic trajectories surrounded by the homoclinic trajectory in Figures 1(b)-1(e). Here, we set where (η 3 , 0) is the intersection of the periodic trajectory and the x-axis.
In addition, in the case of Δ � 0 and q > 0, we have H(x 1 , 0) < H(x 2 , 0) � H(x 3 , 0). And we can give the corresponding Y � − F h (x) function curves and periodic trajectories when the Hamilton energy h 3 satisfies H(x 1 , 0) < h 3 < H(x 2 , 0), which are shown in Figure 5 (note: in the case of Δ � 0 and q < 0, we can give the similar diagrams).

Relationships between the Solitary Wave Solutions and Periodic Wave Solutions of Equation (1)
In this paper, the two forms of bell-shaped solitary wave solutions for equation (1) have been given in Section 3, and the exact periodic wave solutions of equation (1) under different parameters are obtained in Section 4. In this section, we will study the evolutionary relationships between the solitary wave solutions and the periodic wave solutions for equation (1) and reveal the influence by the energy variation of the Hamilton system on the waveform of the solutions. (15). e planar dynamic system (15) is a Hamilton system, and the points on the same trajectory have the same Hamilton energy. When the parameters are fixed, for any h ∈ R, the trajectory

e Relationships between Periodic Wave Solutions, Solitary Wave Solutions for Equation (1), and the Value of Hamilton Energy of System
We take the global phase portrait Figure 1(b) as an example and discuss the relationships between the traveling wave solutions of the form (2) and the value which the Hamilton energy takes.
As shown in Figure 6, these three cases correspond to the three types of trajectories in system (15) in the case of Δ < 0 and q < 0: the homoclinic trajectory, the periodic trajectory surrounding the homoclinic trajectory, and the periodic trajectory contained in the same trajectory. From the discussions in Sections 3 and 4, we know that in Figure 6, the homoclinic trajectories correspond to the bell-shaped solutions u ± S1 (ξ) of equation (14), and the periodic trajectories surrounding the homoclinic trajectories correspond to the periodic solutions u + P4 (ξ) (modulo m 2 < 1/2) of equation (14), and the periodic trajectories contained in the homoclinic trajectories correspond to the periodic solutions u P2 (ξ) and u P3 (ξ). en, we can obtain that, in the case of Δ < 0 and q < 0, the relationships between the value of Hamilton energy h and the solitary wave solutions, periodic wave solutions of equation (1) are as follows: (1) When h ≤ H(x 3 , 0), there is no bounded trajectory in system (15). At this time, the S-S equation (1) has no bounded traveling wave solution of the form (2). (2) When h ∈ (H(x 3 , 0), H(x 1 , 0)], there is a periodic trajectory centered on P 3 and contained in the homoclinic orbit in system (15). At this time, the S-S equation (1) has a cluster of periodic wave solution z + P4 (ξ) � u + P4 (ξ)e i((X/6ε)− (T/108ε 2 )) of the form (2) (modulo m 2 < 1/2), where u + P4 (xi) is given by (69).

e Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of the Symmetric Homoclinic
Trajectories. When Δ < 0 and q � 0, the global phase portrait Figure 1(a) with symmetric homoclinic orbit is given by the qualitative analysis in Section 2. In Section 3, the solitary wave solutions z ± S1 (ξ) of the form (2) of equation (1) which correspond to the symmetric homoclinic orbits are given by (36) and their amplitudes u ± S1 (ξ) � ± ��� � − 2p sec h( ��� � − 4p ξ) correspond to the homoclinic trajectories in the global phase portrait Figure 1(a) (with Hamiltonian energy h 1 � H(x 2 , 0) � 0). In Section 4.1, the periodic wave solutions z ± P1 (ξ) of the form (2) for equation (1) are given which correspond to the homoclinic trajectories inside the closed orbit. And their amplitudes u ± P1 (ξ) are given by (45). In Section 4.3, the periodic wave solutions z ± P4 (ξ) of equation (1) are given, which correspond to the closed orbits surrounding the homoclinic orbit, and their amplitudes u ± P4 (ξ) are given by (78).
When h 1 approaches the Hamilton energy H(x 2 , 0) � 0 which corresponds to the solitary wave solution, we have According to (45), we can obtain . is indicates that when h 1 ⟶ 0 − , the periodic wave solutions converge to the solitary wave solutions; here, periodic wave solutions correspond to the closed trajectories contained in the symmetric homoclinic trajectories, and solitary wave solutions correspond to the homoclinic trajectories.
According to (78), when h 3 ⟶ 0 + , we have k ⟶ 1. So us, we have lim h 3 ⟶ 0 +z ± P4 (ξ) � z ± S1 (ξ), the periodic solutions z ± P4 (ξ) take the solitary wave solutions z ± S1 (ξ) as the limit, where z ± P4 (ξ) correspond to the closed trajectories that surround the symmetric homoclinic trajectories and z ± S1 (ξ) correspond to the homoclinic trajectories. erefore, we have the following theorem. Theorem 6. In the case of Δ < 0 and q � 0, when the Hamilton energy h 1 of system (15) tends to be 0 − , the periodic trajectories centered on P 1 and P 3 in Figure 1(a) will expand into homoclinic trajectories, and the corresponding periodic wave solutions z ± P1 (ξ) of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z ± S1 (ξ); when the Hamilton energy h 3 of the system gradually tends to be 0 + , the periodic trajectories surrounding the symmetric homoclinic trajectories in Figure 1(a) will shrink into Figure 6: Diagram of − F h (x) function curve and trajectories when Δ < 0, q < 0. 12 Complexity homoclinic trajectories, and the corresponding periodic wave solutions z ± P4 (ξ) of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z ± S1 (ξ).

e Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of Asymmetric Homoclinic
Trajectories. According to the analysis of the global phase portraits under different parameters for system (15) in Section 2, we know that there exist two asymmetric homoclinic trajectories in system (15) when Δ < 0, q < 0 or Δ < 0, q > 0. Since the analyses of the two cases are the same, we only need to consider the case of Δ < 0, q < 0. When the Hamilton energy h � H(x 2 , 0), there exist homoclinic trajectories centered on P 1 and P 3 in system (15), which correspond to the bell-shaped solutions z ± S1 (ξ) of the form (2) for equation (1), respectively; when h ∈ (H(x 1 , 0), H(x 2 , 0)), a cluster of periodic trajectories exist inside the homoclinic trajectories centered on P 1 or P 3 , which correspond to the periodic wave solutions z P2 (ξ) and z P3 (ξ) of the form (2) for equation (1); when h > H(x 2 , 0), there exist period trajectories surrounding the homoclinic trajectory in system (15), which correspond to the periodic wave solutions z + P4 (ξ)(m 2 > 1/2) of the form (2) for equation (1).
Next, we consider the relationships between the periodic wave solutions z + P4 (ξ) and z ± S1 (ξ), which correspond to the periodic trajectories surrounding the asymmetric homoclinic trajectories.
For scholars who want to know more about the application of Hamilton energy in the research of solitary wave solution, refer to literatures [35][36][37].

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.