DYNAMICAL BEHAVIOUR AND EXACT SOLUTIONS

Journal of Applied Analysis and Computation Volume 8, Number 1, February 2018, 250–271

Website:http://jaac-online.com/ DOI:10.11948/2018.250

DYNAMICAL BEHAVIOUR AND EXACT SOLUTIONS OF THIRTEENTH ORDER ¨ DERIVATIVE NONLINEAR SCHRODINGER EQUATION∗ Temesgen Desta Leta1 and Jibin Li1,2,† Abstract In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr¨ odinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits. Keywords Coupled integrable system, exact solution, thirteenth order derivative nonlinear Schr¨ odinger equation, homoclinic orbits, hetroclinic orbits, periodic orbits. MSC(2010) 34A05, 34C25-28, 34M55, 35Q51, 35Q53, 58F05, 58F14, 58F30, 35C05, 35C07, 34C60.

1. Introduction Many phenomena in physics, engineering and sciences are described by nonlinear partial differential equations (NPDEs). Exact traveling wave solutions of nonlinear evolution equations is one of the fundamental object of study in mathematical physics. When these exact solutions exist, they can help one to understand the mechanism of the complicated physical phenomena and dynamical process modeled by these nonlinear evolution equations. Derivative nonlinear Schr¨ odinger (DNLS) equations occur frequently in applications, for instance, in models incorporating self-steepening effects in optical pulses [8, 16]. Derivative nonlinear Schrödinger equations constitute a class of models which describe the evolution in physical media that has been drawn considerable attention both in a theoretical context and in many applied disciplines, notably in hydrodynamics, nonlinear optics and the study of Bose-Einstein condensates [2, 6, 19]. In the past decades a vast variety of the powerful and direct methods to find the explicit solutions of NPDE have been developed, such as inverse scattering transform † Corrosponding

author. Email address:[email protected] (J. Li) Department, Zhejiang Normal University, Street, 688 Yingbin Avenue, 321004 Jinhua, China 2 School of Mathematical Sciences, Huaqiao University, 362021 Quanzhou, Fujian, China ∗ The authors were supported by National Natural Science Foundation of China (11471289, 11571318, 11162020). 1 Mathematics

Dynamical behavior and exact solutions of thirteenth order...

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method, Backlund and Darboux transforms method, Lie group method, homotopy perturbation method, planner dynamical method, algebraic method, Jacobi elliptic function expansion method [1, 9, 10, 17, 20, 26] to list a few. In this paper, we apply the planner dynamical method for solving coupled derivatives of nonlinear Schrodinger equations of thirteenth order: iAt + Axx + f 0 (Φ)A + j 0 (Φ)iAx + (g(Φ))x iA −

s|A|xx A = 0, |A|

(1.1)

2

a 4 7 where f = µ2 Φ2 + ν3 Φ3 + ac j = 2aΦ4 and g = α2 Φ2 + β3 Φ3 , are λ Φ − 2λ Φ , analytic functions which depend on the squared amplitude Φ = |A|2 = φ2 + ψ 2 and “ 0 ” stands for the derivative with respect to Φ (see [11, 12] and the reference there in). The constant s > 1 corresponds to the resonant nonlinear Schrödinger (NLS) equation so-called because it admits both fission and fusion resonant solitonic phenomena [11, 21, 23, 25, 27]. In a wide class of NLS equations with underlying Hamiltonian structure was shown to be reducible to the integrable resonant NLS equation [21, 28]. This socalled resonant NLS equation has also been derived in plasma physics where it describes the transmission of long magneto-acoustic waves in a cold collisionless plasma subject to a transverse magnetic field. The resonant NLS equation and its (2+1)-dimensional integrable extension to a resonant Davey-Stewartson system were introduced in a capillarity model context in [25,27]. The nonlinear Schrödinger equation is an example of a universal nonlinear model that describes many physical nonlinear systems [5]. In the setting of optical fiber waveguides, terms involving λ, µ and α are associated with group velocity dispersion, Kerr (cubic) nonlinearity and self-steeping (or, more precisely, an effect which may be converted to selfsteeping form following a gauge transformation). Recent interests on coupled nonlinear Schrödinger systems, on a class of propagating wave patterns for families of derivative nonlinear Schrödinger equations of seventh (septic), ninth (nonic) and thirteenth order, which incorporates de Broglie-Bohm quantum potentials and which admit integrable Ermakov-Ray-Reid sub-systems have bought attention of researchers [4, 29]. A procedure recently employed by [5, 7, 13, 18] (the application of a pair of invariants of motion) is also applied here. To analyze the traveling wave solution with the form:

A(x, t) = [(φ(ξ) + iψ(ξ))]exp(i(νx − λt)), ξ = x − µt,

(1.2)

where µ, ν and λ are related to the nonlinearity induced shifts in three quantities, namely, group delay, carrier frequency and propagation constant respectively. Substituting equation (1.2) into equation (1.1), separating the real and imaginary parts, one obtains the coupled nonlinear integrable system respectively, dφ = q1 , dξ

dq1 = −(f 0 + λ − ν 2 )φ + (j 0 + 2ν − µ)ψ˙ + g 0 (φ2 + ψ 2 )• ψ dξ  !2  ˙ + ψψ) ˙ • ˙ + ψψ ˙ ( φφ φφ  φ, + s − φ2 + ψ 2 φ2 + ψ 2

dψ = q2 , dξ

dq2 = −(f 0 + λ − ν 2 )ψ − (j 0 + 2ν − µ)φ˙ − g 0 (φ2 + ψ 2 )• φ dξ

(1.3)

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 + s

˙ + ψψ) ˙ • (φφ − φ2 + ψ 2

˙ + ψψ ˙ φφ φ2 + ψ 2

!2   ψ.

where, dot indicates a derivative with respect to x − µt. System (1.3) admits two independent integrals of motion and, accordingly, is integrable. Thus, we obtain the first integral of the dynamical invariant in the form of the Hamiltonian: να − µ 2 (2β − 1)ν 3 Φ − Φ 2λ 3λ a(µ − 2ν) 4 a2 7 s Φ˙ 2 ˆ 1, Φ + 2Φ − ≡h − λ λ 4 Φ

ˆ 2H(φ, ψ, q1 , q2 ) =q12 + q22 − (λ − ν 2 )Φ −

(1.4)

ˆ 1 corresponds to the Hamiltonian invariant. The identification where the constant h of constants of motion has been proven as key to the subsequent integration of other physically important nonlinear dynamical systems, such as the Ermakov - Ray - Reid systems with invariant of the type and the second integral [4, 13, 22, 29]: ˆ 2, ˆ ψ, q1 , q2 ) = q1 ψ − q2 φ − a Φ4 − β Φ3 − α Φ2 + 1 (µ − 2ν)Φ = h I(φ, λ2 3λ 4λ 2

(1.5)

ˆ 2 is constant of motion. The pair of integrals of motion equation (1.4) and where, h equation (1.5) allow explicit solution of the nonlinear coupled system (1.3) for φ and ψ in terms of quadrature. ˆˆ ˆ be an invariant manifold family of system (1.3) given by Let N h1 h2 ˆ 1 , Iˆ = h ˆ 2, h ˆ 1, h ˆ 2 ∈ R} ⊂ R4 . ˆ =h ˆ ˆ ˆ = {H N h1 h2

(1.6)

Using the identity ˙ 2 ≡ (φψ˙ − ψ φ) ˙ 2, (φ2 + ψ 2 )(φ˙ 2 + ψ˙ 2 ) − (φφ˙ + ψ ψ)

(1.7)

ˆ 1, h ˆ 2 ) gives rise and combined with equation (1.4) and equation (1.5), for fixed (h 2 dΦ 4Φ ˆ 1 − (λ − ν 2 )Φ − (να − µ) Φ2 − (2β − 1)ν Φ3 − a(µ − 2ν) Φ4 = 2h dξ 1−s 2λ 3λ λ 2 2 4a 4 (µ − 2ν) α 2 β 3 a 4 8 ˆ + 2 Φ − h2 − Φ+ Φ + Φ + Φ λ (1 − s) 1−s 2 4λ 3λ λ (1.8) From equation (1.5), [5] introduced the variable Θ = arctan∆. So that, ψ φ cosΘ = √ , sinΘ = √ , Φ Φ where ∆ =

φ ψ.

We see from equation (1.5) that,

ˆ 2 − 6λ(µ − 2ν)Φ + 3αΦ2 + 4βΦ3 + 12aΦ4 12λh dξ 12λ2 Φ 0 Z ξ Z ξ Z ξ Z 1 dξ α β a ξ 3 ˆ2 + Φdξ + Φ2 dξ + Φ dξ, (1.9) = − (µ − 2ν)ξ + h 2 4λ 0 3λ 0 λ 0 0 Φ Z

Θ(ξ) =

ξ


253

where ξ is a dummy variable of integration. Thus, if we know Φ(ξ) and Θ(ξ) from equation (1.8) and equation (1.9), then we may solve equation (1.1) and system (1.3) to obtain the following solutions: √ A(x, t) = i Φexp [−iΘ + i(νx − λt)] (1.10) and √ φ(ξ) =

√ ΦsinΘ

and ψ(ξ) =

ΦcosΘ.

(1.11)

The corresponding class of exact solutions of equation (1.1), is then given by 1 equation (1.10). Let dΦ dξ = 2 (1 − s)y, then we have a system dΦ 1 = (1 − s)y, dξ 2 dy = β 0 − β 1 Φ − β 2 Φ2 − β 3 Φ3 − β 4 Φ 4 − β 5 Φ5 − β 6 Φ 6 , dξ

(1.12)

where, ˆ1 + h ˆ 2 (µ − 2ν); β0 = 2h h i 1 ˆ2 ; β1 = 8λ(λ − ν 2 ) + λ(µ − 2ν)2 + αh 4λ i 1 h ˆ2 ; (6β − 3(µ − 2ν))α + 4β h β2 = 4λ i 4 h ˆ 2 λ + 1 α2 ; 2νβ − ν + β(µ − 2ν) − 2ah β3 = 3λ 4λ2 5αβ β4 = ; 6λ2 2 β5 = 2 9α2 + 5β 2 ; 3λ 7βa β6 = . 3λ2 Clearly, system (1.12) is a seven parameter system depending on the eight paˆ 1, h ˆ 2 ). It is abound with dynamical system [11,23,24]. rameter group (λ, µ, a, s, ν, δ, h ˆ 1, h ˆ 2 ) such that For a given parameter group (ν, µ) 6= (0, 0), we next take (h β0 ≡ 0,

ˆ 1 = (2ν − µ) × h ˆ 2. h 4

(1.13)

Then, for s = 2 system (1.12) reduces to 1 dΦ = − y, dξ 2 dy = −β1 Φ − β2 Φ2 − β3 Φ3 − β4 Φ4 − β5 Φ5 − β6 Φ6 , dξ

(1.14)

with the first integral 1 e H(Φ, y) = y 2 − Φ2 4

1 1 1 1 1 1 β 1 + β 2 Φ + β 3 Φ2 + β 4 Φ3 + β 5 Φ4 + β 6 Φ5 2 3 4 5 6 7

=e h. (1.15)

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We notice that, Rogers and Chow, [5] did not discuss the dynamical behavior for the cases when; β2 = β4 = β6 = 0 when α = β = 0, β5 6= 0. Moreover, they did not give all possible exact explicit parametric representations of the traveling wave solutions. Incorporating the above given conditions in equation (1.14), we have a new system: 1 dΦ = − y, dξ 2 (1.16) dy 2 4 = −Φ(β1 + β3 Φ + β5 Φ ). dξ In this paper, we use the method of dynamical systems [12,14,15] to investigate the dynamical behavior of system (1.16). In addition we give all possible exact explicit parametric representations of the traveling wave solutions of equation (1.1). This paper is organized as follows. In section 2, we state the dynamical behavior of system (1.16). In section 3, by using the bifurcation of phase portraits we investigate the exact traveling wave solution of equation (1.1) in different parametric regions. In section 4, we give the main result of the study.

2. Bifurcations of phase portraits of system (1.16) We consider system (1.16) which has the hamiltonian H(Φ, y) =

1 1 1 1 2 y − Φ2 ( β1 + β3 Φ2 + β5 Φ4 ) = h. 4 2 4 6

(2.1)

Write ∆ = β32 − 4β1 β5 . Clearly, system (1.16) has always the equilibrium point E0 (0, 0). If ∆ < 0 and β1 < 0 < β5 , then system (1.16) has two equilibrium points √ 1 √ 1 2 2 3+ ∆ and Φ2 = −β2β . If ∆ > 0 E1 (Φ1 , 0) and E2 (Φ2 , 0), where Φ1 = −β32β−5 ∆ 5 and β5 < 0 < β1 , system (1.16) has four equilibrium points E1 (−Φ1 , 0), E2 (−Φ2 , 0), E3 (Φ2 , 0) and E4 (Φ1 , 0). If ∆ = 0, and β1 = 0, then system (1.16) has a double equilibrium point at E12 (Φ1 , 0) = E1 (Φ1 , 0) and E34 (Φ2 , 0) = E2 (Φ2 , 0). ˆ (Φj , 0) be the coefficient matrix of the linearized system of system (1.16) Let M at an equilibrium point Ej . We have ˆ (0, 0) = β1 , J(Φj , y) = detM ˆ (Φj , y) = β1 + 3β3 Φ2 + 5β5 Φ4 . J(0, 0) = detM J(0, 0) = β1 implies that when β1 < 0, the equilibrium point E0 (0, 0) is a center point, when β1 > 0, equilibrium point E0 (0, 0) is a saddle point and while for β1 = 0 and β5 < 0, the equilibrium point E0 (0, 0) is a cusp point. We write that for H(Φ, y) = h given by equation (2.1), h0 = H(0, 0) = 0, h1,2 = H(±Φ1,2 , 0) =

Φ1,2 √ β3 ∆ ± (8β1 β5 − β32 ) . 24β5

(2.2)

For a fixed β1 < 0 or β1 ≥ 0, we change the parameter space β3 and β5√in the parameter plane (β√3 , β5 ). There are two parameter curves (L1 ) : β3 = 34 3β1 β5 and (L2 ) : β3 = 2 β1 β5 , which partition this parameter plane into four regions: (I), (II), (III), (IV ) shown in Fig.1. By using the above information to do qualitative analysis, we have the following bifurcations of phase portraits of system (1.16) in two cases as follows.


255

Figure 1. The parameter regions partitioned by bifurcation curves of (L1 ) and (L2 ).

Case 1. Assume that β1 0. In this case, the equilibrium points E0 (0, 0) is a center. The bifurcations of phase portraits of system (1.16) are shown in Fig.2(a)– (b).

(a) β3 > 0, β5 < 0, ∆ < 0. Figure 2.

(b) β3 = 0, β5 < 0.

Bifurcations of phase portraits of system (1.16), β1 0.

Case 2. Assume that β1 < 0. In this case, system (1.16) has five equilibrium points E0 (0, 0) and Ej (Φj , 0), j = 1, 2, 3, 4, 5. The origin E0 (0, 0), and Ej (Φj , 0), j = 1, 4 are centers, while E2 (Φ2 , 0) and E4 (Φ4 , 0) are saddle points. Specially, in Fig. 3(f), the equilibrium points E1 (−Φ2 , 0) and E2 (Φ2 , 0) are a cusp points. The bifurcations of phase portraits of system (1.16) are shown in Fig. 3(a)–3(f). Case 3. Assume that β1 > 0. When β5 ≤ 0, system (1.16) has three equilibrium points E0 (0, 0) and Ej (Φj , 0), j = 1, 2. The Equilibrium points E1 (Φ1 , 0) and E2 (Φ2 , 0) are center points while E0 (0, 0) is a saddle point. In addition, when β5 > 0, system (1.16) has five equilibrium points E0 (0, 0) and Ej (Φj , 0), j = 1, 2, 3, 4. The Equilibrium points E2 and E3 are center points while E0 , E1 , and E4 are saddle points. The bifurcations of phase portraits of system (1.16) are shown in Fig. 4(a)–4(f).

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(a) β3 > 0, β5 > 0, h2 < 0.

(d) β3 =

8 3

√

3β1 β5 , β5 < 0.

Figure 3.

(b) β5 < β3 < h1 < ∞.

4 3

√

3β1 β5 , 0
0, β5 < 0.

(c) β3 > 0, β5 < 0, h2 < h1 .

(f) β3 < 0, β5 > 0.

Bifurcations of phase portraits of system (1.16), β1 < 0

(a) β3 > 0, β5 > 0, h2 < 0.

√ (d) 0 < β3 < 2 β1 β5 , β5 < 0.

(b) 0 < β3 < 34 0, 0 < h1 < ∞.

√

3β1 β5 , β5
0, β5 < 0.

(c) β3 > 0, β5 < 0, h2 < h1 .

(f) β3 < 0, β5 > 0.

Figure 4. Bifurcations of phase portraits of system (1.16), β1 > 0


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3. Explicit parametric representations of the solutions of system (1.16) We now consider the exact explicit parametric representations of the solutions of √ system (1.16) depending on Φ and Θ. We see from equation (2.1) and the first equation of system (1.14) that r Z Ψ Z Φ dΨ dΦ 2β5 q q = . ξ= 3β 3β 3β 3 6h 6h 1 3 Ψ0 2 Ψ[ Φ0 +Φ2 ( + Φ2 +Φ4 ) +Ψ( 1 + 3β3 Ψ+Ψ2 )] β5

β5

2β5

β5

β5

2β5

(3.1) To find p the exact solutions, we consider all bounded orbits of system (1.16) with Φj = Ψj > 0, for j = 1, 2, 3, 4, M, where Ψj > 0. In this section, we discuss the exact solutions of equation (1.1) in the different regions of parameter plane.

3.1. Explicit parametric representations of the solutions of system (1.16) when β1 0 Consider case 1 in section 2 (see [Fig. 2(a), 2(b)]). In this subsection, we notice that β3 ≥ 0, β5 < 0, ∆ = 0. For h ∈ (h1 , h2 ), the level curves defined by H(Φ, y) = h, has a family of periodic orbits enclosing the origin. 3β3 2 2 3β1 4 = Now, from equation (2.1) we have y 2 = 2β3 5 6h β5 + Φ ( β5 + 2β5 Φ + Φ ) 8β5 8β5 ¯ ) = 3 (Ψ − α0 )Ψ (Ψ − a1 )2 + b21 . Thus, we have the 3 (Ψ − α0 )Ψ(Ψ − γ)(Ψ − γ following parametric representation: 12 α0 B(1 + cn(ω1 ξ, k)) Φ(ξ) = Ψ(ξ) = , (3.2) (B − A1 ) + (A1 + B)cn(ω1 ξ, k) q (A +B)2 −α2 2 2 2 where, A1 = (α0 − b1 ) + a1 , ω1 = 2|β5 3|A1 B , B 2 = a21 + b21 , k 2 = 1 4A1 B 0 , γ )2 γ 2 a21 = − γ−¯ , b21 = (γ+¯ , sn(·, k), cn(·, k), dn(·, k), sd(·, k) are Jacobin elliptic 4 2 functions, E(·, k), is the elliptic integrals of the second kind and Π(·, ·, k), is the elliptic integrals of the third kind [3]. Write that, p

α ¯0 =

A1 + B ¯ α0 B α0 B , β0 = , γ¯0 = . A1 − B A1 − B A1 + B

Thus, we have from equation (1.9) that 1 β(¯ α0 +1) α ¯ 02 Θ(ξ) = (2ν −µ)ξ + × Π arcsin(sn(ω ξ, k)), , k − α ¯ f (ω ξ, k) 1 0 1 1 2 3λ¯ γ0 α ¯ 0 (1− α ¯ 02 ) α ¯ 02 − 1 1 2  ! 12 α ¯0 1 Z Ψ Z Ψ ¯0 + γ¯0 + β ¯ γ ¯ α ¯ α γ ¯ β 0 0 0 α ¯0 ˆ 2 − 0 +  dξ + +h − + dξ γ¯0 1−cn(ω1 ξ, k) 4λ 0 α ¯ 0 1+ α ¯ 0 cn(ω1 ξ, k) 0 a + λ

Z 0

Ψ

β¯0 + αγ¯¯00 γ¯0 − + α ¯0 1+α ¯ 0 cn(ω1 ξ, k)

! 23 dξ,

(3.3)

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where s f1 (t) =

1 − α02 tan−1 k 2 + (k 0 )2 α02

s

! k 2 + (k 0 )2 α02 sd(t, k) , 1 − α02

if

α02 < k2 , −1

α02

α02 = k2 , −1 s "p # p k 2 +(k 0 )2 α02 dn(t, k)+ α02 −1sn(t, k) α02 −1 α02 1 p p ln , if = > k2 . 2 2 2 k 2 +(k 0 )2 α0 α0 − 1 k 2 +(k 0 )2 α02 dn(t, k)− α02 −1sn(t, k) = sd(t, k),

if

α02

Hence, we have the following solution equation (1.1): A(x, t) = i

α0 B(1 + cn(ω1 ξ, k)) (B − A1 ) + (A1 + B)cn(ω1 ξ, k)

41 × exp [−iΘ + i(νx − λt)] , (3.4)

where, Θ(ξ) is given by Eq. equation (3.3).

3.2. Explicit parametric representations of the solutions of system (1.16) when β1 < 0 In this section, we consider case 2 of section 2 for β1 < 0, (see [Fig. 3(a)–3(h)]). (1) β3 > 0, β5 > 0, h ∈ (0, h1 ] (see [Fig. 3(a)]) (i) Corresponding to the family of periodic orbits defined by H(Φ, y) = h, h ∈ (0, h1 ), we have from equation (2.1) that y 2 = 8β3 5 (r1 − Ψ)(r2 − Ψ)Ψ(Ψ − r4 ), where r4 < 0 < r2 < r1 . Thus, we have the following parametric representation of the family of periodic orbits of system (1.16): √

1 Φ(ξ) = ± Ψ(ξ) = ± r4 1 − 1 − α12 sn2 (ω2 ξ, k) q β5 r1 (r2 −r4 ) 2 2 r1 −r4 2 where, α12 = r2r−r , k = α . , ω = 2 1 r2 −r4 6 4 p

12 ,

(3.5)

It follows from equation (1.9) that, ! ˆ2 1 h βr4 β α12 Θ(ξ) = (2ν − µ) − √ + ξ+ Π arcsinh(sn(ω2 ξ, k)), 2 ,k 2 r4 3λ 3λ α1 − 1 21 √ Z ˆ2 α r4 Ψ h sn(ω2 ξ, k) 1 + √ ln + 1− dξ α1 r4 cn(ω2 ξ, k)+dn(ω2 ξ, k) 4λ 0 1−α12 sn2 (ω2 ξ, k) 3 Z 32 ar42 Ψ 1 + 1− dξ. (3.6) λ 0 1 − α12 sn2 (ω2 ξ, k) Hence, we have the following solution equation (1.1): A(x, t) = i r4 1 −

1 2 1 − α1 sn2 (ω2 ξ, k)

where, Θ(ξ) is given by equation (3.6).

14 × exp [−iΘ + i(νx − λt)] ,

(3.7)


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(ii) The level curve defined by H(Φ, y) = h1 , have two hetroclinic orbits. In this case Φ2 = −Φ1 and the origin is a center. Now, from equation (2.1) we have 8β5 6h 3β1 2 3β3 3 8β5 y2 = Ψ+ Ψ + Ψ + Ψ4 = (Ψ2 − Ψ)2 Ψ(Ψ − r1 ), 3 β5 β5 2β5 3 where 0 < r1 < Ψ2 = Φ22 . Then, the hetroclinic orbit has a parametric representations: Φ(ξ) = ±

p

Ψ(ξ) = ± Ψ2 −

2Ψ2 (Ψ2 − r1 ) r1 cosh(

p

(Ψ2 − r1 )Ψ2 ξ) + (2Ψ2 − r1 )

! 21 .

(3.8)

It follows from equation (1.9) that, p r (Ψ2 −r1 )Ψ2 1 β 2β p Ψ2 −r1 Θ(ξ) = (2ν −µ)+ Ψ2 ξ − tan ξ (Ψ2 −r1 )Ψ2 arctan 2 3λ 3λ Ψ2 2 s p ˆ 2 Z Ψ r1 cosh( (Ψ2 − r1 )Ψ2 ξ) + r1 − 2Ψ2 h p +√ dξ Ψ2 0 r1 cosh( (r1 − Ψ2 )Ψ2 ξ) + Ψ2 ! 21 p √ Z r1 cosh( (Ψ2 − r1 )Ψ2 ξ + Ψ2 α Ψ2 Ψ p + dξ 4λ r1 cosh( (Ψ2 − r1 )Ψ2 ξ) + (2Ψ2 − r1 ) 0 ! 23 p √ Z r1 cosh( (Ψ2 − r1 )Ψ2 ξ + Ψ2 a Ψ2 Ψ p + dξ. (3.9) λ r1 cosh( (Ψ2 − r1 )Ψ2 ξ) + (2Ψ2 − r1 ) 0 Hence, we have the following solution equation (1.1): ! 41 2Ψ2 (Ψ2 − r1 ) p ×exp [−iΘ + i(νx−λt)] , A(x, t) = i Ψ2 − r1 cosh( (Ψ2 − r1 )Ψ2 ξ) + (2Ψ2 − r1 ) (3.10) where, Θ(ξ) is given√by equation (3.9). (2) β5 < β3 < 43 3β1 β5 , 0 < h1 < ∞ (see [Fig. 3(b)]). (i) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (0, h1 ), there exist three families of periodic orbits of system (1.16), enclosing the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0), E0 (0, 0) respectively. Now, equation (3.1) has the forms r Z Ψ 8β5 dΨ p ξ= 3 (Ψ − r3 )(Ψ − r2 )Ψ(r1 − Ψ) r2 and

r

8β5 ξ= 3

Z 0

Ψ

dΨ p , Ψ(r3 − Ψ)(r2 − Ψ)(r1 − Ψ)

where 0 < r3 < r2 < Φ1 < r1 . Therefore, the periodic orbits enclosing the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0) has a parametric representations: 21 p r2 − r3 , (3.11) Φ(ξ) = ± Ψ(ξ) = ± r3 + 1 − α12 sn2 (ω3 ξ, k)

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q

β5 r2 (r1 −r3 ) r1 −r2 1 −r2 )r3 2 . where k 2 = (r (r1 −r3 )r2 , α2 = r1 −r3 , ω3 = 6 Thus, we have from equation (1.9) that 1 βr4 β(r2 − r3 ) Θ(ξ) = (2ν − µ) + ξ+ × Π arcsinh (sn (ω3 ξ, k)) , α22 , k 2 3λ 3λ 12 32 Z Z Ψ r2 − r3 1−α22 sn2 (ω3 ξ, k) a Ψ ˆ r3 + + h2 dξ + dξ. r2 −r3 α22 sn2 (ω3 ξ, k) λ r2 1 − α22 sn2 (ω3 ξ, k) r2 (3.12)

Hence, we have the following solution equation (1.1): A(x, t) = i r3 +

r2 − r3 1 − α22 sn2 (ω3 ξ, k)

14 × exp [−iΘ + i(νx − λt)] ,

(3.13)

where, Θ(ξ) is given by equation (3.12). The periodic orbits enclosing the equilibrium points E0 (0, 0) has the parametric representation: p Φ(ξ) = ± Ψ(ξ) = ± r1 −

r1 1 − α22 sn2 (ω3 ξ, k)

21 ,

(3.14)

r3 1 −r2 )r3 2 where k 2 = (r (r1 −r3 )r2 , α3 = |r3 −r1 | . Thus, we have from equation (1.9) that βr1 1 βr1 (2ν − µ) + Π arcsinh(sn(ω3 ξ, k)), α32 , k Θ(ξ) = ξ− 2 3λ 3λ s √ Z s ˆ2 Z Ψ α r1 Ψ 1 h 1 dξ + 1− +√ dξ r1 0 4λ 0 1 − α32 sn2 (ω3 ξ, k) 1 − 1−α2 sn12 (ω ξ,k) 3

+

a λ

Z

Ψ r1 −

0

3

r1 1 − α32 sn2 (ω3 ξ, k)

32 dξ.

(3.15)

Hence, we have the following solution equation (1.1): A(x, t) = i r1 −

r1 2 1 − α3 sn2 (ω3 ξ, k)

14 × exp [−iΘ + i(νx − λt)] ,

(3.16)

where, Θ(ξ) is given by equation (3.15). (ii) The level curves defined by H(Φ, y) = h1 there exist two homoclinic orbits of system (1.16), enclosing the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0); two hetroclinic orbits connecting E2 (−Φ2 , 0) and E3 (Φ2 , 0), enclosing the equilibrium point E0 (0, 0). Corresponding to the two homoclinic orbits, equation (3.1) becomes r Z ΨM 8β5 dΨ p ξ=± , 3 (Ψ − Ψ ) (ΨM − Ψ)Ψ Ψ 3 where, 0 < Φ3 < Φ4 < ΦM , and ω4 =

q

2β5 Ψ3 |Ψ3 −ΨM | . 3


261

Thus, we obtain the parametric representations of the homoclinic orbit: p Φ(ξ) = ± Ψ(ξ) = ± Ψ3 +

2Ψ3 (ΨM − Ψ3 ) ΨM cosh(ω4 ξ) − (ΨM − 2Ψ3 )

21 .

(3.17)

Thus, we have from equation (1.9) that s  1 |Ψ −Ψ | β β p 1 3 M  Ψ3 (ΨM −Ψ3 )arctan Θ(ξ) = (2ν −µ)+ Ψ3 ξ + tanh ω4 ξ 2 3λ 3λ Ψ3 2 1 ΨM cosh(ω4 ξ) + (2Ψ3 − ΨM ) 2 dξ Ψ3 ΨM cosh(ω4 ξ) + ΨM Ψ3 ) Ψ 1 Z ΨM Ψ3 ΨM cosh(ω4 ξ) + ΨM Ψ3 ) 2 α dξ + 4λ Ψ ΨM cosh(ω4 ξ) + (2Ψ3 − ΨM ) 3 Z Ψ3 ΨM cosh(ω4 ξ) + ΨM Ψ3 ) 2 a ΨM + dξ, λ Ψ ΨM cosh(ω4 ξ) + (2Ψ3 − ΨM ) ˆ2 +h

Z

ΨM

(3.18)

Hence, we have the following solution equation (1.1): A(x, t) = i Ψ3 +

2Ψ3 (ΨM − Ψ3 ) ΨM cosh(ω4 ξ) − (ΨM − 2Ψ3 )

41 × exp [−iΘ + i(νx − λt)] , (3.19)

where, Θ(ξ) is given by equation (3.18). Next, for the two hetroclinic orbits, equation (3.1) becomes r Z Ψ dΨ 8β5 p ξ=± . 3 (Ψ − Ψ) (ΨM − Ψ)Ψ 0 3 Thus, we obtain the parametric representation of a hetroclinic orbits as follows: p Φ(ξ) = ± Ψ(ξ) = ± Ψ3 −

2Ψ3 (ΨM − Ψ3 ) ΨM cosh(ω4 ξ) + (ΨM − 2Ψ3 )

21 , ξ ∈ (0, ∞). (3.20)

Thus, we have from equation (1.9) that ! r Ψ3 1 βΨ3 β p 1 Θ(ξ) = (2ν −µ)+ ξ+ Ψ3 (ΨM −Ψ3 )arctan tanh( ω5 ξ) 2 3λ 3λ ΨM −Ψ3 2 21 Z Ψ α 2(ΨM − Ψ3 )Ψ3 + Ψ3 − 4λ 0 ΨM cosh(ω5 ξ) + (ΨM − 2ψ3 ) 23 Z a Ψ 2(ΨM − Ψ3 )Ψ3 + Ψ3 − dξ λ 0 ΨM cosh(ω5 ξ) + (ΨM − 2ψ3 ) ˆ 2, − Σ(ξ)h (3.21) where ω5 =

q

2β5 Ψ3 |ΨM −Ψ3 | . 3

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Hence, we have the following solution equation (1.1): 41 2Ψ3 (ΨM − Ψ3 ) A(x, t) = i Ψ3 − × exp [−iΘ + i(νx − λt)] , ΨM cosh(ω4 ξ) + (ΨM − 2Ψ3 ) (3.22) where, Θ(ξ) is given by equation (3.21). (iii) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (h1 , ∞), there exists a family of periodic orbits of system (1.16), enclosing five equilibrium points. It has √ the same parametric representation as equation (3.4). (3) β3 = 38 3β1 β5 , β5 < 0 (see [Fig. 3 (d)]) In this case, we have that h2 < 0 < h1 . (i) Corresponding to the level curves defined by H(Φ, y), h ∈ (h2 , 0], there exists a family of two periodic orbits of system (1.16), enclosing the equilibrium point E1 (−Φ1 , 0) and E4 (Φ1 , 0). It has the same parametric representation as equation (3.13). (ii) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (0, h1 ), there exist three families of periodic orbits of system (1.16), enclosing the equilibrium points E1 (−Φ1 , 0), E4 (Φ1 , 0) and E0 (0, 0) respectively. They have the same parametric representation as equation (3.13) and equation (3.16). (iii) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist two homoclinic orbits of system (1.16), enclosing the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0), and two hetroclinic orbits, enclosing the equilibrium point E0 (0, 0). They have the same parametric representation as equation (3.19) and equation (3.22). (iv) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (h1 , ∞), there exists a family of periodic orbits of system (1.16), enclosing five equilibrium points. It has the same parametric representation as equation (3.4). (4) β3 > 0, β5 < 0. (see [Fig. 3(e)]) In this case, we have that ∆ = 0, h2 < 0 < h1 . (i) For h ∈ (h2 , 0), the level curves defined by H(Φ, y) = h, there exists a family of periodic orbits of system (1.16), enclosing the equilibrium point E0 (0, 0). It has the same parametric representation as equation (3.4). (ii) For h ∈ (0, h1 ), the level curves defined by H(Φ, y) = h, there exist three families of periodic orbits of system (1.16), enclosing the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0), and E0 (0, 0), respectively. They have the same parametric representation as equation (3.19) and equation (3.22). (iii) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist two homoclinic orbits of system (1.16), enclosing the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0) and two hetroclinic orbits, enclosing the equilibrium point E0 (0, 0). They have the same parametric representation as equation (3.13) and equation (3.16). (iv) For h ∈ (h1 , ∞), to the level curves defined by H(Φ, y) = h, there exists a family of periodic orbits of system (1.16), enclosing five equilibrium points. It has the same parametric representation as equation (3.4). (5) β3 < 0, β5 > 0. (see [Fig. 3(f)]) In this case, we have S that Φ1 = Φ2 , h2 = h1 . (i) For h ∈ (0, h1 ) (h1 , ∞) the level curves defined by H(Φ, y) = h, there exists a family of periodic orbits of system (1.16), enclosing the equilibrium point E0 (0, 0). It has the same solution as equation (3.4). (ii) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist two hetroclinic orbits connecting the equilibrium point (cusp) E2 (−Φ2 , 0) and E3 (Φ2 , 0).


263

For two hetroclinic orbits, equation (3.1) becomes r Z Ψ 8β5 dΨ p , ξ=± 3 0 (Ψ2 − Ψ) (Ψ2 − Ψ)Ψ √ where, Φ2 = ± Ψ2 . Then, we have the following parametric representation of Φ(ξ): p p Φ(ξ) = ± Ψ(ξ) = ± Ψ2 1 −

4 4 − (Ψ2 ξ)2

21 , ξ ∈ (0, ∞).

(3.23)

Thus, we have from equation (1.9) that Θ(ξ) =

β 1 (2ν − µ) + 2 2λ

ˆ2 h ξ+ |Ψ2 |

p

4 − (Ψ2 ξ)2 − 2ln

β|Ψ2 | p α|Ψ2 | p a + 4−(Ψ2 ξ)2 + 4−(Ψ2 ξ)2 + 3λ 4λ λ

Z 0

Ψ

2−

p

s 1−

4 − (Ψ2 ξ)2 Ψ2 ξ

4 4 − (Ψ2 ξ)2

!!

3 dξ. (3.24)

Hence, we have the following solution equation (1.1):

4 A(x, t) = i 1 − 4 − (Ψ2 ξ)2

41 × exp [−iΘ + i(νx − λt)] ,

(3.25)


3.3. Explicit parametric representations of the solutions of system (1.16) when β1 > 0 In this section we consider case 4 of section 2, for β1 > 0. (see [Fig. 4(a)–4(g)]) (1) β3 > 0, β5 > 0, h2 < h1 = 0. (see [Fig. 4(a)]) (i) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (h2 , h1 ), there exist two families of periodic orbits of system (1.16). Equation (3.1) has the form r Z Ψ dΨ 8β5 p ξ= , 3 (Ψ − r4 )Ψ(Ψ − r2 )(r1 − Ψ) r2 where r4 < 0 < r2 < r1 . It gives rise to the parametric representations of two families of the periodic orbits of system (1.16) as follows: 12 r2 Φ(ξ) = ± Ψ(ξ) = ± , (3.26) 1 − α32 sn2 (ω3 ξ, k) q −r2 )|r4 | β5 r1 −r2 1 2 = where k 2 = (rr11(r , α < 1, ω = 6 4 r1 2 6 r1 (r2 − r4 ). 2 −r4 ) Thus, we have from equation (1.9) that ! ˆ 2 Z Ψq 1 βr2 h 2 Θ(ξ) = (2ν −µ)ξ + Π arcsin (sn (ω6 ξ, k)) , α4 , k + √ 1−α42 sn2 (ω6 ξ, k)dξ 2 3λ r2 r2 p

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T. Leta & J. Li

+

α√ r2 4λ

Z

Ψ

r2

dξ p

1−

α42 sn2 (ω6 ξ, k)

+

a λ

3

r22

Z

Ψ

r2

dξ p 3

1−

α42 sn2 (ω6 ξ, k)

.

(3.27) Hence, we have the following solution equation (1.1): A(x, t) = i

r2 1 − α42 sn2 (ω6 ξ, k)

14 × exp [−iΘ + i(νx − λt)] ,

(3.28)

where, Θ(ξ) is given by equation (3.27). (ii) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist two homoclinic orbits of system (1.16) to the multiple equilibrium point at E0 (0, 0). Equation (3.1) has the form r

8β5 ξ=± 3

Z

ΨM

dΨ q Ψ Ψ

Ψ

4 3 |β3 |

. −Ψ

Hence, we obtain the following parametric representations of system (1.16): Φ(ξ) = ±

p

Ψ(ξ) = ±

12|β3 | 9 + 16β32 ξ 2

21 .

(3.29)

Thus, we have from equation (1.9) that " !# p ˆ2 q 9+16β32 ξ 2 h 1 β 2 2 9+16β3 ξ +3ln ξ + Θ(ξ) = − (µ−2ν)+ √ + arctan(4β3 ξ) 2 2 4β3 3λ 2 3β3 ! p √ √ Z 9 + 16β32 ξ 2 a 3 12β3 ΨM α 3 dξ p + . (3.30) + √ ln ξ + 3 4β3 λ 8λ β3 9 + 16β32 ξ 2 Ψ Hence, we have the following solution equation (1.1): A(x, t) = i

12|β3 | 9 + 16β32 ξ 2

41 × exp [−iΘ + i(νx − λt)] ,

(3.31)

where, Θ(ξ) is given by equation (3.30). (iii) For h ∈ (0, ∞), the level curves defined by H(Φ, y) = h, there exists a family of periodic orbits of system (1.16), enclosing three equilibrium points. It has the same parametric representation of solution as equation (3.4). √ (2) 0 < β3 < 34 3β1 β5 , β5 < 0. (see [Fig. 4(b)]) (i) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (h2 , 0), there exist two families of periodic orbits of system (1.16). The parametric representations of solution of equation (1.1) is the same as equation (3.28). (ii) For h = 0, to the level curves defined by H(Φ, y) = h, there exist two homoclinic orbits of system (1.16). Equation (3.1) has the form r

8β5 ξ= 3

Z

ΨM

Ψ

dΨ p , Ψ (ΨM − Ψ)(Ψ − r4 )


265

√

√

+ ∆ 3+ ∆ , ΨM = −3β4β , ∆ = 9β32 + 12β1 |β5 |. It gives where r4 < 0 < ΨM , r4 = − 3β34β 5 5 rise to the parametric representations of homoclinic orbit of system (1.3) as follows:

 21

 p  Φ(ξ) = ± Ψ(ξ) = ±  √

∆ cosh

3β1 q

3β1 4β5 ξ

+ 3β3

  .

(3.32)

Thus, we have from equation (1.9) that ! p 2 3β1 |β5 | 1 β p √ Θ(ξ) = (2ν − µ)ξ + arctan tanh 2 3β3 + ∆ 18λβ1 |β5 |   12 Z ΨM 3β1 α   q + √  dξ 3β1 4λ Ψ ∆ cosh ξ + 3β 3 4β5 v s ! u ˆ 2 Z ΨM u √ h 3β1 t 3β3 + ∆cosh +√ ξ dξ 4β5 6β1 Ψ   32 Z ΨM a 3β1   q + √  dξ. 3β1 λ Ψ ∆ cosh 4β5 ξ + 3β3

1 4

s

3β1 ξ |β5 |

!

(3.33)

Hence, we have the following solution of equation (1.1):  14

  A(x, t) = i  √

∆ cosh

3β1 q

3β1 4β5 ξ

+ 3β3

  × exp [−iΘ + i(νx − λt)] ,

(3.34)

where, Θ(ξ) is given by equation (3.33). (iii) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (0, ∞), there exists a family of periodic orbits of system (1.16), enclosing three equilibrium points. It has the same parametric representation as equation (3.4). (3) β3 > 0, β5 < 0. (see [Fig. 4(c)]) In this case h2 < 0 < h1 , Φ1 = −Φ4 and Φ2 = −Φ3 . (i) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (h2 , 0), there exist two families of periodic orbits of system (1.16). The parametric representations of solution of equation (1.1) is the same as equation (3.28). (ii) Corresponding to the level curves defined by H(Φ, y) = 0, there exist two homoclinic orbits of system (1.16) with the following parametric representation: p Φ(ξ) = ± Ψ(ξ) = ±

2ΨM ΨL (ΨL − ΨM ) cosh(ω7 ξ) + (ΨM + ΨL )

12 ,

(3.35)

where 0 < ΨM < ΨL . Thus, we have from equation (1.9) that 1 β Θ(ξ) = (2ν − µ)ξ + √ arccoth 2 3λ ΨM ΨL

r

ΨM ΨL

! tanh

p 1 ΨM ΨL ξ 2

266

T. Leta & J. Li ΨM p ˆ2 h (ΨM + ΨL ) + (ΨL − ΨM )cosh(ω7 ξ)dξ 2ΨM ΨL Ψ 21 Z ΨM α 2ΨM ΨL + 4λ Ψ (ΨL − ΨM ) cosh(ω7 ξ) + (ΨM + ΨL ) 32 Z ΨM 2ΨM ΨL a dξ. + λ Ψ (ΨL − ΨM ) cosh(ω7 ξ) + (ΨM + ΨL )

+√

Z

(3.36)

Hence, we have the following solution equation (1.1): A(x, t) = i

2ΨM ΨL (ΨL − ΨM ) cosh(ω7 ξ) + (ΨM + ΨL )

14 × exp [−iΘ + i(νx − λt)] , (3.37)

q ΨL where, ω7 = β5 ΨM and Θ(ξ) is given by equation (3.36). 6 (iii) For h ∈ (0, h1 ), the level curves defined by H(Φ, y) = h, there exists a family of periodic orbits of system (1.16), enclosing three equilibrium points E1 (−Φ1 , 0), E2 (Φ2 , 0), and E0 (0, 0). Now, equation (3.1) has the forms r Z Ψ 8β5 dΨ p ξ= , 3 (Ψ − Ψl )Ψ(r1 − Ψ)(ΨL − Ψ) 0 where Ψl < 0 < r1 < ΨL . Therefore, the periodic orbits has a parametric representations: 12 p Ψl Φ(ξ) = ± Ψ(ξ) = ± Ψl − , (3.38) 1 − α22 sn2 (ω4 ξ, k) q β5 ΨL (r1 −Ψl ) Ψl r1 2 2 where α52 = r1 −Ψ , ω = , k = α 1 − 8 5 6 ΨL . l Thus, we have from equation (1.9) that βΨl βΨl 1 (2ν − µ) + ξ− Π arcsinh(sn(ω8 ξ, k)), α52 , k Θ(ξ) = 2 3λ 3λ s √ Z s Z Ψ ˆ2 h 1 α Ψl Ψ 1 +√ dξ + 1− dξ 1 2 4λ 1 − α5 sn2 (ω8 ξ, k) 1 − 1−α2 sn2 (ω8 ξ,k) Ψl 0 0 5 23 Z a Ψ Ψl dξ. (3.39) + Ψl − λ 0 1 − α52 sn2 (ω8 ξ, k) Hence, we have the following solution equation (1.1): A(x, t) = i Ψl −

Ψl 1 − α52 sn2 (ω8 ξ, k)

41 × exp [−iΘ + i(νx − λt)] ,

(3.40)

where, Θ(ξ) is given by equation (3.39). (iv) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist two hetroclinic orbits, enclosing the equilibrium points E2 (−Φ2 , 0), E0 (0, 0) and E3 (Φ2 , 0) connecting the equilibrium points E1 (−Φ1 , 0) and E4 (Φ1 , 0).


267

Then, we have the following parametric representation as system (1.16), p Φ(ξ) = ± Ψ(ξ) = ± Ψ1 −

2Ψ1 (rM − Ψ1 ) rM cosh (ω9 ξ) − (2Ψ1 − rM )

12 ,

(3.41)

q

where ω9 = β5 Ψ1 |r6M −Ψ1 | , 0 < rM < Ψ1 . Thus, from equation (1.9) we have s ! 1 Ψ1 βΨ1 β 1 Θ(ξ) = (2ν − µ) + ξ− arctan tanh ω9 ξ 2 3λ 3λ |rM − Ψ1 | 2 21 Z Ψ rM cosh (ω9 ξ) − (2Ψ1 − rM ) ˆ2 dξ +h Ψ1 rM cosh (ω9 ξ) − Ψ1 rM 0 12 Z Ψ Ψ1 rM cosh (ω9 ξ) − Ψ1 rM α dξ + 4λ 0 rM cosh (ω9 ξ) − (2Ψ1 − rM ) 23 Z a Ψ Ψ1 rM cosh (ω9 ξ) − Ψ1 rM + dξ. (3.42) λ 0 rM cosh (ω9 ξ) − (2Ψ1 − rM )

Hence, we have the following solution equation (1.1): A(x, t) = i Ψ1 −

2Ψ1 (rM − Ψ1 ) rM cosh (ω9 ξ) − (2Ψ1 − rM )

41 × exp [−iΘ + i(νx − λt)] , (3.43)

where, Θ(ξ) is given √ by equation (3.42). (4) 0 < β3 < 2 β1 β5 , β5 < 0. (see [Fig. 4 (d)]) In this case 0 < h1 < ∞, Φ4 = −Φ1 and Φ3 = −Φ2 . (i) Corresponding to the level curves defined by H(Φ, y) = h, h ∈ (0, h1 ), there exist two families of periodic orbits of system (1.16). The parametric representations of solution of equation (1.1) is the same as equation (3.28). (ii) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist pairs of hetroclinic orbits connecting E0 (0, 0) and E1 (±Φ1 , 0), enclosing the equilibrium points E2 (±Φ2 , 0). Then, we have the following parametric representation as system (1.16). p Φ(ξ) = ± Ψ(ξ) = ±

Ψ4 1 + exp(Ψ4 ξ)

12 .

(3.44)

Thus, from equation (1.9) we have √ 1 βΨ4 α Ψ4 β (2ν − µ) + − ξ− ln(1 + exp(Ψ4 ξ)) 2 3λ 2λ 3λ ˆ2 Z Ψ p p α h − √ ln 1 + 1 + exp(Ψ4 ξ) + √ 1 + exp(Ψ4 ξ)dξ 2λ Ψ4 Ψ4 r0 32 Z a Ψ Ψ4 + dξ. (3.45) λ r0 1 + exp(Ψ4 ξ)

Θ(ξ) =

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T. Leta & J. Li

Hence, we have the following solution equation (1.1) A(x, t) = i

Ψ4 1 + exp(Ψ4 ξ)

41 × exp [−iΘ + i(νx − λt)] ,

(3.46)

where, r0 ∈ (0, Ψ4 ) and Θ(ξ) is given by equation (3.45). (5) β3 > 0, β5 < 0. (see [Fig. 4(e)]) In this case h2 < 0 < h1 , Φ4 = −Φ1 and Φ3 = −Φ2 . (i) For h ∈ (h2 , h1 ), the level curves defined by H(Φ, y) = h, there exist a pair of periodic orbits, enclosing the equilibrium points E2 (±Φ2 , 0). Then, we have the following parametric representation as system (1.16). p Φ(ξ) = ± Ψ(ξ) = ± r2 −

r2 − r4 1 − α62 sn(Ω1 ξ, k)

21 ,

(3.47)

q β5 (r2 −r3 )|r4 | r2 3| 2 2 where r4 < Ψ2 < r3 < 0 < r2 , α62 = |rr42 −r k = α and Ω = . 1 6 −r3 |r4 | 6 Thus, from equation (1.9) we have 1 βr2 β (2ν − µ) + ξ+ (r4 − r2 ) × Π arcsinh (sn (Ω1 ξ, k)) , α62 , k Θ(ξ) = 2 3λ 3λΩ1 12 1 Z Ψ Z Ψ α r4 − r2 α62 sn(Ω1 ξ, k) 2 1 − α62 sn(Ω1 ξ, k) ˆ + h2 dξ + dξ r4 − r2 α62 sn(Ω1 ξ, k) 4λ r4 1 − α62 sn(Ω1 ξ, k) r4 3 Z a Ψ r4 − r2 α62 sn(Ω1 ξ, k) 2 + dξ. (3.48) λ r4 1 − α62 sn(Ω1 ξ, k) Hence, we have the following solution equation (1.1): A(x, t) = i r2 −

r2 − r4 1 − α62 sn(Ω1 ξ, k)

14 × exp [−iΘ + i(νx − λt)] ,

(3.49)

where, Θ(ξ) is given by equation (3.48). (ii) Corresponding to the level curves defined by H(Φ, y) = h1 , there exist two homoclinic orbits, enclosing the equilibrium points E2 (−Φ2 , 0) and E3 (Φ2 , 0) at E1 (−Φ1 , 0) and E4 (Φ1 , 0), respectively. Then, we have the following parametric representation of system (1.16). p Φ(ξ) = ± Ψ(ξ) = ± Ψ1 +

2Ψ1 (ΨM − Ψ1 ) ΨM cosh (Ω2 ξ) − (2Ψ1 − ΨM )

12 ,

(3.50)

q where Ψ1 < ΨM < 0 and Ω2 = β5 Ψ1 (Ψ6M −Ψ1 ) . Thus, from equation (1.9) we have ! r 1 βΨ1 β Ψ1 1 Θ(ξ) = (2ν − µ) + ξ+ arctan tanh Ω2 ξ 2 3λ 3λ ΨM − Ψ1 2 1 Z Ψ Ψ1 ΨM cosh (Ω2 ξ) − 4Ψ21 + 3Ψ1 ΨM 2 ˆ2 +h dξ ΨM cosh (Ω2 ξ) − (2Ψ1 − ΨM ) 0


269

21 Z Ψ ΨM cosh (Ω2 ξ) − (2Ψ1 − ΨM ) α + 4λ 0 Ψ1 ΨM cosh (Ω2 ξ) − 4Ψ21 + 3Ψ1 ΨM 23 Z Ψ a ΨM cosh (Ω2 ξ) − (2Ψ1 − ΨM ) + dξ. λ 0 Ψ1 ΨM cosh (Ω2 ξ) − 4Ψ21 + 3Ψ1 ΨM

(3.51)

Hence, we have the following solution equation (1.1):

2Ψ1 (ΨM − Ψ1 ) A(x, t) = i Ψ1 + ΨM cosh (Ω2 ξ) − (2Ψ1 − ΨM )

14 × exp [−iΘ + i(νx − λt)] , (3.52)


4. Conclusion To sum up, we have proved the following Theorems: Theorem 4.1. Suppose that the parametric conditions β5 6= 0, β2 = β4 = β6 = 0 of system (1.14) holds. Depending on the changes of system parameters, the bifurcations of phase portraits of system (1.16) are shown in Fig.1–Fig.4. Theorem 4.2. (i) By using the method of dynamical systems we have found fourteen solutions depending on change of parameters regions of (β3 , β5 ) for β1 < 0 and β1 > 0, corresponding to the periodic, homoclinic and hetroclinic orbits of system (1.16). The thirteenth order derivative nonlinear Schr¨ odinger equation (1.1) has sixteen exact solutions given by equations (3.4), (3.7), (3.10), (3.13), (3.16), (3.19), (3.22), (3.25), (3.28), (3.31), (3.34), (3.37), (3.40), (3.43), (3.46), (3.49) √ and (3.52). (ii) √ System (1.3) has sixteen exact explicit solutions φ(ξ) = Φ sin Θ, and ψ(ξ) = Φ cos Θ, where φ(ξ) and ψ(ξ) are given in section 1.

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