Dynamical behavior and exact solution in invariant manifold for a septic derivative nonlinear Schrödinger equation

Nonlinear Dynamics, Mar 2017

In this paper, we consider a pulse dynamics in nonlinear optics (fiber-optic communications) in the presence of both self-steepening and septic nonlinear effects. Propagating profiles of the septic derivative nonlinear Schrödinger model which are isolated via coupled integrable invariants of motion, that admits exact solution, are investigated by a method of dynamical systems. By investigating the dynamical behavior and bifurcation of phase portraits of the traveling wave system, we obtain possible explicit exact parametric representations of solutions under different parameter conditions.

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Dynamical behavior and exact solution in invariant manifold for a septic derivative nonlinear Schrödinger equation

Dynamical behavior and exact solution in invariant manifold for a septic derivative nonlinear Schrödinger equation Temesgen Desta Leta 0 1 Jibin Li 0 1 0 J. Li School of Mathematical Sciences, Huaqiao University , Quanzhou 362021, Fujian , People's Republic of China 1 This research was partially supported by the National Natural Science Foundation of China (11471289 , 11571318, 11162020) In this paper, we consider a pulse dynamics in nonlinear optics (fiber-optic communications) in the presence of both self-steepening and septic nonlinear effects. Propagating profiles of the septic derivative nonlinear Schrödinger model which are isolated via coupled integrable invariants of motion, that admits exact solution, are investigated by a method of dynamical systems. By investigating the dynamical behavior and bifurcation of phase portraits of the traveling wave system, we obtain possible explicit exact parametric representations of solutions under different parameter conditions. Coupled integrable system; Exact solution; Septic derivative nonlinear Schrödinger equation; Homoclinic orbits; Heteroclinic orbits; Periodic orbits - 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 [1–4]. Moreover, nonlinear Schrödinger (NLS) equations incorporating cubic and quintic order nonlinearity are of fundamental interest in the physics of nonlinear optics [4,5]. It is natural to consider models in nonlinear dynamics combining these two features, and indeed such systems likewise have physical applications [6–8]. 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 [9,10]. In this paper, we shall consider the generalized class of higher-order derivative nonlinear Schrödinger equations of the type: + ψ 2), “ ” stands for the derivative with respect to Φ and s > 1 corresponds to the so-called resonant nonlinear Schrödinger equation because it admits both fission and fusion resonant solitonic phenomena [11,12]. A procedure recently employed by [13–16] (the application of a pair of invariants of motion) is also applied here. where A2 be the squared amplitude, x and t are the normalized spatial coordinate and retarded time, respectively. To analyze the traveling wave solution with the form: where c, ν, and λ; are related to the nonlinearity induced shifts in three quantities, namely, group delay, carrier frequency and propagation constant respectively. Substituting Eq. (2) into Eq. (1), separating the real and imaginary parts, one obtains the coupled nonlinear integrable system = −( f + λ − ν2)φ + ( j + 2ν − μ)ψ˙ = −( f + λ − ν2)ψ − ( j + 2ν − μ)φ˙ where dot indicates a derivative with respect to x − μt . Substituting Φ4 + μ2 Φ2 + ν3 Φ3, j = 2aΦ4, g = 2δ Φ2, in system (3), that admits two independent integrals of motion and, accordingly, is integrable. Thus, the first integral becomes: Φ2 − (λ − ν2)Φ − 4s ΦΦ˙2 where the constant Hˆ = hˆ1 corresponds to the Hamiltonian invariant. And the second integral I (φ, ψ, q1, q2) = q1ψ − q2φ − λa2 Φ4 − 4λδ2 Φ2 + 21 (μ − 2ν)Φ, where I (φ, ψ, q1, q2) = hˆ2 is constant of motion. The pair of integrals of motion (5) and (6) allow explicit solution of the nonlinear coupled system (3) for φ and ψ in terms of quadrature. It is easy to see that system (3) has at most 17 equilibrium points at P0(0, 0, 0, 0), Pa j (±φr j , 0, 0, 0), Pb j (0, ±ψr j , 0, 0), j = 1, 2, 3, 4, where φr j = ψr j = X j are four positive real roots of the algebraic equation δ¯1Φ4 − δ¯2Φ3 + δ¯3Φ2 − δ¯4Φ + δ¯0 = 0 Let Nhˆ1hˆ2 be an invariant manifold family of system (3) given by Nhˆ1hˆ2 = Hˆ = hˆ1, I = hˆ2, hˆ1, hˆ2 ∈ R ⊂ R4. (7) Using the identity combined with (5) and (6), for fixed (hˆ1, hˆ2) gives rise Φ 2hˆ1 − (λ − ν2)Φ Φ2 − 3νλ Φ3 − aλ (μ − 2ν) Φ4 hˆ2 − 21 (μ − 2ν)Φ + 4δλ Φ2 + aλ Φ4 For Eq. (1), Rogers and Chow [13] introduced the variable Θ = arctan . So that, cosΘ = √ψΦ , sinΘ = √φΦ , where = ψφ . We see from (5) that, where ξ is a dummy variable of integration. Thus, if we know Φ(ξ ) and Θ(ξ ) from Eqs. (9) and (10), then we may solve Eq. (1) and system (3) to obtain the following solutions: √ A(x , t ) = i Φexp [−i Θ + i (ν x − λt )] √ΦsinΘ and ψ (ξ ) = √ΦcosΘ The corresponding class of exact solutions of Eq. (1), is then given by Eq. (11). Let ddΦξ = 21 (1 − s)y, then we have a system 2 − b3Φ3 − b5Φ5 − b7Φ7. Clearly, system (13) is a six-parameter system depending on the nine-parameter group (λ, μ, a, s, ν, δ, p, hˆ1, hˆ2). It is abound with dynamical system. For a given parameter group (ν, μ) = (0, 0), we next take (hˆ1, hˆ2) such that (2ν − μ) b0 ≡ 0, hˆ1 = 4 Then, system (13) reduces to with the first integral 2 − b3Φ3 − b5Φ5 − b7Φ7, clearly, for p = 2, 3, we have ≡ −b1Φ − b pΦ p − b2 p−1Φ2 p−1 S2(Φ) = −Φ b1 + b2Φ + b3Φ2 + b5Φ4 , S3(Φ) = −Φ b1 + b3Φ2 + b5Φ4 + b7Φ6 . We notice that the exact traveling wave solution and bifurcation of the cubic–quintic was previously analyzed in [16–18]. Though, in this paper we give the dynamical behavior and the exact solution for the case of the septic order nonlinearities in the invariant manifold Nhˆ1hˆ2 by using Eq. (9). In order to consider the dynamics for system (17), when we apply a transfor1 mation Φ = ϕ p−1 , and substituting ϕ˙ = 21 (1 − s)y in system (17), then we obtain a new system: 4( p − 1)ϕ Obviously, system (18) is a singular traveling wave system of the first class [19–21] with a singular straight lines ϕ = 0. Thus, by the theory of singular systems, if an orbit of system (18) contacts the singular straight line at the origin, then along this orbit a phase point only takes finite “time interval” to arrive at the origin. The existence of the singular straight line leads to a dynamical behavior of solutions. Now, consider the associated regular system of (18), which has the same invariant curve solutions given as: where dξ = −2( p − 1)ϕdζ , for p = 1. For s = 2, system (18) has the first integral 2b2 p+1 ϕ3 + b2 p−1 ϕ2 + p2+bp1 ϕ + b1 3 p − 1 p = h. For the corresponding traveling wave Eq. (1) and systems (3), the following problems are considered here. What are the dynamical behavior of traveling wave solutions? How do the traveling wave solutions depend on the parameters of these systems? As far as we know, these problems have not been considered before for septic order nonlinearity. We shall consider the existence and dynamical behavior of the bounded traveling wave solutions of Eq. (1) and system (3) in different regions of their parametric spaces, by using the methods of dynamical systems developed by [22,23]. Then, by calculating Eq. (8) and applying in Eqs. (9) and (10), we obtain the exact solutions of Eq. (1) and system (3). This paper is organized as follows. In Sect. 2, we consider the bifurcations of phase portraits of system (18) for p = 2. Corresponding to some phase portraits given in Sect. 2, in Sect. 3, we give some possible exact solutions of Eq. (1), under different parametric conditions and different h values. In Sect. 4, we consider the dynamical behavior and bifurcation of phase portraits of system (18) for p = 3. Corresponding to some phase portraits given in Sect. 4, in Sect. 5, we give some possible exact solutions of Eq. (1), under different parametric conditions. Lastly, in Sect. 6, we state the main result. 2 Bifurcations of phase portraits of system (18) when p = 2 First, we consider system (18) which has a Hamiltonian for p = 2 given as ≡ h. Obviously, system (18) has always has an equilibrium point E0(0, 0). Write, P3(ϕ) = b1+b2ϕ+b3ϕ2+b5ϕ3. Clearly, P3(ϕ) = b2 +2b3ϕ +3b5ϕ2. When ϕ = r1,2 = −b33±b5√ , = b32 − 3b2, we have P3(r1,2) = 0. To investigate the equilibrium points of (18), we need to find all zeros of the function P3(ϕ). The cubic polynomial P3(ϕ) has three simple real zeros if and only if 1 3 1 2 2 1 Q(b1, b2, b3) ≡ 27 b2 − 108 b2b3 − 6 b1b2b3 If Q(b1, b2, b3) = 0, then P3(ϕ) has one simple zero ϕ1 and one double real zero. When Q(b1, b2, b3) > 0, P3(ϕ) has only one real zero. For a given fixed b1 in the (b2, b3)-parametric plane, the function Q(b2, b3) = 0 defines a quartic curves shown in Fig. 1a, b, which has three branches and partitions the (b2, b3)-parameter plane in to three regions. It is easy to prove that for b1 > 0, the curves defined by 2 1 Q(b2, b3) = 0 has a cusp point at (3b13 , 3b13 ). While for b1 < 0, the curves defined by Q(b2, b3) = 0 has 2 1 a cusp point at (3b13 , −3b13 ). In the regions (II), (III) for b1 > 0 and regions IˆI , I Iˆ I for b1 < 0, there exists three real zeros of P3(ϕ). Let Mˆ (ϕ j , 0) be the coefficient matrix of the linearized system of (18) at an equilibrium point E j . We have J (0, 0) = detMˆ (0, 0) = 0, J (ϕ j , 0) = detMˆ (ϕ j , 0) = −8ϕ 2j(4φ3 2 j + 3b2φ j + 2b1φ j + b0). We write that h0 = H2(0, 0) = 0, h j = H2(ϕ j , 0). By using the above information to do qualitative analysis, we have the following bifurcations of the phase portraits of system (12) with different cases, when b1 < 0, and b1 > 0, respectively, as follows. Fig. 1 The partition of the (b2, b3)-parametric plane of system (18). a b0 > 0, b b0 < 0 Case 1 Assume that b1 > 0, Q(b1, b2, b3) < 0. When b5 > 0, system (18) has two equilibrium points E0(0, 0) and E1(φ1, 0). We have the phase portrait shown in Fig. 2a–c. Case 2 Assume that b1 > 0, Q1(b2, b3) < Q2 (b2, b3). In this case, P3(φ) has three real zeros φ j , j = 1, 2, 3. System (18) has four equilibrium points E0(0, 0) and E j (φ j , 0), j = 1, 2, 3. When b5 > 0, the origin E0(0, 0) and E3 are saddle points; E1 and E2 are center points. The bifurcations of phase portraits of system (18) are shown in Fig. 3a–h. Case 3 Assume that b1 < 0, Q1(b2, b3) < 0 < Q2(b2, b3), 0 < 3b2 < b32. System (18) has four equilibrium points E0(0, 0) and E j (φ j , 0), j = 1, 2, 3. When b5 > 0, the Equilibrium points E1 and E2 are saddle points; E0 and E3 are center points. The bifurcations of phase portraits of system (18) are shown in Fig. 4a–h. Case 4 Assume that b1 < 0, 0 < 3b2 < b32, Q(b2, b3) > 0. In this case, we have φ1 = φ2 = r1. P3(φ) has a double real zero φ12 = r1 and a simple real zero φ3. System (18) has three equilibrium points E12(r1, 0), E0(0, 0) and E3(φ3, 0). When b5 > 0, the origin E0(0, 0) is center point, E12 is a cusp and E3 is a saddle point. The bifurcations of phase portraits of system (18) are shown in Fig. 5a–c. 3 Explicit parametric representations of the solutions of system (18) when p = 2 We now consider the exact explicit parametric representations of the solutions of system (18) depending on Fig. 3 Bifurcations of phase portraits of system (18), when b1 > 0. a r2 < 0 < r1, h1 < h2 < 0, b r2 < r1 < 0, h1 < h2 < 0, c r2 < r1 < 0, 0 < h2 < h1, d r2 < 0 < r1, h2 = h3, e r1 < 0 < r2, h2 < h1 < 0, f r1 < r2 < 0, h1 = h3, g r2 < 0 < r1, h2 < 0 < h1, h r2 < 0 < r1, h2 < 0 < h1, i r2 = 0 < r1, h1 < h2 = 0 Fig.4 Bifurcationsofphaseportraitsofsystem(18),whenb1 < 0. a r2 < 0 < r1,h2 < 0 < h1, b r2 < b2 < r1,0 < h1 < h2, c r2 < 0 < r1,h2 < 0 < h1, d r2 < 0 < r1,0 < h1 < h2, e r2 < 0 < r1,h2 < 0 < h1, f r2 < 0 < r1,0 < h1 < h2, g r1 < r2 < 0,0 < h1 < h2, h r2 = r1 > 0,h1 = h2 > 0 √Φ and Θ. We see from (21) and the first equation of system (18) that To find the exact solutions, we consider all bounded orbits of system (18) with Φ = ϕ > 0, for p = 2. In this section, we discuss the exact solutions of Eq. (1) in the different regions of parameter plane (b2, b3) for a given b1 < 0 and b1 > 0. 3.1 Consider case 1 in Sect. 2 (see Fig. 2a, b) For h = 0, the level curves defined by H2(Φ, y) = h, b1 > 0, there exists a homoclinic orbit of system (18) to the saddle point E0(0, 0) enclosing the equilibrium point E1(Φ1, 0). In this case, (21) can be written as y2 = 43 b1Φ2 + 98 b2Φ3 + 23 b3Φ4 + 185 b5Φ5 = ( 81b55 )Φ2(Φ −Φm )[(Φ −bˆ1)2 +aˆ12]. Hence, (21) implies we have the following parametric representation of Φ(ξ ): 2 A1 Φ(ξ ) = (Φm + A1) − 1 + cn(ω0ξ, k) . Thus, we have from (10) that 1 1 a 2 (2ν − μ) + 2 δ(Φm + A1) + 4 (Φm + A1)3 ξ 3.2 Consider case 2 in Sect. 2 for b1 > 0 (see Fig. 3a–h) (1) The case of r1 < 0 < r2; h3 < 0 (see Fig. 3a). For h2 = 0, the level curves defined by H2(φ, y) = 0, there exist two homoclinic orbits of system (18) with “eight shape” to the origin E0(0, 0). In this case, (21) can be written as y2 = 81b55 Φ2(Φ − Φl )(Φ − Φm )(ΦM − Φ), where Φl < Φm < Φ1 < 0 < Φ2 < ΦM . Hence, for the right homoclinic orbit, we have the following parametric representation of Φ(ξ ): Φ(ξ ) = ΦM − (ΦM − Φm )sn2(ω1ξ, k). Φl − Φm A(x , t ) = i Φl − dn2(ω1ξ, k) where Θ(ξ ) is given by (31) and I1 = k12 E (arcsin (sn(ω1ξ, k)), k) − kk22 sn(ω1ξ, k)cn(ω1ξ, k), I2 = k14 2(2 − k2)E (arcsin(sn(ω1ξ, k)), k) − k2sn(ω1ξ, k)cn (ω1ξ, k)(k2nd2(ω1ξ, k) + 4 − 2k2) , I3 = 5k1 2 (5 − 4k2)I2, k = √1 − k2 and sn(·, k), cn(·, k), dn(·, k), sd (·, k) are Jacobin elliptic functions and (·, ·, k), is the elliptic integrals of the third kind [24]. (2) The case of r2 < r1 < 0, h1 < h2 < 0 (see Fig. 3b). Corresponding to the level curve defined by H2(Φ, y) = h, h ∈ (h1, h2), there exist two homoclinic orbit to the equilibrium E2(Φ2, 0) with “figure eight” of system (18). Now, (23) has the form (Φ − Φ2)√(Φ − Φm )(ΦM − Φ)(ΦL − Φ) Corresponding to the right homoclinic orbits we obtain from (10), that 21 (2ν − μ) + 21 δΦM + a4 ΦM3 ξ hˆ2 arcsin(sn(ω1ξ, k)), ΦM − Φm , k + ΦM ΦM + 41 3aΦM2 − 2δ (ΦM − Φm )β1 − a4 [3ΦM β2 + (ΦM − Φm )β3] (ΦM − Φm )2. From Eqs. (27) and (28), we have the following solution of Eq. (1): 1 A(x , t ) = i ΦM − (ΦM − Φm )sn2(ω1ξ, k) 2 × exp [−i Θ + i (ν x − λt )] where Θ(ξ ) is given by (28), α12 = 1 + 3k2 sn3(ω1ξ, k)cn(ω1ξ, k)dn(ω1ξ, k), + 4(1 + k2)β2 − 3β1 . For the left homoclinic orbit, we have the following parametric representation of Φ(ξ ): Φl − Φm . Φ(ξ ) = Φl − dn2(ω1ξ, k) And, corresponding to the left homoclinic orbits we obtain from (10), that 21 (2ν − μ) + 21 δΦl + a4 Φl3 ξ 2δ − 3aΦl2 (Φl − Φm )I1 − 1 where Φm < Φ1 < Φ2 < Φ3 < ΦM < 0 < ΦL . Using the above integrals, corresponding to the right homoclinic orbit, we have the following parametric representation of Φ(ξ ): ΦM − ΦL . Φ(ξ ) = ΦL + dn2(ω2ξ, k) Thus, corresponding to the right homoclinic orbits we obtain from (10), that 1 1 a 3 2 (2ν − μ) + 2 δΦL + 4 ΦL ξ ΦM − ΦL A(x , t ) = i ΦL + dn2(ω2ξ, k) Hence, we have the following solution of Eq. (1): 1 2 5k1 2 (5 − 4k2)I5. Corresponding to the left homoclinic orbit, we have the following parametric representation of Φ(ξ ): Φ(ξ ) = Φm + (ΦM − Φm )sn2(ω2ξ, k). Thus we obtain from (10) that Θ(ξ ) = 21 (2ν − μ) + 21 δΦM + a4 ΦM3 hˆ2 arcsin(sn(ω2ξ, k)), Φm − ΦM , k + Φm ΦM 1 + 4 2δ − 3aΦM2 (ΦM − Φm )β4 + a4 [3ΦM β5 − (ΦM − Φm )β6] (ΦM − Φm )2. Thus, we have the following solution of Eq. (1): 1 A(x , t ) = i Φm + (ΦM − Φm )sn2(ω2ξ, k) 2 β6 = 51k2 sn3(ω2ξ, k)cn(ω2ξ, k)dn(ω2ξ, k) + 54k2 (1 + k2)β5 − 3β4. Equations (33) and (36) give rise to the profiles of solitary waves shown in Fig. 6b, c. (i) Corresponding to the level curves defined by H2(Φ, y) = 0, there exist a periodic orbits of system (18). (23) has the form Φ Φ(ξ ) = r1 + (r2 − r1)sn2(ω3ξ, k). From Eq. (10) we have Θ(ξ ) = 21 (2ν − μ) + 21 δr1 + a4 r13 ξ arcsin(sn(ω3ξ, k)), r1 − r2 , k which gives rise to a parametric representation of Eq. (1). Now, (23) has the form r2 (Φ2 − Φ)√(Φ − r2)(r1 − Φ)(Φ2 − Φ) Therefore, we obtain the following parametric representation of a periodic orbits: Φ(ξ ) = r2 + (r1 − r2)sn2(ω4ξ, k). Thus we obtain from (10) that 21 (2ν − μ) + 21 δr2 + a4 r23 ξ 1 + 3k2 sn3(ω3ξ, k)cn(ω3ξ, k)dn(ω3ξ, k), 1 β9 = 5k2 sn3(ω3ξ, k)cn(ω3ξ, k)dn(ω3ξ, k) Φ(ξ ) = r1 + 1 −Φsnm2(−ω3r1ξ, k) . Hence, from Eqs. (42) and (43), we have the following solution of Eq. (1): Φm − r1 A(x , t ) = i r1 + 1 − sn2(ω3ξ, k) Corresponding to the curves defined by H2(Φ, y) = h h ∈ (0, h1), we have y2 = ( 81b55 )(Φ − r2)(r1 − Φ)(Φ2 − Φ)3, where r2 < Φ1 < r1 < 0 < Φ2 = Φ3. Equation (18) has a periodic orbit enclosing E1(Φ1, 0), 1 β12 = 5k2 sn3(ω4ξ, k)cn(ω4ξ, k)dn(ω4ξ, k) 1 + 5k2 4(1 + k2)β11 − 3β10 . (5) The case of r2 < 0 < r1, h2 < 0 < h1 (see Fig. 3g). For h = 0 and 3b2b5 < b32 the level curves defined by H2(Φ, y) = 0, we have from Eq. (21), y2 = 8b5 (Φ − Φm )Φ3(ΦL − Φ), where Φm < Φ1 < 0 < 15 ΦL . Equation (18) has a degenerate homoclinic orbit at a cusp E0(0, 0), enclosing E1(Φ1, 0). Now, (23) has the form Φm Φ√(Φ − Φm )Φ(ΦL − Φ) Therefore, we have the following parametric representation of Φ(ξ ): enclosing the equilibrium point E2(Φ2, 0). Now (23) can be written as Φ(ξ ) = Φm (1 − sn2(ω2ξ, k)). Thus we obtain from (10) that hˆ2 (E (arcsin(sn(ω2ξ, k)), k) + Φm − dn(ω2ξ, k)tn(ω2ξ, k)) . Therefore, we have the following solution of Eq. (1): 1 A(x , t ) = i Φm (1 − sn2(ω2ξ, k)) 2 where Θ(ξ ) is given by (49) and k2 = ΦLΦ−mΦm , I7 = k12 E (arcsin(sn(ω2ξ, k)), k), I8 = 31k2 2(1 + k2)I7 + k2 sn3(ω2ξ, k)cn(ω2ξ, k)dn(ω2ξ, k), I9 = 51k2 sn3(ω2ξ, 1 k)cn(ω2ξ, k)dn(ω2ξ, k) + 5k2 4(1 + k2)I8 − 3I7 . (6) The case of r2 < 0 < r1, h2 < 0 < h1 (see Fig. 3h). Corresponding to the level curve defined by H2(Φ, y) = 0, there exist two homoclinic orbit to the origin E0(0, 0) with “figure eight” of (18). Now, (23) has the form Φ√(Φ − Φm )(ΦM − Φ)(ΦL − Φ) Φm Φ√(Φ − Φm )(ΦM − Φ)(ΦL − Φ) where Φm < Φ1 < 0 < Φ2 < ΦM < ΦL . Using the above integrals, corresponding to the right and left homoclinic orbit we have the same parametric representation and solution of Eq. (1) as (35) and (38), respectively. (7) The case of r2 = 0 < r1, h1 < h2 = 0 (see Fig. 3i). In this case, corresponding to the level curve defined by H2(Φ, y) = 0, there exists a homoclinic orbit to the origin E0(0, 0), enclosing the equilibrium point E1(Φ1, 0) and there are two heteroclinic orbits connecting the equilibrium point E3(Φ3, 0) and E0(0, 0), Φm (Φ3 − Φ)Φ√Φ − Φm (i) Corresponding to the left homoclinic orbit, we have the following parametric representation of Φ(ξ ): Φ(ξˆ ) = Φm sech2(ξˆ ), ξˆ ∈ (0, ∞), we obtain from Eq. (10) that 1 δ Θ(tˆ) = 2 (2ν − μ)tˆ + 2 Φm tanh(ξˆ ) sinh(ξˆ ) 4sinh(ξˆ ) 4sinh(ξˆ ) × cosh(ξˆ ) + 5cosh3(ξˆ ) + 15cosh2(ξˆ ) where, ξˆ = tanh−1 Φ(ξˆ)−Φm . Hence, we have the Φm following solution of Eq. (1): where Θ(tˆ) is given by (52). (ii) Corresponding to the heteroclinic orbit, we have the following parametric representation of Φ(ξ ): Φ(ξˆ ) = Φm + (Φm − Φ3)tanh2(tˆ), we have from Eq. (10) that 1 1 a 3 2 (2ν − μ) + 2 δΦm + 4 Φm 1 tanh5(ξˆ ) − 4 tanh3(ξˆ ) − tanh(ξˆ ) tanh3(ξˆ ) − 4tanh(ξˆ ) Φm (Φm − Φ3)2 where ξˆ = tanh−1 ΦΦ(ξmˆ)−−ΦΦ3m . Thus, we have the following solution of Eq. (1): Fig. 7 The profiles of kink and anti-kink waves of system (18). a Kink wave, b anti-kink wave 1 A(x , t ) = i Φm + (Φm − Φ3)tanh2(ξˆ ) 2 Equation (54) give rise to the profiles of kink and anti-kink waves shown in Fig. 7a, b. 3.3 Consider case 3 in section 2, for b1 < 0 (see Fig. 4a–h). In this case, the phase portraits are determined by making a shift transformation, such that a saddle point becomes the origin, we can obtain the parametric representations of some solutions of heteroclinic and homoclinic orbits. (1) The case of r1 < r2 < 0, 0 < h1 < h2 (see Fig. 4g). For h ∈ (h2, +∞), the level curve defined by H2(Φ, y) = h, there are two heteroclinic orbits connecting the equilibrium point E2(Φ2, 0) and E3(Φ3, 0), enclosing the equilibrium point E0(0, 0) and there exists a homoclinic orbit to the equilibrium point E2(Φ2, 0), enclosing the equilibrium point E1(Φ1, 0). Now (23) can be written as Φm (Φ3 − Φ)(Φ − Φ2)√Φ − Φm (i) For the heteroclinic orbit, we have the following parametric representation of Φ(ξ ): Φ(ξˆ ) = Φm + (Φ2 − Φm )tanh2(ξˆ ), we have from Eq. (10) that 1 1 Θ(tˆ) = 2 (2ν − μ) + 2 δΦm + a4 Φm3 ξˆ tanh−1 tanh3(ξˆ ) − tanh(ξˆ ) (Φm − Φ2)2 + a4 51 tanh5(ξˆ ) − 41 tanh3(ξˆ ) − tanh(ξˆ ) Φ(ξˆ)−Φm . Φ2−Φm (ii) Corresponding to the homoclinic orbit of system (18) to the equilibrium point E2(Φ2, 0) enclosing the equilibrium point E1(φ1, 0), we obtain a parametric representation of Φ(ξ ): Φ(ξˆ ) = Φm + (Φm − Φ2)tanh2(ξˆ ), we have from Eq. (10) that Therefore, we have the following solution of Eq. (1): 1 A(x , t ) = i Φm + (Φm − Φ2)tanh2(ξˆ ) 2 81b55 ξ = ΦΦm (Φ1 − Φ)d2Φ√Φ − Φm , where Φm < 0 < Φ1. Thus, we have parametric representation: tanh3(ξ ) − tanh(ξˆ ) (Φm − Φ1)2 + a4 15 tanh5(ξˆ ) − 41 tanh3(ξˆ ) − tanh(ξˆ ) (Φm − Φ1)3. Hence, we have the following solution of Eq. (1): 1 A(x , t ) = i Φm + (Φ1 − Φm )tanh2(ξˆ ) 2 where Θ(ξˆ ) is given by (64) and ξˆ = tanh−1 Φ(ξˆ)−Φm . Φ1−Φm 3.4 Consider case 4 in Sect. 2, for b1 < 0 (see Fig. 5a–c) (i) Corresponding to the level curves defined by H (φ, y) = 0 in (21), there exist two heteroclinic orbits of system (18) connecting the equilibrium point (cusp point) E12(r1, 0) and the saddle equilibrium point E3(Φ3, 0) (see Fig. 5a). In this case, (23) can be written as where Φ0 ∈ (r1, Φ3) and r1 < 0 < Φ3. Completing the above integral, we have the following parametric representation of Φ(ξ ): Φ(ξˆ ) = r1 + (r1 − Φ3)tanh2(ξˆ ), we obtain from Eq. (10) that 21 (2ν − μ) + 21 δr1 + a4 r13 ξˆ tanh3(ξˆ ) − tanh(ξˆ ) (Φ3 − r1)2 − a4 15 tanh5(ξ ) − 41 tanh3(ξˆ ) − tanh(ξˆ ) where, ξˆ = tanh−1 Φ(tˆ)−r1 . Hence, we have the r1−Φ3 following solution of Eq. (1): A(x , t ) = i r1 + (r1 − Φ3)tanh2(ξˆ ) where Θ(ξˆ ) is given by (67). (ii) Corresponding to a homoclinic orbit of system (18) to the saddle point E3(Φ3, 0) enclosing the origin E0(0, 0) (see Fig. 5b). Hence, we obtain a similar parametric representation Eq. (1) as Eq. (65). (iii) Corresponding to the curves defined by H2(Φ, y) = h1 = h2 (see Fig. 5c), we have from Eq. (21) that y2 = ω5(Φ − r1)3(ΦL − Φ)(ΦM − Φ), where r1 < Φ3 < ΦM < 0 < ΦL . Equation (18) has a homoclinic orbit to the cusp point E12(Φ1, 0). Then (23) has the form Φ It gives rise to the parametric representations of a homoclinic orbit of system (18) as follows: (Φ − r1)√(Φ − r1)(ΦM − Φ)(ΦL − Φ) ΦM − ΦL . Φ(ξ ) = ΦL + dn2(ω5ξ, k) Thus we obtain from (10) that + Φhˆ2L ΦΦML − k2 1 − 4 2δ − 3aΦL2 (ΦM − ΦL )I4 + a4 (3ΦL I6 − (ΦM − ΦL )I5) (ΦM − ΦL )2. Therefore, we have the following solution of Eq. (1): ΦM − ΦL A(x , t ) = i ΦL + dn2(ω5ξ, k) 4 The bifurcations of phase portraits of system (18) when p = 3 In order to obtain the exact parametric representations of some orbits in the septic order nonlinearity of derivative of Schrödinger equation of system (18), we consider the case of p = 3 with Hamiltonian given as: ≡ h. Obviously, if = b32 − 3b2b4 < 0, system (18) has only one equilibrium point E0(0, 0). If > 0, b7 = 0, system (18) has three equilibrium points E−b053(±0b7,√0)., EIf1(ϕ1=,0)0, atnhden EE21(ϕa2n,d0)E, 2 wbheecroemeϕ1a,2dou=ble equilibrium point. Let M (ϕ j , 0) be the coefficient matrix of the linearized system of (18) at an equilibrium point (ϕ j , 0), j = 1, 2. We have J (ϕ j , 0) = det M (ϕ j , 0) = 16ϕ 3j(b3 + 2b5ϕ j + 3b7ϕ 2j), j = 1, 2 and J (0, 0) = det M (0, 0) = 0. We write for H3(ϕ, y) given by (9), h0 = H3(0, 0), and h j = H3(ϕ j , 0). By using the above information to do qualitative analysis, we have the following bifurcations of phase portraits of system (18) shown in Fig. 8a–l. 5 Explicit parametric representations of the solutions of system (18) when p = 3 In this section, we discuss the parametric representations of the solutions of system (18). Here, we take that p = 3. We see from (72) and the first equation of (18) that ϕh + 16b1ϕ2 + 8b3ϕ3 + 136 b5ϕ4 + 4b7ϕ5 ≡ 43b7 ϕϕ0 √F5d(ϕϕ, h) . (73) Thus, we shall find all possible exact explicit parametric representations for all bounded functions Φ = √ϕ where ϕ > 0, determined by Eq. (18) in different parametric region of the (b3, b5)-parameter space. 3 5.1 The case of 23 √3b3b7 < 3b12 , 0 < h1 < h2 (see Fig. 8a) For h = h1, the level curve defined by H3(Φ, y) = h, Eq. (18) has a homoclinic orbit to the cusp point E23(Φ2, 0), enclosing the Equilibrium point E1(Φ1, 0). In this case F5 = (Φ − Φm )(Φ2 − Φ)3Φ, where Φm < Φ1 < Φ2 = Φ3 < 0. Then Eq. (73) has the form Φm (Φ2 − Φ)√(Φ − Φm )(Φ2 − Φ)Φ It gives rise to the parametric representations of a homoclinic orbit of system (18) as follows: 1 Φm + (Φ2 − Φm )sn2(ω6ξ, k) 2 . Fig. 8 Bifurcations of phase portraits of system (18), when p = 3. a 23 √3b3b7 < b5, 0 < h1 < h2, b b3 < 2√b3b7 h1 < h2, c 23 √3b3b7 < 3b123 , h1 < h2, d 43 √3b3b7 = 2√b3b7, 0 < 4 √ h2 < h1, e b3 < 0 < b7, h1 < 0 < h2, f − 3 3b3b7 < 3 3 3b12 < b5, h2 < h1. g − 43 √−3b3b7 < b3 < 3b12 , h2 < = 0, h1 = h2 Thus, we obtain from (10), that r1 + (r2 − r1)sn2(ω7ξ, k) 3 r1 + (r2 − r1)sn2(ω7ξ, k) 2 dξ. Φm + (Φ2 − Φm )sn2(ω6ξ, k) Therefore, we have the following solution of Eq. (1): 1 A(x , t ) = i Φm + (Φ2 − Φm )sn2(ω6ξ, k) 2 3 5.2 The case of −3b12 < √|b3b7| h1 < h2 (see Fig. 8c). (i) For h ∈ (h1, h2) the level curves defined by H3(Φ, y) = h in (72), there exist a periodic orbits of system (18) enclosing an equilibrium point of E1(Φ1, 0). In this case we have F5 = (Φ −r2)(r1 − Φ)Φ(Φ3 −Φ)2, where r2 < Φ1 < r1 < 0 < Φ2 < Φ3. Now, (73) can be written as r2 (Φ3 − Φ)√(Φ − r2)(r1 − Φ)Φ Completing the above integral, we can get a periodic orbit of Eq. (1): 1 Φ(ξ ) = r1 + (r2 − r1)sn2(ω7ξ, k) 2 , we obtain from Eq. (10) that Θ(ξ ) = 21 (2ν − μ)ξ + 21 δ (r1 + (r2 − r1) Hence, from Eqs. (77) and (78) we have the following solution of Eq. (1): ×exp [−i Θ + i (ν x − λt )] , (79) where k2 = | r2r−2r1 |, ω7 = √4 √3|br27| . (ii) Corresponding to the level curves defined by H3(Φ, y) = h, h ∈ (h1, h2) in (72), there exist a homoclinic orbits of system (18) at the equilibrium points E0(0, 0), enclosing the equilibrium point E2(Φ2, 0). Now, (73) can be written as 0 (Φ3 − Φ)√(Φ − r2)(Φ − r1)Φ Completing the above integral, we have the following parametric representation of Eq. (1): 1 2 we obtain from Eq. (10) that 1 δ Θ(ξ ) = 2 (2ν − μ)ξ + 2 |r1| 1 A(x , t ) = i r1 1 − 1 − sn2(ω7ξ, k) where Θ(ξ ) is given by (81) and Φ3¬0. (iii) For h = 0, the level curves defined by H2(Φ, y) = h, Eq. (18) has a homoclinic orbit to the origin (cusp point) E0(0, 0) enclosing an equilibrium point of E1(Φ1, 0). In this case we have F5 = (Φm − we obtain from Eq. (10) that Hence, we have the following solution of Eq. (1): 1 A(x , t ) = i Φm (1 − sn2(ω8ξ, k)) 4 where Θ(ξ ) is given by (84) and k2 = ΦL|Φ−mΦm , ω8 = √3(4Φ√Lb−7Φm) . 5.3 The case of b3 < 0 < b5, h1 < 0 < h2 (see Fig. 8d) For h = 0, the level curve defined by H3(Φ, y) = h, there exist two homoclinic orbit to the origin E0(0, 0) with “figure eight” of (18). In this case we have F5 = (Φ − Φm )Φ2(ΦM − Φ)(ΦL − Φ), where Φm < Φ1 < 0 < Φ2 < ΦM < ΦL . Now, (73) has the form Φ√(Φ − Φm )(ΦM − Φ)(ΦL − Φ) Hence, we have the following solution of Eq. (1): 5.4 The case of − 43 √−3b3b7 < b3 < 3b132 , h2 < 0 < h1 (see Fig. 8g) ΦM − ΦL ΦL + dn2(ω8ξ, k) For h = h1 the curves defined by H2(Φ, y) = h, we have y2 = 81b55 (Φ − r1)Φ(Φ2 − Φ)3, where we obtain from Eq. (10) that Hence, we have the following solution of Eq. (1): 1 (ΦM − ΦL ) 4 A(x , t ) = i ΦL + dn2(ω8ξ, k) where Θ(ξ ) is given by (87) and corresponding to the left homoclinic orbit, we have the following parametric representation of Eq. (1): 1 Φm + (ΦM − Φm )sn2(ω8ξ, k) 2 . Thus we obtain from Eq. (10) that 1 Φm + (ΦM − Φm )sn2(ω8ξ, k) 2 dξ 1 Φm + (ΦM − Φm )sn2(ω8ξ, k) − 2 dξ 3 Φm + (ΦM − Φm )sn2(ω8ξ, k) 2 dξ. 1 Φ(ξ ) = √r1 1 − sn2(ω4ξ, k) 2 , r1 < Φ1 < 0 < Φ2 = Φ3. Equation (18) has a periodic orbit enclosing E1(Φ1, 0), which gives rise to a periodic orbit of system (18). Now, (23) has the form Hence, we have the following solution of Eq. (1): A(x , t ) = i r1 1 − sn2(ω4ξ, k) where Θ(ξ ) is given by (93) and k2 = Φ|2r−1|r2 . 5.5 The case of 3b123 < 2√b3b7, 0 < h1 < h2 (see Fig. 8h) For h ∈ (h1, 0), the level curve defined by H3(Φ, y) = h, there exist two homoclinic orbit to the equilibrium point E2(Φ2, 0) of system (18). Here, the origin is E0(0, 0) is a high order equilibrium point. In this case we have F5 = (Φ − Φm )(Φ − Φ2)2Φ(ΦL − Φ), where Φm < Φ1 < Φ2 < Φ3 < 0 < ΦL . Now, (73) has the form Φm (Φ2 − Φ)√(Φ − Φm )Φ(ΦL − Φ) we obtain from Eq. (10) that 3 Φm (1 − sn2(ω8ξ, k)) 2 dξ. (96) Hence, we have the following solution of Eq. (1): Φ2. and corresponding to the right homoclinic orbit, we have the following parametric representation of Eq. (1): we obtain from Eq. (10) that Hence, we have the following solution of Eq. (1): 1 A(x , t ) = i ΦL (1 − dn2(ω8ξ, k) 5.6 The case of 3b123 < 2√b3b7, 0 < h1 < h2 (see Fig. 8k). For h = h2, the level curves defined by H (Φ, y) = h, in (72), we have a homoclinic orbit at a cusp point E12(Φ1, 0) of system (18) enclosing E3(Φ3, 0). In this (Φ − Φ1)√(Φ − Φ1)(ΦM − Φ)Φ It gives rise to the parametric representations of Eq. (1) as follows: Thus we obtain from (10) that δ√ΦM tanh−1 k sn(ω6ξ, k) − cn(ω2ξ, k) + 2k k sn(ω6ξ, k) + cn(ω2ξ, k) a 3 hˆ2 sinh−1(ω2ξ, k) + 4k 3 Φ M2 + √ΦM (2 − k2)tanh−1 k sn(ω6ξ, k) − cn(ω2ξ, k) k sn(ω6ξ, k) + cn(ω2ξ, k) a 3 − 4k 3 Φ M2 k k2sn(ω2ξ, k)cn(ω2ξ, k)nd2(ω2ξ, k). Therefore, we have the following solution of Eq. (1): 1 A(x , t ) = i ΦM nd2(ω6ξ, k) 4 5.7 The case of b3 < 0 < b5, Fig. 8l). = 0, h1 = h2 (see For h = h1, the level curves defined by H (Φ, y) = h, in (72), we have a homoclinic orbit to the multiple equilibrium point (cusp point) (Φ1, 0) and Φ1 = Φ2of system (18) enclosing E3(Φ3, 0). In this case we have 3 F5 = 3(Φ − Φ1)4Φ, where b2 = 3(b1) 2 , b3 = −3(b1) 2 and Φ1 < Φ3 < ΦM < 0. Now, (73) can be written as 0 (Φ2 − Φ)2√Φ It gives rise to the parametric representations of Eq. (1) as follows: and by using Eq. (10) we have 1 δ Θ((ξˆ )) = 2 (2ν − μ)ξˆ + 2 1 + sinh(ξˆ ) 1 − sinh(ξˆ ) 1 + sinh(ξˆ ) 1 − sinh(ξˆ ) Hence, we have the following solution of Eq. (1): 6 Conclusion To sum up, we have proved the following Theorems. Theorem 1 Depending on the changes of system parameters, the bifurcations of phase portraits of system (18) are shown in Figs. 1, 2, 3, 4, 5 and 6 (i) Depending on change of parameters regions of (b2, b3, b4) for b1 < 0 and b1 > 0, we have found 18 solutions corresponding to the periodic, homoclinic and heteroclinic orbits of system (18) with p = 2. The septic derivative nonlinear Schrödinger equation (1) has 18 exact solutions given by (26), (29), (32), (35), (38), (41), (44), (47), (50), (53), (56), (59), (62), (65), (68), and (71). (ii) System (3) has 18 exact explicit solutions Φ(ξ ) = √ΦsinΘ, andψ (ξ ) = √ΦcosΘ, where Φ(ξ ) and ψ (ξ ) are given in Sect. 3. Theorem 2 (i) Depending on change of parameters regions of (b2, b3, b4) for b1 < 0 and b1 > 0, we have found 11 solutions corresponding to the periodic, homoclinic and heteroclinic orbits of system (18) with p = 3. 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Dynamical behavior and exact solution in invariant manifold for a septic derivative nonlinear Schrödinger equation, Nonlinear Dynamics, 2017, DOI: 10.1007/s11071-017-3468-3