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

Abstract

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.

1 Introduction

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 [1,2,3,4].

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 [6,7,8].

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:

$$\begin{aligned}&iA_t+A_{xx}+f'(\varPhi )A+j'(\varPhi )iA_x+(g(\varPhi ))_xiA\nonumber \\&\quad -\frac{s|A|_{xx}}{|A|}A=0, \end{aligned}$$

(1)

where (f, g and j are arbitrary analytic functions which depend on the squared amplitude $\varPhi = |A|^2 = \phi ^2 + \psi ^2$), “$\ ^{\prime }$” stands for the derivative with respect to $\varPhi$ 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 [13,14,15,16] (the application of a pair of invariants of motion) is also applied here. where $A^2$ 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:

$$\begin{aligned} A(x,t)= & {} \left[ (\phi (\xi )+i\psi (\xi ))\right] \text{ exp }\left( i\left( \nu x-\lambda t\right) \right) ,\nonumber \\ \xi= & {} x-ct \end{aligned}$$

(2)

where $c,\nu , \text{ and } \lambda$; 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

$$\begin{aligned} \frac{\hbox {d}\phi }{\hbox {d}\xi }= & {} q_1,\nonumber \\ \frac{\hbox {d}q_1}{\hbox {d}\xi }= & {} -(f{'}+\lambda -\nu ^2)\phi +(j'+2\nu -\mu ){\dot{\psi }}\nonumber \\&+\,g'{(\phi ^2+\psi ^2)^{\bullet }}\psi \nonumber \\&+\,s\left[ \frac{{({\dot{\phi }}\phi +{\dot{\psi }}\psi )}^{\bullet }}{\phi ^2+\psi ^2}- \left( \frac{{\dot{\phi }}\phi +{\dot{\psi }}\psi }{\phi ^2+\psi ^2}\right) ^2\right] \phi .\nonumber \\ \frac{\hbox {d}\psi }{\hbox {d}\xi }= & {} q_2,\nonumber \\ \frac{\hbox {d}q_2}{\hbox {d}\xi }= & {} -(f{'}+\lambda -\nu ^2)\psi -(j'+2\nu -\mu ){\dot{\phi }}\nonumber \\&-\,g'{(\phi ^2+\psi ^2)^{\bullet }}\phi \nonumber \\&+\,s\left[ \frac{({\dot{\phi }}\phi +{\dot{\psi }}\psi )^{\bullet }}{\phi ^2+\psi ^2}-\left( \frac{{\dot{\phi }}\phi +{\dot{\psi }}\psi }{\phi ^2+\psi ^2}\right) ^2\right] \psi . \end{aligned}$$

(3)

where dot indicates a derivative with respect to $x-\mu t$.

Substituting

$$\begin{aligned} f=\frac{ac}{\lambda }\varPhi ^4+\frac{\mu }{2}\varPhi ^2+\frac{\nu }{3}\varPhi ^3, j=2a\varPhi ^4, \ \ g=\frac{\delta }{2}\varPhi ^2,\nonumber \\ \end{aligned}$$

(4)

in system (3), that admits two independent integrals of motion and, accordingly, is integrable. Thus, the first integral becomes:

$$\begin{aligned}&{\hat{H}}(\phi ,\psi ,q_1,q_2)=\frac{1}{2}\left[ q_{1}^2+q_{2}^2+\frac{a}{\lambda ^2}(\mu -2\nu )\varPhi ^4\right. \nonumber \\&\quad \left. -\frac{\nu }{3\lambda }\varPhi ^3+\left( \frac{\mu }{4\lambda }+\frac{p\delta }{2\lambda ^2}\right) \varPhi ^2-(\lambda -\nu ^2)\varPhi - \frac{s}{4}\frac{{{\dot{\varPhi }}}^2}{\varPhi }\right] \nonumber \\ \end{aligned}$$

(5)

where the constant ${\hat{H}}={\hat{h}}_1$ corresponds to the Hamiltonian invariant.

And the second integral

$$\begin{aligned} I(\phi ,\psi ,q_1,q_2)= & {} q_1\psi -q_2\phi -\frac{a}{\lambda ^2}\varPhi ^4\nonumber \\&-\frac{\delta }{4\lambda ^2}\varPhi ^2+\frac{1}{2}(\mu -2\nu )\varPhi , \end{aligned}$$

(6)

where $I(\phi ,\psi ,q_1,q_2)={\hat{h}}_{2}$ is constant of motion. The pair of integrals of motion (5) and (6) allow explicit solution of the nonlinear coupled system (3) for $\phi$ and $\psi$ in terms of quadrature.

It is easy to see that system (3) has at most 17 equilibrium points at $P_0(0,0,0,0)$, $P_{a_j}(\pm \phi _{r_j},0,0,0)$, $P_{b_{j}}(0,\pm \psi _{r_j},0,0)$, $j= 1, 2, 3, 4$, where $\phi _{r_j}=\psi _{r_j}=\sqrt{X_j}$ are four positive real roots of the algebraic equation ${\bar{\delta }}_1\varPhi ^4-{\bar{\delta }}_2\varPhi ^3+{\bar{\delta }}_3\varPhi ^2 -{\bar{\delta }}_4\varPhi +{\bar{\delta }}_0=0$ Let $N_{{\hat{h}}_{1}{\hat{h}}_{2}}$ be an invariant manifold family of system (3) given by

$$\begin{aligned} N_{{\hat{h}}_{1}{\hat{h}}_{2}}=\left\{ {\hat{H}}={\hat{h}}_1, \ I={\hat{h}}_2, \ {\hat{h}}_1, \ {\hat{h}}_2 \in \mathbf {R}\right\} \subset \mathbf{{R}}^4. \end{aligned}$$

(7)

Using the identity

$$\begin{aligned} (\phi ^2+\psi ^2)({{\dot{\phi }}}^2+{{\dot{\phi }}}^2)-(\phi {\dot{\phi }}+\psi {\dot{\psi }})^2\equiv (\phi {\dot{\psi }}-\psi {\dot{\phi }})^2\nonumber \\ \end{aligned}$$

(8)

combined with (5) and (6), for fixed $({\hat{h}}_{1},{\hat{h}}_{2})$ gives rise

$$\begin{aligned}&\left( \frac{\hbox {d}\varPhi }{\hbox {d}\xi }\right) ^2=\frac{4}{1-s}\varPhi \left[ 2{\hat{h}}_{1}- (\lambda -\nu ^2)\varPhi \right. \nonumber \\&\quad \left. +\left( \frac{p\delta -\mu }{2\lambda }\right) \varPhi ^2-\frac{\nu }{3\lambda }\varPhi ^3-\frac{a}{\lambda }\left( \mu -2\nu \right) \varPhi ^4\right] \nonumber \\&\quad -\frac{4}{1-s}\left[ {\hat{h}}_2-\frac{1}{2}(\mu -2\nu )\varPhi +\frac{\delta }{4\lambda }\varPhi ^2+\frac{a}{\lambda }\varPhi ^4\right] ^2\nonumber \\ \end{aligned}$$

(9)

For Eq. (1), Rogers and Chow [13] introduced the variable $\varTheta =\text{ arctan }\Delta$. So that, $\text{ cos }\varTheta =\frac{\psi }{\sqrt{\varPhi }}, \text{ sin }\varTheta =\frac{\phi }{\sqrt{\varPhi }}$, where $\Delta =\frac{\phi }{\psi }$. We see from (5) that,

$$\begin{aligned} \varTheta (\xi )= & {} \int _{0}^{\xi }\frac{4\lambda {\hat{h}}_2-2(\mu -2\nu )\lambda \varPhi +\delta \varPhi ^2+4a\varPhi ^4}{4\lambda \varPhi }\hbox {d}\xi \nonumber \\= & {} -\frac{1}{2}(\mu -2\nu )\xi +{\hat{h}}_2\int _{0}^{\xi }\frac{\hbox {d}\xi }{\varPhi }+\frac{\delta }{4\lambda }\int _{0}^{\xi }\varPhi \hbox {d}\xi \nonumber \\&+\,\frac{a}{\lambda }\int _{0}^{\xi }\varPhi ^3 \hbox {d}\xi \end{aligned}$$

(10)

where $\xi$ is a dummy variable of integration. Thus, if we know $\varPhi (\xi )$ and $\varTheta (\xi )$ from Eqs. (9) and (10), then we may solve Eq. (1) and system (3) to obtain the following solutions:

$$\begin{aligned} A(x,t)=i\sqrt{\varPhi }\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] \end{aligned}$$

(11)

and

$$\begin{aligned} \phi (\xi )=\sqrt{\varPhi }\text{ sin }\varTheta \quad \text{ and } \quad \psi (\xi )=\sqrt{\varPhi }\text{ cos }\varTheta \end{aligned}$$

(12)

The corresponding class of exact solutions of Eq. (1), is then given by Eq. (11). Let $\frac{\hbox {d}\varPhi }{\hbox {d}\xi }=\frac{1}{2}(1-s)y$, then we have a system

$$\begin{aligned} \frac{\hbox {d}\varPhi }{\hbox {d}\xi }= & {} \frac{1}{2}(1-s)y,\nonumber \\ \frac{\hbox {d}y}{\hbox {d}\xi }= & {} b_0-b_1\varPhi -b_2\varPhi ^2-b_3\varPhi ^3-b_5\varPhi ^5-b_7\varPhi ^7.\nonumber \\ \end{aligned}$$

(13)

where

$$\begin{aligned} b_0= & {} 2{\hat{h}}_1+{\hat{h}}_2(\mu -2\nu );\nonumber \\ b_1= & {} \frac{1}{\lambda }\left[ (\mu -2\nu )^2\lambda +2\delta {\hat{h}}_2+4(\lambda -\nu ^2)\lambda \right] ;\nonumber \\ b_2= & {} \frac{1}{4\lambda }\left[ 6\mu -6p\delta -3\delta (\mu -2\nu )\right] ;\nonumber \\ b_3= & {} \frac{1}{6\lambda ^2}\left[ 3\delta ^2+48a{\hat{h}}_2\lambda +8\nu ^2\lambda \right] ;\nonumber \\ b_5= & {} \frac{4}{2\lambda ^2}a\delta ;\nonumber \\ b_7= & {} \frac{8}{\lambda ^2}a^2. \end{aligned}$$

Clearly, system (13) is a six-parameter system depending on the nine-parameter group $(\lambda ,\mu ,a,s,\nu ,\delta ,p,{\hat{h}}_1,{\hat{h}}_2)$. It is abound with dynamical system. For a given parameter group $(\nu ,\mu )\ne (0,0)$, we next take $({\hat{h}}_1,{\hat{h}}_2)$ such that

$$\begin{aligned} b_0\equiv 0, \quad {\hat{h}}_1=\frac{(2\nu -\mu )}{4}\times {\hat{h}}_2. \end{aligned}$$

(14)

Then, system (13) reduces to

$$\begin{aligned} \frac{\hbox {d}\varPhi }{\hbox {d}\xi }= & {} \frac{1}{2}(1-s)y\nonumber \\ \frac{\hbox {d}y}{\hbox {d}\xi }= & {} -b_1\varPhi -b_2\varPhi ^2-b_3\varPhi ^3-b_5\varPhi ^5-b_7\varPhi ^7,\nonumber \\ \end{aligned}$$

(15)

with the first integral

$$\begin{aligned} {\tilde{H}}(\varPhi ,y)= & {} \frac{1}{4}(1-s)y^2+\left[ \frac{1}{2}b_1\varPhi ^2+\frac{1}{3}b_2\varPhi ^3\right. \nonumber \\&\left. +\frac{1}{4}b_3\varPhi ^4+\frac{1}{6}b_5\varPhi ^6+\frac{1}{8}b_7\varPhi ^8\right] . \end{aligned}$$

(16)

Equation (15) is equivalent to the planar dynamical system

$$\begin{aligned} \frac{\hbox {d}\varPhi }{\hbox {d}\xi }= & {} \frac{1}{2}(1-s)y,\nonumber \\ \frac{\hbox {d}y}{\hbox {d}\xi }= & {} S_p(\varPhi )\nonumber \\\equiv & {} -b_1\varPhi -b_p\varPhi ^p-b_{2p-1}\varPhi ^{2p-1}\nonumber \\&-b_{2p+1}\varPhi ^{3p-2}, \end{aligned}$$

(17)

clearly, for $p=2,3$, we have

$$\begin{aligned} S_2(\varPhi )= & {} -\varPhi \left( b_1+b_2\varPhi +b_{3}\varPhi ^{2}+b_{5}\varPhi ^{4}\right) ,\nonumber \\ S_3(\varPhi )= & {} -\varPhi \left( b_1+b_3\varPhi ^2+b_{5}\varPhi ^{4}+b_{7}\varPhi ^{6}\right) . \end{aligned}$$

We notice that the exact traveling wave solution and bifurcation of the cubic–quintic was previously analyzed in [16,17,18]. 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 $N_{{\hat{h}}_{1}{\hat{h}}_{2}}$ by using Eq. (9). In order to consider the dynamics for system (17), when we apply a transformation $\varPhi =\varphi ^{\frac{1}{p-1}}$, and substituting ${\dot{\varphi }}=\frac{1}{2}(1-s)y$ in system (17), then we obtain a new system:

$$\begin{aligned} \frac{\hbox {d}\varphi }{\hbox {d}\xi }= & {} \frac{1}{2}(1-s)y,\nonumber \\ \frac{\hbox {d}y}{\hbox {d}\xi }= & {} \frac{(p-2)\left( {1-s}\right) ^2y^2-4(p-1)^2(b_1\varphi ^{2}+b_p\varphi ^{3}+ b_{2p-1}\varphi ^4+b_{2p+1}\varphi ^5)}{4(p-1)\varphi }. \end{aligned}$$

(18)

Obviously, system (18) is a singular traveling wave system of the first class [19,20,21] with a singular straight lines $\varphi =0$. 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:

$$\begin{aligned} \frac{\hbox {d}\varphi }{\hbox {d}\zeta }= & {} 2(1-s)(p-1)\varphi y,\nonumber \\ \frac{\hbox {d}y}{\hbox {d}\zeta }= & {} (p-2)(1-s)^2y^2-4(p-1)^2\varphi ^{2}\nonumber \\&\times \left( b_1+b_p\varphi +b_{2p-1}\varphi ^2+b_{2p+1}\varphi ^3\right) . \end{aligned}$$

(19)

where $\hbox {d}\xi =-2(p-1)\varphi d\zeta$, for $p\ne 1$.

For $s=2$, system (18) has the first integral

$$\begin{aligned}&H_p(\varphi ,y)=\frac{3}{2}(p-1)\varphi ^{-\frac{2(p-2)}{p-1}}y^{2}-2(p-1)^2 \varphi ^{\frac{2}{p-1}}\nonumber \\&\quad \left( \frac{2b_{2p+1}}{3p-1}\varphi ^3+\frac{b_{2p-1}}{p}\varphi ^2+ \frac{2b_p}{p+1}\varphi +b_1\right) =h.\nonumber \\ \end{aligned}$$

(20)

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

$$\begin{aligned} H_2(\varphi ,y)= & {} \frac{3}{2}y^2-2\varphi ^2\left[ b_1+\frac{2}{3}b_2\varphi +\frac{1}{2}b_3\varphi ^2+\frac{1}{5}b_5\varphi ^3\right] \nonumber \\\equiv & {} h. \end{aligned}$$

(21)

Obviously, system (18) has always has an equilibrium point $E_0(0,0)$. Write, $P_3(\varphi )=b_1+b_2\varphi +b_{3}\varphi ^2+b_{5}\varphi ^3$. Clearly, $P_{3}^{\prime }(\varphi )=b_2+2b_{3}\varphi +3b_{5}\varphi ^2$. When $\varphi =r_{1,2}=\frac{-b_{3}\pm \sqrt{\Delta }}{3b_{5}}, \Delta =b_{3}^2-3b_2$, we have $P_{3}^{\prime }(r_{1,2})=0$. To investigate the equilibrium points of (18), we need to find all zeros of the function $P_3(\varphi )$. The cubic polynomial $P_3(\varphi )$ has three simple real zeros if and only if

$$\begin{aligned} Q(b_1,b_2,b_{3})\equiv & {} \frac{1}{27}b_2^3-\frac{1}{108}b_2^2b_3^2-\frac{1}{6}b_1b_2b_3\nonumber \\&+\,\frac{1}{4}b_1^2+\frac{1}{27}b_1b_3^3<0. \end{aligned}$$

(22)

If $Q(b_1,b_2,b_{3})=0$, then $P_3(\varphi )$ has one simple zero $\varphi _{1}$ and one double real zero. When $Q(b_1,b_2,b_{3})>0$, $P_3(\varphi )$ has only one real zero.

For a given fixed $b_1$ in the $(b_2,b_{3})$-parametric plane, the function $Q(b_2,b_{3})=0$ defines a quartic curves shown in Fig. 1a, b, which has three branches and partitions the $(b_2,b_{3})$-parameter plane in to three regions. It is easy to prove that for $b_1>0$, the curves defined by $Q(b_2,b_{3})=0$ has a cusp point at $(3b_{1}^\frac{2}{3},3b_{1}^\frac{1}{3})$. While for $b_1<0$, the curves defined by $Q(b_2,b_{3})=0$ has a cusp point at $(3b_{1}^\frac{2}{3},-3b_{1}^\frac{1}{3})$. In the regions (II), (III) for $b_{1}>0$ and regions ${\hat{II}}, {\hat{III}}$ for $b_{1}<0$, there exists three real zeros of $P_3(\varphi )$.

Fig. 1

The partition of the $(b_2,b_3)$-parametric plane of system (18). a $b_0 >0$, b $b_0 <0$

Fig. 2

Bifurcations of phase portraits of system (18), when $b_1>0$. a $r_2< 0< r_1, h_2 < 0$, b $r_2 = r_1< 0, h_1 = h_2 < 0$, c $0< r_2< r_1, b_2< h_2 < 0$

Let ${\hat{M}}(\varphi _j, 0)$ be the coefficient matrix of the linearized system of (18) at an equilibrium point $E_j$. We have $J(0,0)=\text{ det }{\hat{M}}(0,0)=0,\, J(\varphi _j,0)=\text{ det }{\hat{M}}(\varphi _j,0) =-8\varphi _{j}^2(4\phi _j^3+3b_2\phi _j^2+2b_1\phi _j+b_0)$. We write that $h_0=H_2(0,0)=0, \ h_j=H_2(\varphi _j,0)$. 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 $b_1<0$, and $b_1>0$, respectively, as follows.

Case 1 Assume that $b_1>0, Q(b_1,b_2,b_3)<0$. When $b_5>0$, system (18) has two equilibrium points $E_0(0,0)$ and $E_1(\phi _1,0)$. We have the phase portrait shown in Fig. 2a–c.

Case 2 Assume that $b_1>0, Q_1(b_2,b_3)<Q_2(b_2,b_3)$. In this case, $P_3(\phi )$ has three real zeros $\phi _j, j=1,2,3$. System (18) has four equilibrium points $E_0(0,0)$ and $E_j(\phi _j,0), j=1,2,3$. When $b_5>0$, the origin $E_0(0,0)$ and $E_3$ are saddle points; $E_1$ and $E_2$ are center points. The bifurcations of phase portraits of system (18) are shown in Fig. 3a–h.

Fig. 3

Bifurcations of phase portraits of system (18), when $b_1>0$. a $r_2<0<r_1, h_1<h_2<0$, b $r_2<r_1<0, h_1<h_2<0$, c $r_2<r_1<0, 0<h_2<h_1$, d $r_2<0<r_1, h_2=h_3$, e $r_1<0<r_2, h_2<h_1<0$, f $r_1<r_2<0, h_1=h3$, g $r_2< 0< r_1, h_2< 0 < h_1$, h $r_2< 0< r_1, h_2< 0 < h_1$, i $r_2 = 0< r_1, h_1 < h_2 = 0$

Case 3 Assume that $b_1<0, Q_1(b_2,b_3)<0<Q_2(b_2,b_3),0<3b_2<b_3^2$. System (18) has four equilibrium points $E_0(0,0)$ and $E_j(\phi _j,0), j=1,2,3$. When $b_5>0$, the Equilibrium points $E_1$ and $E_2$ are saddle points; $E_0$ and $E_3$ are center points. The bifurcations of phase portraits of system (18) are shown in Fig. 4a–h.

Fig. 4

Bifurcations of phase portraits of system (18), when $b_1<0$. a $r_2<0<r_1, h_2<0<h_1$, b $r_2<b_2<r_1, 0<h_1<h_2$, c $r_2<0<r_1, h_2<0<h_1$, d $r_2<0<r_1, 0<h_1<h_2$, e $r_2<0<r_1, h_2<0<h_1$, f $r_2<0<r_1, 0<h_1<h_2$, g $r_1< r_2< 0, 0< h_1 < h_2$, h $r_2 = r_1> 0, h_1 = h_2 > 0$

Case 4 Assume that $b_1<0, 0<3b_2<b_3^2, Q(b_2,b_3)>0$. In this case, we have $\phi _1=\phi _2=r_1$. $P_3(\phi )$ has a double real zero $\phi _{12}=r_1$ and a simple real zero $\phi _3$. System (18) has three equilibrium points $E_{12}(r_1,0), E_0(0,0)$ and $E_3(\phi _3,0)$. When $b_5>0$, the origin $E_0(0,0)$ is center point, $E_{12}$ is a cusp and $E_3$ is a saddle point. The bifurcations of phase portraits of system (18) are shown in Fig. 5a–c.

Fig. 5

Bifurcations of phase portraits of system (18) when $b_1<0$. a $h_2 < r_1$, b $r_1 < h_2$, c $r_2< r_1< 0, h_2 = h_3 < 0$

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 $\sqrt{\varPhi }$ and $\varTheta$. We see from (21) and the first equation of system (18) that

$$\begin{aligned} \xi= & {} \int _{\varPhi _0}^{\varPhi }\frac{\hbox {d}\varPhi }{y(\varPhi )}\nonumber \\= & {} \int _{\varPhi _0}^{\varPhi }\frac{\hbox {d}\varPhi }{\sqrt{\frac{2}{3}h+ \frac{8}{15}b_5\varPhi ^5+\frac{2}{3}b_3\varPhi ^4+\frac{8}{9}b_2\varPhi ^3+\frac{4}{3}b_1\varPhi ^2}}.\nonumber \\ \end{aligned}$$

(23)

To find the exact solutions, we consider all bounded orbits of system (18) with $\varPhi =\varphi >0$, for $p = 2$. In this section, we discuss the exact solutions of Eq. (1) in the different regions of parameter plane $(b_2,b_3)$ for a given $b_1<0$ and $b_1>0$.

3.1 Consider case 1 in Sect. 2 (see Fig. 2a, b)

For $h=0$, the level curves defined by $H_2(\varPhi ,y)=h, b_1>0$, there exists a homoclinic orbit of system (18) to the saddle point $E_0(0,0)$ enclosing the equilibrium point $E_1(\varPhi _1,0)$. In this case, (21) can be written as $y^2=\frac{4}{3}b_1\varPhi ^2+\frac{8}{9}b_2\varPhi ^3+\frac{2}{3}b_3\varPhi ^4+\frac{8}{15}b_5\varPhi ^5= (\frac{8b_5}{15})\varPhi ^2(\varPhi -\varPhi _m)[(\varPhi -{\hat{b}}_1)^2+{\hat{a}}_1^2]$. Hence, (21) implies we have the following parametric representation of $\varPhi (\xi )$:

$$\begin{aligned} \varPhi (\xi )=(\varPhi _m+A_1)-\frac{2A_1}{1+\text{ cn }(\omega _0\xi ,k)}. \end{aligned}$$

(24)

Thus, we have from (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta (\varPhi _m+ A_1)+\frac{a}{4}(\varPhi _m+ A_1)^3\right) \xi \nonumber \\&-\,\left( \delta +\frac{3a}{2}(\varPhi _m+ A_1)^2A_1\right) \nonumber \\&\times \, E(\text{ arccos }(\text{ cn }(\omega _0\xi ,k)),k)-\frac{a}{4} (\varPhi _m+ A_1)^3\rho _1\nonumber \\&-\,3a(\varPhi _m+A_1)A_1^2\rho _2+\frac{{\hat{h}}_2}{\varPhi _m+A_1}\Psi _0. \end{aligned}$$

(25)

Hence, we have the following solution Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+A_1-\frac{2A_1}{1+\text{ cn }(\omega _0\xi ,k)}\right) ^\frac{1}{2}\nonumber \\&\times \,\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] \end{aligned}$$

(26)

where $\varTheta (\xi )$ is given by (25), $A_1^2=({\hat{b}}_1-\varPhi _m)^2+{\hat{a}}_1^2, k^2=\frac{A_1+{\hat{b}}_1-\varPhi _m}{2A_1}, \omega _0=\frac{\sqrt{8b_5}}{\sqrt{15A_1}}$, $\alpha _r^2=\frac{\varPhi _m+A_1}{\varPhi _m-A_1}$, $\Psi _0=\frac{\varPhi _m-A_1}{2\varPhi _m}\Big [\Pi (\text{ arcsin }(\text{ sn }(\omega _0\xi ,k)), \frac{\alpha _r^2}{\alpha _r^2-1},k)-\alpha _r^2\text{ sn }(\omega _0\xi ,k)\Big ]$, $\rho _1=\frac{1}{5}k^2F(\text{ arccos }(\text{ cn }(\omega _0\xi ,k)),k)-\frac{6}{5}k^2E(\text{ arccos }(\text{ cn }(\omega _0\xi ,k)),k)+\frac{2}{5}(4k^2+1)\rho _2+\frac{\text{ sn }(\omega _0,k)\text{ cn }(\omega _0\xi ,k)}{5(1+\text{ cn }(\omega _0\xi ,k))^3}$, $\rho _2=\frac{1}{3}\Big [(4k^2+1)E(\text{ arccos }(\text{ cn }(\omega _0\xi ,k)),k)+\frac{\text{ sn }(\omega _0\xi ,k)\text{ cn }(\omega _0\xi ,k)}{(1+\text{ cn }(\omega _0\xi ,k))^2}\Big ]$.

3.2 Consider case 2 in Sect. 2 for $b_1>0$ (see Fig. 3a–h)

• (1) The case of $r_1<0<r_2; \ h_3<0$ (see Fig. 3a).

For $h_2=0$, the level curves defined by $H_2(\phi ,y)=0$, there exist two homoclinic orbits of system (18) with “eight shape” to the origin $E_0(0,0)$. In this case, (21) can be written as $y^2=\frac{8b_5}{15}\varPhi ^2(\varPhi -\varPhi _l)(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )$, where $\varPhi _l<\varPhi _m<\varPhi _1<0<\varPhi _2<\varPhi _M$.

Hence, for the right homoclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

$$\begin{aligned} \varPhi (\xi )=\varPhi _M-(\varPhi _M-\varPhi _m){\text{ sn }^2(\omega _1\xi ,k)}. \end{aligned}$$

(27)

Corresponding to the right homoclinic orbits we obtain from (10), that

$$\begin{aligned} \varTheta (\xi )= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _M+\frac{a}{4}\varPhi _M^3\right) \xi \nonumber \\&+\, \frac{{\hat{h}}_2}{\varPhi _M}\Pi \left( \arcsin (\text{ sn }(\omega _1\xi ,k)),\frac{\varPhi _M-\varPhi _m}{\varPhi _M},k\right) \nonumber \\&+\,\frac{1}{4}\left( 3a\varPhi _M^2-2\delta \right) (\varPhi _M-\varPhi _m)\beta _1\nonumber \\&-\, \frac{a}{4}\left[ 3\varPhi _M\beta _2+(\varPhi _M-\varPhi _m)\beta _3\right] (\varPhi _M-\varPhi _m)^2.\nonumber \\ \end{aligned}$$

(28)

From Eqs. (27) and (28), we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _M-(\varPhi _M-\varPhi _m){\text{ sn }^2(\omega _1\xi ,k)}\right) ^\frac{1}{2}\nonumber \\&\times \,\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] \end{aligned}$$

(29)

where $\varTheta (\xi )$ is given by (28), $\alpha _1^2=\frac{\varPhi _M-\varPhi _m}{\varPhi _M}, k^2=\frac{\varPhi _M-\varPhi _m}{\varPhi _M-\varPhi _l}, \omega _1=\sqrt{\frac{2b_5(\varPhi _M-\varPhi _l)}{15}}$

$$\begin{aligned} \beta _1= & {} \frac{1}{k^2}E(\text{ arcsin }(\text{ sn }(\omega _1\xi ,k)),k),\\ \beta _2= & {} \frac{1}{3k^2}2(1+k^2)\beta _1\\&+\,\frac{1}{3k^2}\text{ sn }^{3}(\omega _1\xi ,k)\text{ cn }(\omega _1\xi ,k)\text{ dn }(\omega _1\xi ,k),\\ \beta _3= & {} \frac{1}{5k^2}\left[ \text{ sn }^{3}(\omega _1\xi ,k)\text{ cn }(\omega _1\xi ,k)\text{ dn }(\omega _1\xi ,k)\right. \\&\left. +\,4(1+k^2)\beta _2-3\beta _1\right] . \end{aligned}$$

For the left homoclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

$$\begin{aligned} \varPhi (\xi )=\varPhi _l-\frac{\varPhi _l-\varPhi _m}{\text{ dn }^{2}(\omega _1\xi ,k)}. \end{aligned}$$

(30)

And, corresponding to the left homoclinic orbits we obtain from (10), that

$$\begin{aligned}&\varTheta (\xi )=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _l+\frac{a}{4}\varPhi _{l}^3\right) \xi \nonumber \\&\quad + \frac{{\hat{h}}_2}{\varPhi _l}\left( \frac{\varPhi _l}{\varPhi _m}-1\right) \Pi \left( \arcsin (\text{ sn }(\omega _1\xi ,k)),\right. \nonumber \\&\quad \left. \frac{\varPhi _l}{\varPhi _m}k^2,k\right) +\frac{1}{4}\left( 2\delta -3a\varPhi _{l}^2\right) (\varPhi _l-\varPhi _m)I_1 \nonumber \\&\quad +\left( \frac{3a}{4}\varPhi _lI_2-\frac{a}{4}(\varPhi _l-\varPhi _m)I_3\right) (\varPhi _l-\varPhi _m)^2. \end{aligned}$$

(31)

Hence, from Eqs. (30) and (31), we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _l-\frac{\varPhi _l-\varPhi _m}{\text{ dn }^{2}(\omega _1\xi ,k)}\right) ^\frac{1}{2}\nonumber \\&\times \,\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] \end{aligned}$$

(32)

where $\varTheta (\xi )$ is given by (31) and $I_1=\frac{1}{k^{\prime 2}}E\left( \arcsin (\text{ sn }(\omega _1\xi ,k)),k\right) -\frac{k^2}{k^{\prime 2}}\text{ sn }(\omega _1\xi ,k)\text{ cn }(\omega _1\xi ,k)$, $I_2=\frac{1}{k^{\prime 4}}\left[ 2(2-k^2)E\left( \arcsin (\text{ sn }(\omega _1\xi ,k)),k\right) -k^2\text{ sn }(\omega _1\xi ,k)\text{ cn }(\omega _1\xi ,k)(k^2\text{ nd }^2(\omega _1\xi ,k)+4-2k^2)\right]$, $I_3=\frac{1}{5k^{\prime 2}}(5-4k^2)I_2, k^{\prime }=\sqrt{1-k^2}$ and $\text{ sn }(\cdot ,k), \text{ cn }(\cdot ,k)$, $\text{ dn }(\cdot ,k),\text{ sd }(\cdot ,k)$ are Jacobin elliptic functions and $\Pi (\cdot , \cdot ,k)$, is the elliptic integrals of the third kind [24].

• (2) The case of $r_2<r_1<0, \ h_1<h_2<0$ (see Fig. 3b).

Corresponding to the level curve defined by $H_2(\varPhi ,y)=h, \ h \in (h_1,h_2)$, there exist two homoclinic orbit to the equilibrium $E_2(\varPhi _2,0)$ with “figure eight” of system (18). Now, (23) has the form

$$\begin{aligned}&\left( \sqrt{\frac{8b_5}{15}}\right) \xi \\&\;\;=\int _{\varPhi }^{\varPhi _M}\frac{\hbox {d}\varPhi }{(\varPhi -\varPhi _2)\sqrt{(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}}, \end{aligned}$$

and

$$\begin{aligned}&\left( \sqrt{\frac{8b_5}{15}}\right) \xi \\&\;\;=\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _2-\varPhi )\sqrt{(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}}, \end{aligned}$$

where $\varPhi _m<\varPhi _1<\varPhi _2<\varPhi _3<\varPhi _M<0<\varPhi _L$. Using the above integrals, corresponding to the right homoclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

$$\begin{aligned} \varPhi (\xi )=\varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _2\xi ,k)}. \end{aligned}$$

(33)

Thus, corresponding to the right homoclinic orbits we obtain from (10), that

$$\begin{aligned}&\varTheta (\xi )=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _L+\frac{a}{4}\varPhi _{L}^3\right) \xi \nonumber \\&\quad +\frac{{\hat{h}}_2}{\varPhi _L}\left( \frac{\varPhi _L}{\varPhi _M}-1\right) \Pi \left( \arcsin (\text{ sn }(\omega _2\xi ,k)),\alpha _{2}^2,k\right) \nonumber \\&\quad -\frac{1}{4}\left( 2\delta -3a\varPhi _{L}^2\right) (\varPhi _M-\varPhi _L)I_4\nonumber \\&\quad +\frac{3a}{4}\varPhi _L(\varPhi _M-\varPhi _L)^2I_5-(\varPhi _M-\varPhi _L)^3I_6. \end{aligned}$$

(34)

Fig. 6

Three profiles of the solitary waves of system (18). a $H_2(\varPhi ,y)=0$, b $H_2(\varPhi ,y)=h_2$, right orbit, c $H_2(\varPhi ,y)=h_2$, left orbit

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _2\xi ,k)}\right) ^\frac{1}{2}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(35)

where $\varTheta (\xi )$ is given by (34) and $\alpha _{2}^2=\frac{\varPhi _M-\varPhi _m}{\varPhi _2-\varPhi _m}, I_4=\frac{k^2}{k^{\prime 2}}\left[ \beta _4-\text{ sn }(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)\right] , I_5=\frac{k^2}{k^{\prime 4}}2(2-k^2)\beta _4-\frac{k^2}{k^{\prime 4}} \text{ sn }(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)(k^2(\text{ nd }^2(\omega _2\xi ,k)-2)+4), I_6=\frac{1}{5k^{\prime 2}}(5-4k^2)I_5$.

Corresponding to the left homoclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

$$\begin{aligned} \varPhi (\xi )=\varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _2\xi ,k). \end{aligned}$$

(36)

Thus we obtain from (10) that

$$\begin{aligned}&\varTheta (\xi )=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _M+\frac{a}{4}\varPhi _M^3\right) \xi \nonumber \\&\quad + \frac{{\hat{h}}_2}{\varPhi _m}\Pi \left( \arcsin (\text{ sn }(\omega _2\xi ,k)),\frac{\varPhi _m-\varPhi _M}{\varPhi _M},k\right) \nonumber \\&\quad +\frac{1}{4}\left( 2\delta -3a\varPhi _M^2\right) (\varPhi _M-\varPhi _m)\beta _4\nonumber \\&\quad + \frac{a}{4}\left[ 3\varPhi _M\beta _5-(\varPhi _M-\varPhi _m)\beta _6\right] (\varPhi _M-\varPhi _m)^2.\nonumber \\ \end{aligned}$$

(37)

Thus, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _2\xi ,k)\right) ^\frac{1}{2}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(38)

where $\varTheta (\xi )$ is given by (37), and $k^2=\frac{\varPhi _M-\varPhi _m}{\varPhi _2-\varPhi _m}$, $\omega _2=\sqrt{\frac{2b_5(\varPhi _L-\varPhi _m)}{15}}$, $\beta _4=\frac{1}{k^2}E\left( \arcsin (\text{ sn }(\omega _2\xi ,k)),k\right)$, $\beta _5=\frac{1}{3k^2}2(1+k^2)\beta _4+\frac{1}{3k^2}\text{ sn }^{3}(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)\text{ dn }(\omega _2\xi ,k)$, $\beta _6=\frac{1}{5k^2}\text{ sn }^{3}(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)\text{ dn }(\omega _2\xi ,k)+ \frac{4}{5k^2}(1+k^2)\beta _5-3\beta _4$. Equations (33) and (36) give rise to the profiles of solitary waves shown in Fig. 6b, c.

• (3) The case of $r_2<r_1<0, \ h_1=0$ (see Fig. 3c).

• (i) Corresponding to the level curves defined by $H_2(\varPhi ,y)=0$, there exist a periodic orbits of system (18). (23) has the form

  $$\begin{aligned} \omega _3\xi =\int _{r_2}^{\varPhi }\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -r_1)(r_2-\varPhi )(\varPhi _m-\varPhi )}}, \end{aligned}$$

  where, $r_2<\varPhi _1<r_1<\varPhi _m<\varPhi _2<0$. It gives rise to the parametric representations as follows:

  $$\begin{aligned} \varPhi (\xi )=r_1+(r_2-r_1)\text{ sn }^2(\omega _3\xi ,k). \end{aligned}$$

  (39)

  From Eq. (10) we have

  $$\begin{aligned} \varTheta (\xi )= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta r_1+\frac{a}{4}r_1^3\right) \xi \nonumber \\&+\frac{{\hat{h}}_2}{r_1}\Pi \left( \arcsin (\text{ sn }(\omega _3\xi ,k)),\frac{r_1-r_2}{r_1},k\right) \nonumber \\&-\frac{1}{4}\left( 2\delta +3ar_1^2\right) (r_1-r_2)\beta _7\nonumber \\&-\frac{a}{4}\left[ 3r_2\beta _8+(r_2-r_1)\beta _9\right] (r_2-r_1)^2.\nonumber \\ \end{aligned}$$

  (40)

  Hence, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( r_1+(r_2-r_1)\text{ sn }^2(\omega _3\xi ,k)\right) ^\frac{1}{2}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] \end{aligned}$$

  (41)

  where $\varTheta (\xi )$ is given by (40) and $k^2=\frac{r_1-r_2}{\varPhi _m-r_2}$, $\omega _3=\sqrt{\frac{2b_5(\varPhi _m-r_2)}{15}}$,

  $$\begin{aligned} \beta _7= & {} \frac{1}{k^2}E\left( \arcsin (\text{ sn }(\omega _3\xi ,k)),k\right) ,\\ \beta _8= & {} \frac{1}{3k^2}2(1+k^2)\beta _7\\&+\frac{1}{3k^2}\text{ sn }^{3}(\omega _3\xi ,k)\text{ cn }(\omega _3\xi ,k)\text{ dn }(\omega _3\xi ,k),\\ \beta _9= & {} \frac{1}{5k^2}\text{ sn }^{3}(\omega _3\xi ,k)\text{ cn }(\omega _3\xi ,k)\text{ dn }(\omega _3\xi ,k)\\&+\frac{1}{5k^2}\left[ 4(1+k^2)\beta _8-3\beta _7\right] . \end{aligned}$$

• (ii) Corresponding to the level curves defined by $H_2(\varPhi ,y)=0$, there exists a homoclinic orbit of system (18) to the saddle point $E_0(0,0)$ enclosing the equilibrium point $E_2(\varPhi _2,0)$. Equation (23) has the form

  $$\begin{aligned} \omega _3\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -r_1)(\varPhi -r_2)(\varPhi -\varPhi _m)}}. \end{aligned}$$

  It gives rise to the parametric representations a homoclinic orbit of system (18) as follows:

  $$\begin{aligned} \varPhi (\xi )=r_1+\frac{\varPhi _m-r_1}{1-\text{ sn }^2(\omega _3\xi ,k)}. \end{aligned}$$

  (42)

  Thus we obtain from (10) that

  $$\begin{aligned} \varTheta (\xi )= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta r_1+\frac{a}{4}r_1^3\right) \xi \nonumber \\&+\left( 3r_1\Gamma _1+\frac{a}{4}(\varPhi _m-r_1)\Gamma _2\right) (\varPhi _m-r_1)^2.\nonumber \\&+ \frac{{\hat{h}}_2}{\varPhi _m}\Psi _1+\frac{1}{4}(3ar_1^2+2\delta )(\varPhi _m-r_1)\nonumber \\&\Pi \left( \arcsin (\text{ sn }(\omega _3\xi ,k)),\frac{r_1}{\varPhi _m},k\right) . \end{aligned}$$

  (43)

  Hence, from Eqs. (42) and (43), we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( r_1+\frac{\varPhi _m-r_1}{1-\text{ sn }^2(\omega _3\xi ,k)}\right) ^\frac{1}{2}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

  (44)

  where $\Gamma _1=\frac{2(2k^2-1)}{3k^{\prime 4}} E\left( \arcsin (\text{ sn }(\omega _3\xi ,k)),k\right) + \frac{(2-4k^2+k^{\prime 2}\text{ nc }^2(\omega _3\xi ,k))}{3k^{\prime 4}}\text{ dn }(\omega _3\xi ,k)\text{ tn }(\omega _3\xi ,k)$, $\Gamma _2=\frac{1}{5k^{\prime 2}}\left[ (3k^2+4(1-2k^2))\Gamma _1 +\text{ dn }(\omega _3\xi ,k)\text{ tn }(\omega _3\xi ,k)\text{ nc }^4(\omega _3\xi ,k)\right]$ and $\varPhi _1$ is a function of $\xi$. Note: $\Psi _1=\int _{\varPhi _m}^{\varPhi }\frac{1-\text{ sn }^2(\omega _3\xi ,k)}{1-\alpha _3^2\text{ sn }^2(\omega _3\xi ,k)}\hbox {d}\xi$.

• (4) The case of $r_2<0<r_1, \ h_2=h_3$ (see Fig. 3d).

Corresponding to the curves defined by $H_2(\varPhi ,y)=h \ h \in (0,h_1)$, we have $y^2=(\frac{8b_5}{15})(\varPhi -r_2)(r_1-\varPhi )(\varPhi _2-\varPhi )^3$, where $r_2<\varPhi _1<r_1<0<\varPhi _2=\varPhi _3$. Equation (18) has a periodic orbit enclosing $E_1(\varPhi _1,0)$, which gives rise to a parametric representation of Eq. (1). Now, (23) has the form

$$\begin{aligned} \omega _4\xi =\int _{r_2}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _2-\varPhi )\sqrt{(\varPhi -r_2)(r_1-\varPhi )(\varPhi _2-\varPhi )}}. \end{aligned}$$

Therefore, we obtain the following parametric representation of a periodic orbits:

$$\begin{aligned} \varPhi (\xi )=r_2+(r_1-r_2)\text{ sn }^2(\omega _4\xi ,k). \end{aligned}$$

(45)

Thus we obtain from (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta r_2+\frac{a}{4} r_2^3\right) \xi \nonumber \\&+ \frac{{\hat{h}}_2}{r_2}\Pi \left( \arcsin (\text{ sn }(\omega _4\xi ,k)),\left( \frac{r_2-r_1}{r_2}\right) ,k\right) \nonumber \\&+\frac{1}{4}\left( 2\delta +3ar_2^2\right) (r_1-r_2)\beta _{10}\nonumber \\&+\frac{a}{4}\left[ 3r_2\beta _{11}+(r_1-r_2)\beta _{12}\right] (r_1-r_2)^2.\nonumber \\ \end{aligned}$$

(46)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( r_2+(r_1-r_2)\text{ sn }^2(\omega _4\xi ,k)\right) ^\frac{1}{2}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \end{aligned}$$

(47)

where $\varTheta (\xi )$ is given by (46), $k^2=\frac{r_1-r_2}{\varPhi _2-r_2}, \beta _{10}=\frac{1}{k^2}E\left( \arcsin (\text{ sn }(\omega _4\xi ,k)),k\right)$,

$$\begin{aligned} \omega _4= & {} \frac{4\sqrt{{2b_4}}}{\sqrt{15(\varPhi _2-r_2)}},\\ \beta _{11}= & {} \frac{1}{3k^2}\left[ 2(1+k^2)\beta _{10}\right. \\&\left. +\text{ sn }^{3}(\omega _4\xi ,k)\text{ cn }(\omega _4\xi ,k)\text{ dn }(\omega _4\xi ,k)\right] ,\\ \beta _{12}= & {} \frac{1}{5k^2}\text{ sn }^{3}(\omega _4\xi ,k)\text{ cn }(\omega _4\xi ,k)\text{ dn }(\omega _4\xi ,k)\\&+\frac{1}{5k^2}\left[ 4(1+k^2)\beta _{11}-3\beta _{10}\right] . \end{aligned}$$

• (5) The case of $r_2<0<r_1, \ h_2<0<h_1$ (see Fig. 3g).

For $h=0 \text{ and } \quad 3b_2b_5<b_3^2$ the level curves defined by $H_2(\varPhi ,y)=0$, we have from Eq. (21), $y^2=\left( \frac{8b_5}{15}\right) (\varPhi -\varPhi _m)\varPhi ^3(\varPhi _L-\varPhi )$, where $\varPhi _m<\varPhi _1<0<\varPhi _L$. Equation (18) has a degenerate homoclinic orbit at a cusp $E_0(0,0)$, enclosing $E_1(\varPhi _1,0)$. Now, (23) has the form

$$\begin{aligned} \omega _2\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -\varPhi _m)\varPhi (\varPhi _L-\varPhi )}}. \end{aligned}$$

Therefore, we have the following parametric representation of $\varPhi (\xi )$:

$$\begin{aligned} \varPhi (\xi )=\varPhi _m(1-\text{ sn }^2(\omega _2\xi ,k)). \end{aligned}$$

(48)

Thus we obtain from (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _m+\frac{a}{4}\varPhi _m^3\right) \xi \nonumber \\&-\,\frac{1}{4}\left( 2\delta +3a\varPhi _m^3\right) I_{7}+\frac{a}{4}\varPhi _m^3 \left( 3I_{8}-I_{9}\right) \nonumber \\&+\,\frac{{\hat{h}}_2}{\varPhi _m}\left( E\left( \arcsin (\text{ sn }(\omega _2\xi ,k)),k\right) \right. \nonumber \\&\left. -\,\text{ dn }(\omega _2\xi ,k)\text{ tn }(\omega _2\xi ,k)\right) . \end{aligned}$$

(49)

Fig. 7

The profiles of kink and anti-kink waves of system (18). a Kink wave, b anti-kink wave

Therefore, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\sqrt{\varPhi _m}\left( (1-\text{ sn }^2(\omega _2\xi ,k))\right) ^\frac{1}{2}\nonumber \\&\times \,\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \end{aligned}$$

(50)

where $\varTheta (\xi )$ is given by (49) and $k^2=\frac{\varPhi _m}{\varPhi _L-\varPhi _m}$, $I_7=\frac{1}{k^2}E\left( \arcsin (\text{ sn }(\omega _2\xi ,k)),k\right)$, $I_8=\frac{1}{3k^2}2(1+k^2)I_7+k^2\text{ sn }^{3}(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)\text{ dn }(\omega _2\xi ,k)$, $I_9=\frac{1}{5k^2}\text{ sn }^{3}(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)\text{ dn }(\omega _2\xi ,k)+\frac{1}{5k^2}\left[ 4(1+k^2)I_8-3I_7\right]$.

• (6) The case of $r_2<0<r_1, \ h_2<0<h_1$ (see Fig. 3h).

Corresponding to the level curve defined by $H_2(\varPhi ,y)=0$, there exist two homoclinic orbit to the origin $E_0(0,0)$ with “figure eight” of (18). Now, (23) has the form

$$\begin{aligned} \omega _2\xi =\int _{\varPhi }^{\varPhi _M}\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}} \end{aligned}$$

and

$$\begin{aligned} \omega _2\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}}, \end{aligned}$$

where $\varPhi _m<\varPhi _1<0<\varPhi _2<\varPhi _M<\varPhi _L$. 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 $r_2=0<r_1, \ h_1<h_2=0$ (see Fig. 3i).

In this case, corresponding to the level curve defined by $H_2(\varPhi ,y)=0$, there exists a homoclinic orbit to the origin $E_0(0,0)$, enclosing the equilibrium point $E_1(\varPhi _1,0)$ and there are two heteroclinic orbits connecting the equilibrium point $E_3(\varPhi _3,0)$ and $E_0(0,0)$, enclosing the equilibrium point $E_2(\varPhi _2,0)$. Now (23) can be written as

$$\begin{aligned} \left( \sqrt{\frac{8b_5}{15}}\right) \xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _3-\varPhi )\varPhi \sqrt{\varPhi -\varPhi _m}}. \end{aligned}$$

• (i) Corresponding to the left homoclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

  $$\begin{aligned} \varPhi ({\hat{\xi }})=\varPhi _m\text{ sech }^2({\hat{\xi }}), \quad {\hat{\xi }}\in (0,\infty ), \end{aligned}$$

  (51)

  we obtain from Eq. (10) that

  $$\begin{aligned} \varTheta ({\hat{t}})= & {} \frac{1}{2}(2\nu -\mu ){\hat{t}}+\frac{\delta }{2} \varPhi _m\text{ tanh }({\hat{\xi }})\nonumber \\&+\frac{{\hat{h}}_2}{2\varPhi _m}\text{ sinh }({\hat{\xi }})\text{ cosh }({\hat{\xi }})+\frac{a}{4}\varPhi _m^3\nonumber \\&\times \left[ \frac{\text{ sinh }({\hat{\xi }})}{\text{ cosh }({\hat{\xi }})}+\frac{4\text{ sinh }({\hat{\xi }})}{5\text{ cosh }^3({\hat{\xi }})}+ \frac{4\text{ sinh }({\hat{\xi }})}{15\text{ cosh }^2({\hat{\xi }})}\right. \nonumber \\&\left. +\frac{1}{2}\text{ ln }\left( \frac{1+\text{ sinh }({\hat{\xi }})}{1-\text{ sinh }({\hat{\xi }})}\right) \right] . \end{aligned}$$

  (52)

  where, ${\hat{\xi }}=\text{ tanh }^{-1}\sqrt{\frac{\varPhi ({\hat{\xi }})-\varPhi _m}{\varPhi _m}}$. Hence, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\sqrt{\varPhi _m}\text{ sech }({\hat{t}})\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \nonumber \\ \end{aligned}$$

  (53)

  where $\varTheta ({\hat{t}})$ is given by (52).

• (ii) Corresponding to the heteroclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

  $$\begin{aligned} \varPhi ({\hat{\xi }})=\varPhi _m+(\varPhi _m-\varPhi _3)\text{ tanh }^2({\hat{t}}), \end{aligned}$$

  (54)

  we have from Eq. (10) that

  $$\begin{aligned}&\varTheta ({\hat{\xi }})=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _m+\frac{a}{4}\varPhi ^3_m\right) {\hat{\xi }}\nonumber \\&\quad +\frac{{\hat{h}}_2}{\varPhi _3}\sqrt{\frac{\varPhi _m-\varPhi _3}{\varPhi _m}}\text{ tanh }^{-1}\left( \sqrt{\frac{\varPhi _m-\varPhi _3}{\varPhi _m}}\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad +\frac{1}{4}\left[ 3a\varPhi _m^2-2\delta \right] (\varPhi _m-\varPhi _3)\text{ tanh }({\hat{\xi }})\nonumber \\&\quad -\frac{3a}{4}\left( \text{ tanh }^3({\hat{\xi }})-4\text{ tanh }({\hat{\xi }})\right) \varPhi _m(\varPhi _m-\varPhi _3)^2\nonumber \\&\quad -\frac{a}{4}\left( \frac{1}{5}\text{ tanh }^5({\hat{\xi }})-\frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad (\varPhi _3-\varPhi _m)^3 \end{aligned}$$

  (55)

  where ${\hat{\xi }}=\text{ tanh }^{-1}\sqrt{\frac{\varPhi ({\hat{\xi }})-\varPhi _m}{\varPhi _m-\varPhi _3}}$. Thus, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _m-\varPhi _3)\text{ tanh }^2({\hat{\xi }})\right) ^\frac{1}{2}\nonumber \\&\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \end{aligned}$$

  (56)

  where $\varTheta ({\hat{\xi }})$ is given by (55).

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 $b_1<0$ (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 $r_1<r_2<0, \ 0<h_1<h_2$ (see Fig. 4g).

For $h \in (h_2,+\infty )$, the level curve defined by $H_2(\varPhi ,y)=h$, there are two heteroclinic orbits connecting the equilibrium point $E_2(\varPhi _2,0)$ and $E_3(\varPhi _3,0)$, enclosing the equilibrium point $E_0(0,0)$ and there exists a homoclinic orbit to the equilibrium point $E_2(\varPhi _2,0)$, enclosing the equilibrium point $E_1(\varPhi _1,0)$. Now (23) can be written as

$$\begin{aligned} \left( \sqrt{\frac{8b_5}{15}}\right) \xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _3-\varPhi )(\varPhi -\varPhi _2)\sqrt{\varPhi -\varPhi _m}}, \end{aligned}$$

where, $\varPhi _m<\varPhi _1<\varPhi _2<0<\varPhi _3$.

• (i) For the heteroclinic orbit, we have the following parametric representation of $\varPhi (\xi )$:

  $$\begin{aligned} \varPhi ({\hat{\xi }})=\varPhi _m+(\varPhi _2-\varPhi _m)\text{ tanh }^2({\hat{\xi }}), \end{aligned}$$

  (57)

  we have from Eq. (10) that

  $$\begin{aligned} \varTheta ({\hat{t}})= & {} \left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _m+\frac{a}{4}\varPhi _m^3\right) {\hat{\xi }}\nonumber \\&+\frac{{\hat{h}}_2}{\varPhi _2-2\varPhi _m}\sqrt{\frac{\varPhi _2-\varPhi _m}{\varPhi _m}}\nonumber \\&\text{ tanh }^{-1}\left( \sqrt{\frac{\varPhi _2-\varPhi _m}{\varPhi _m}}\text{ tanh }({\hat{\xi }})\right) \nonumber \\&+\frac{1}{4}\left[ 3a\varPhi _m^2-2\delta \right] (\varPhi _m-\varPhi _2)\text{ tanh }({\hat{\xi }})\nonumber \\&-\frac{3a}{4}\varPhi _m\left( \frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) (\varPhi _m-\varPhi _2)^2\nonumber \\&+\frac{a}{4}\left( \frac{1}{5}\text{ tanh }^5({\hat{\xi }})-\frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) \nonumber \\&(\varPhi _m-\varPhi _2)^3. \end{aligned}$$

  (58)

  Hence, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _2-\varPhi _m)\text{ tanh }^2({\hat{\xi }})\right) ^\frac{1}{2}\nonumber \\&\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

  (59)

  where $\varTheta ({\hat{\xi }})$ is given by (58) and ${\hat{\xi }}=\text{ tanh }^{-1}\sqrt{\frac{\varPhi ({\hat{\xi }})-\varPhi _m}{\varPhi _2-\varPhi _m}}$.

• (ii) Corresponding to the homoclinic orbit of system (18) to the equilibrium point $E_{2}(\varPhi _2,0)$ enclosing the equilibrium point $E_{1}(\phi _{1},0)$, we obtain a parametric representation of $\varPhi (\xi )$:

  $$\begin{aligned} \varPhi ({\hat{\xi }})=\varPhi _m+(\varPhi _m-\varPhi _2)\text{ tanh }^2({\hat{\xi }}), \end{aligned}$$

  (60)

  we have from Eq. (10) that

  $$\begin{aligned}&\varTheta ({\hat{\xi }})=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _m+\frac{a}{4}\varPhi _m^3\right) {\hat{\xi }}\nonumber \\&\quad +\frac{{\hat{h}}_2}{\varPhi _2}\sqrt{\frac{\varPhi _m-\varPhi _2}{\varPhi _m}}\text{ tanh }^{-1}\left( \sqrt{\frac{\varPhi _m-\varPhi _2}{\varPhi _m}}\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad +\frac{1}{4}\left[ 3a\varPhi _m^2-2\delta \right] (\varPhi _2-\varPhi _m)\text{ tanh }({\hat{\xi }})\nonumber \\&\quad -\frac{3a}{4}\varPhi _m\left( \frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) (\varPhi _2-\varPhi _m)^2\nonumber \\&\quad -\frac{a}{4}\left( \frac{1}{5}\text{ tanh }^5({\hat{\xi }})-\frac{1}{4}\text{ tanh }^3({\hat{\xi }})\right. \nonumber \\&\quad \left. -\text{ tanh }({\hat{\xi }})\right) (\varPhi _2-\varPhi _m)^3. \end{aligned}$$

  (61)

  Therefore, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _m-\varPhi _2)\text{ tanh }^2({\hat{\xi }})\right) ^\frac{1}{2}\nonumber \\&\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

  (62)

  where $\varTheta ({\hat{\xi }})$ is given by (61) and ${\hat{\xi }}=\text{ tanh }^{-1}\sqrt{\frac{\varPhi ({\hat{\xi }})-\varPhi _m}{\varPhi _2-\varPhi _m}}$.

• (2) The case of $r_2=r_1>0, \ h_1=h_2>0$ (see Fig. 4h).

For $h=0.1$, the level curve defined by $H_2(\varPhi ,y)=h$, there exists a degenerated homoclinic orbit to the multiple equilibrium point $(\varPhi _1,0)$, (23) becomes that

$$\begin{aligned} \left( \sqrt{\frac{8b_5}{15}}\right) \xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _1-\varPhi )^2\sqrt{\varPhi -\varPhi _m}}, \end{aligned}$$

where $\varPhi _m<0<\varPhi _1$. Thus, we have parametric representation:

$$\begin{aligned} \varPhi ({\hat{\xi }})=\varPhi _m+(\varPhi _1-\varPhi _m)\text{ tanh }^2({\hat{\xi }}), \end{aligned}$$

(63)

we obtain from Eq. (10) that

$$\begin{aligned}&\varTheta ({\hat{\xi }})=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _m+\frac{a}{4}\varPhi _m^3\right) {\hat{\xi }}\nonumber \\&\quad +\frac{{\hat{h}}_2}{\varPhi _1-2\varPhi _m}\sqrt{\frac{\varPhi _1-\varPhi _m}{\varPhi _m}}\text{ tanh }^{-1}\nonumber \\&\left( \sqrt{\frac{\varPhi _1-\varPhi _m}{\varPhi _m}}\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad +\frac{1}{4}\left[ 3a\varPhi _m^2-2\delta \right] (\varPhi _m-\varPhi _1)\text{ tanh }({\hat{\xi }})\nonumber \\&\quad -\frac{3a}{4}\varPhi _m\left( \frac{1}{4}\text{ tanh }^3(\xi )-\text{ tanh }({\hat{\xi }})\right) (\varPhi _m-\varPhi _1)^2\nonumber \\&\quad +\frac{a}{4}\left( \frac{1}{5}\text{ tanh }^5({\hat{\xi }})-\frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad (\varPhi _m-\varPhi _1)^3. \end{aligned}$$

(64)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _1-\varPhi _m)\text{ tanh }^2({\hat{\xi }})\right) ^\frac{1}{2}\nonumber \\&\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \end{aligned}$$

(65)

where $\varTheta ({\hat{\xi }})$ is given by (64) and ${\hat{\xi }}=\text{ tanh }^{-1}\sqrt{\frac{\varPhi ({\hat{\xi }})-\varPhi _m}{\varPhi _1-\varPhi _m}}$.

3.4 Consider case 4 in Sect. 2, for $b_1<0$ (see Fig. 5a–c)

• (i) Corresponding to the level curves defined by $H(\phi ,y)=0$ in (21), there exist two heteroclinic orbits of system (18) connecting the equilibrium point (cusp point) $E_{12}(r_1,0)$ and the saddle equilibrium point $E_{3}(\varPhi _3,0)$ (see Fig. 5a). In this case, (23) can be written as

  $$\begin{aligned} \left( \sqrt{\frac{8b_5}{15}}\right) \xi =\int _{\varPhi _0}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _3-\varPhi )(\varPhi -r_1)^{\frac{3}{2}}}, \end{aligned}$$

  where $\varPhi _0 \in \ (r_1,\varPhi _3)$ and $r_1<0<\varPhi _3$. Completing the above integral, we have the following parametric representation of $\varPhi (\xi )$:

  $$\begin{aligned} \varPhi ({\hat{\xi }})=r_1+(r_1-\varPhi _3)\text{ tanh }^2({\hat{\xi }}), \end{aligned}$$

  (66)

  we obtain from Eq. (10) that

  $$\begin{aligned}&\varTheta ({\hat{\xi }})=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta r_1+\frac{a}{4}r_1^3\right) {\hat{\xi }}\nonumber \\&\quad +\frac{{\hat{h}}_2}{\varPhi _2}\sqrt{\frac{r_1-\varPhi _3}{r_1}}\text{ tanh }^{-1}\left( \sqrt{\frac{r_1-\varPhi _3}{r_1}}\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad +\frac{1}{4}\left[ 3ar_1^2-2\delta \right] (\varPhi _3-r_1)\text{ tanh }({\hat{\xi }})\nonumber \\&\quad -\frac{3a}{4}r_1\left( \frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) (\varPhi _3-r_1)^2\nonumber \\&\quad -\frac{a}{4}\left( \frac{1}{5}\text{ tanh }^5(\xi )-\frac{1}{4}\text{ tanh }^3({\hat{\xi }})-\text{ tanh }({\hat{\xi }})\right) \nonumber \\&\quad (\varPhi _3-r_1)^3, \end{aligned}$$

  (67)

  where, ${\hat{\xi }}=\text{ tanh }^{-1}\sqrt{\frac{\varPhi ({\hat{t}})-r_1}{r_1-\varPhi _3}}$. Hence, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( r_1+(r_1-\varPhi _3)\text{ tanh }^2({\hat{\xi }})\right) \nonumber \\&\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \end{aligned}$$

  (68)

  where $\varTheta ({\hat{\xi }})$ is given by (67).

• (ii) Corresponding to a homoclinic orbit of system (18) to the saddle point $E_{3}(\varPhi _3,0)$ enclosing the origin $E_{0}(0,0)$ (see Fig. 5b). Hence, we obtain a similar parametric representation Eq. (1) as Eq. (65).

• (iii) Corresponding to the curves defined by $H_2(\varPhi ,y)=h_1=h_2$ (see Fig. 5c), we have from Eq. (21) that $y^2=\omega _5(\varPhi -r_1)^3(\varPhi _L-\varPhi )(\varPhi _M-\varPhi )$, where $r_1<\varPhi _3<\varPhi _M<0<\varPhi _L$. Equation (18) has a homoclinic orbit to the cusp point $E_{12}(\varPhi _1,0)$. Then (23) has the form

  $$\begin{aligned}&\omega _5\xi \\&\;=\int _{\varPhi }^{\varPhi _M}\frac{\hbox {d}\varPhi }{(\varPhi -r_1)\sqrt{(\varPhi -r_1)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}}. \end{aligned}$$

  It gives rise to the parametric representations of a homoclinic orbit of system (18) as follows:

  $$\begin{aligned} \varPhi (\xi )=\varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _5\xi ,k)}. \end{aligned}$$

  (69)

  Thus we obtain from (10) that

  $$\begin{aligned}&\varTheta (\xi )=\left( \frac{1}{2}(2\nu -\mu )+\frac{1}{2}\delta \varPhi _L+\frac{a}{4}\varPhi _{L}^3\right) \xi \nonumber \\&\quad + \frac{{\hat{h}}_2}{\varPhi _L}\left( \frac{\varPhi _L}{\varPhi _M}-k^2\right) \Pi \left( \arcsin (\text{ sn }(\omega _2\xi ,k)),\frac{\phi _L}{\varPhi _M},k\right) \nonumber \\&\quad -\frac{1}{4}\left( 2\delta -3a\varPhi _{L}^2\right) (\varPhi _M-\varPhi _L)I_4 \nonumber \\&\quad +\frac{a}{4}\left( 3\varPhi _LI_6-(\varPhi _M-\varPhi _L)I_5\right) (\varPhi _M-\varPhi _L)^2.\nonumber \\ \end{aligned}$$

  (70)

  Therefore, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( \varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _5\xi ,k)}\right) ^\frac{1}{2}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

  (71)

  where $\varTheta (\xi )$ is given by (39), and $k^2=\frac{\varPhi _M-r_1}{\varPhi _L-r_1}$, $\omega _5=\frac{4\sqrt{{2b_5}}}{\sqrt{15(\varPhi _L-r_1)}}$, $I_4=\frac{k^2}{k^{\prime 2}}\left[ \beta _4-\text{ sn }(\omega _5\xi ,k)\text{ cn }(\omega _2\xi ,k)\right]$, $I_5=\frac{1}{5k^{\prime 2}}(5-4k^2)I_6$, $I_6=\frac{k^2}{k^{\prime 4}}\left[ 2(2-k^2)\beta _4-\text{ sn }(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)(k^2(\text{ nd }^2(\omega _2\xi ,k)-2)+4)\right]$.

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:

$$\begin{aligned} H_3(\varphi ,y)= & {} 3\frac{y^2}{\varphi }-16\varphi \left[ b_1+\frac{1}{2}b_3\varphi \right. \nonumber \\&\left. +\frac{1}{3}b_5\varphi ^2+\frac{1}{4}b_7\varphi ^3\right] \equiv h. \end{aligned}$$

(72)

Obviously, if $\Delta =b_3^2-3b_2b_4<0$, system (18) has only one equilibrium point $E_0(0,0)$. If $\Delta >0,\ b_7\ne 0$, system (18) has three equilibrium points $E_0(0,0), E_1(\varphi _1,0)$ and $E_2(\varphi _2,0)$, where $\varphi _{1,2}=\frac{-b_5\pm \sqrt{\Delta }}{3b_7}$. If $\Delta =0$, then $E_1$ and $E_2$ become a double equilibrium point.

Let $M(\varphi _{j},0)$ be the coefficient matrix of the linearized system of (18) at an equilibrium point $(\varphi _{j},0), j=1,2$. We have $J(\varphi _{j},0)=\det M(\varphi _{j},0)=16\varphi _{j}^3(b_3+2b_5\varphi _{j}+3b_7\varphi _{j}^2), j= 1,2$ and $J(0,0)=\det M(0,0)= 0$.

We write for $H_3(\varphi ,y)$ given by (9), $h_0=H_3(0,0)$, and $h_{j}=H_3(\varphi _j,0)$. 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.

Fig. 8

Bifurcations of phase portraits of system (18), when $p=3$. a $\frac{2}{3}\sqrt{3b_3b_7}<b_5, \ 0<h_1<h_2$, b $b_3<2\sqrt{b_3b_7} \ h_1<h_2$, c $\frac{2}{3}\sqrt{3b_3b_7}<3b_1^\frac{3}{2}, h_1<h_2$, d $\frac{4}{3}\sqrt{3b_3b_7}=2\sqrt{b_3b_7}, 0<h_2<h_1$, e $b_3<0<b_7, \ h_1<0<h_2$, f $-\frac{4}{3}\sqrt{3b_3b_7}<3b_1^\frac{3}{2}<b_5, h_2<h_1$. g $-\frac{4}{3}\sqrt{-3b_3b_7}<b3< 3b_1^\frac{3}{2}, h_2<0<h_1$, h $3b_1^\frac{3}{2}<2\sqrt{b_3b_7}, 0<h_1<h_2$, i $\left( b_3,b_5\right) =\left( 3b_1^\frac{3}{2},-3b_1^\frac{3}{2}\right) , h_2<h_1$, j $-3b_1^\frac{3}{2}<b_5<3b_1^\frac{3}{2}, h_2=h_1$, k $3b_1^\frac{3}{2}<2\sqrt{b_3b_7}, 0<h_1<h_2$, l $b_3<0<b_5, \Delta =0, h_1=h_2$

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

$$\begin{aligned} \frac{1}{\sqrt{3}}\xi= & {} \int _{\varphi _0}^{\varphi }\frac{\hbox {d}\phi }{\sqrt{\varphi h+16b_1\varphi ^2+8b_3\varphi ^3+\frac{16}{3}b_5\varphi ^4+4b_7\varphi ^5}}\nonumber \\\equiv & {} \sqrt{\frac{4b_7}{3}}\int _{\varphi _0}^{\varphi }\frac{\hbox {d}\varphi }{\sqrt{F_5(\varphi ,h)}}. \end{aligned}$$

(73)

Thus, we shall find all possible exact explicit parametric representations for all bounded functions $\varPhi =\sqrt{\varphi }$ where $\varphi >0$, determined by Eq. (18) in different parametric region of the $(b_3,b_5)$-parameter space.

5.1 The case of $\frac{2}{3}\sqrt{3b_3b_7}<3b_1^\frac{3}{2}, \ 0<h_1<h_2$ (see Fig. 8a)

For $h=h_1$, the level curve defined by $H_3(\varPhi ,y)=h$, Eq. (18) has a homoclinic orbit to the cusp point $E_{23}(\varPhi _2,0)$, enclosing the Equilibrium point $E_{1}(\varPhi _1,0)$. In this case $F_5=(\varPhi -\varPhi _m)(\varPhi _2-\varPhi )^3\varPhi$, where $\varPhi _m<\varPhi _1<\varPhi _2=\varPhi _3<0$. Then Eq. (73) has the form

$$\begin{aligned} \omega _6\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _2-\varPhi )\sqrt{(\varPhi -\varPhi _m)(\varPhi _2-\varPhi )\varPhi }}. \end{aligned}$$

It gives rise to the parametric representations of a homoclinic orbit of system (18) as follows:

$$\begin{aligned} \varPhi (\xi )=\left( \varPhi _m+(\varPhi _2-\varPhi _m)\text{ sn }^2(\omega _6\xi ,k)\right) ^\frac{1}{2}. \end{aligned}$$

(74)

Thus, we obtain from (10), that

$$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi +\frac{1}{2}\delta \int _{\varPhi _m}^{\varPhi }\big (\varPhi _m+(\varPhi _2-\varPhi _m)\nonumber \\&\times \,\text{ sn }^2(\omega _6\xi ,k)\big )^\frac{1}{2}\hbox {d}\xi \nonumber \\&+\, {\hat{h}}_2\int _{\varPhi _m}^{\varPhi }\left( \frac{1}{\varPhi _m+(\varPhi _2-\varPhi _m)\text{ sn }^2(\omega _6\xi ,k)}\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+\,\frac{a}{4}\int _{\varPhi _m}^{\varPhi }\left( \varPhi _m+(\varPhi _2-\varPhi _m)\text{ sn }^2(\omega _6\xi ,k)\right) ^\frac{3}{2}\hbox {d}\xi .\nonumber \\ \end{aligned}$$

(75)

Therefore, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _2-\varPhi _m)\text{ sn }^2(\omega _6\xi ,k)\right) ^\frac{1}{2}\nonumber \\&\times \,\text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t),\right] \end{aligned}$$

(76)

where $\varTheta (\xi )$ is given by (75), and $\ k^2=\frac{\varPhi _2-\varPhi _m}{|\varPhi _m|}, \ \omega _6=4\sqrt{\frac{b_7}{3|\varPhi _m|}}$.

5.2 The case of $-3b_1^\frac{3}{2}<\sqrt{|b_3b_7|} \ h_1<h_2$ (see Fig. 8c).

• (i) For $h \in (h_1, h_2)$ the level curves defined by $H_3(\varPhi ,y)=h$ in (72), there exist a periodic orbits of system (18) enclosing an equilibrium point of $E_{1}(\varPhi _1,0)$. In this case we have $F_5=(\varPhi -r_2)(r_1-\varPhi )\varPhi (\varPhi _3-\varPhi )^2$, where $r_2<\varPhi _1<r_1<0<\varPhi _2<\varPhi _3$. Now, (73) can be written as

  $$\begin{aligned} \omega _7\xi =\int _{r_2}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _3-\varPhi )\sqrt{(\varPhi -r_2)(r_1-\varPhi )\varPhi }}. \end{aligned}$$

  Completing the above integral, we can get a periodic orbit of Eq. (1):

  $$\begin{aligned} \varPhi (\xi )=\left( r_1+(r_2-r_1)\text{ sn }^2(\omega _7\xi ,k)\right) ^\frac{1}{2}, \end{aligned}$$

  (77)

  we obtain from Eq. (10) that

  $$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi +\frac{1}{2}\delta \int _{r_2}^{\varPhi }\left( r_1+(r_2-r_1)\right. \nonumber \\&\left. \text{ sn }^2(\omega _7\xi ,k)\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+ {\hat{h}}_2\int _{r_2}^{\varPhi }\left( \frac{1}{r_1+(r_2-r_1)\text{ sn }^2(\omega _7\xi ,k)}\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+\frac{a}{4}\int _{r_2}^{\varPhi }\left( r_1+(r_2-r_1)\text{ sn }^2(\omega _7\xi ,k)\right) ^\frac{3}{2}\hbox {d}\xi .\nonumber \\ \end{aligned}$$

  (78)

  Hence, from Eqs. (77) and (78) we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( r_1+(r_2-r_1)\text{ sn }^2(\omega _7\xi ,k)\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

  (79)

  where $\ k^2=|\frac{r_2-r_1}{r_2}|, \ \omega _7=\frac{4\sqrt{{b_7}}}{\sqrt{3|r_2|}}$.

• (ii) Corresponding to the level curves defined by $H_3(\varPhi ,y)=h, h \in (h_1, h_2)$ in (72), there exist a homoclinic orbits of system (18) at the equilibrium points $E_{0}(0,0)$, enclosing the equilibrium point $E_{2}(\varPhi _2,0)$. Now, (73) can be written as

  $$\begin{aligned} \omega _7\xi =\int _{0}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _3-\varPhi )\sqrt{(\varPhi -r_2)(\varPhi -r_1)\varPhi }}. \end{aligned}$$

  Completing the above integral, we have the following parametric representation of Eq. (1):

  $$\begin{aligned} \varPhi (\xi )=\sqrt{r_1}\left( 1-\frac{1}{1-\text{ sn }^2(\omega _7\xi ,k)}\right) ^\frac{1}{2}, \end{aligned}$$

  (80)

  we obtain from Eq. (10) that

  $$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi +\frac{\delta }{2} \sqrt{|r_1|} \int _{0}^{\varPhi }\nonumber \\&\left( 1-\frac{1}{1-\text{ sn }^2(\omega _7\xi ,k)}\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+ \frac{{\hat{h}}_2}{\sqrt{r_1}}\int _{0}^{\varPhi }\left( 1-\frac{1}{1-\text{ sn }^2(\omega _7\xi ,k)}\right) ^{-\frac{1}{2}}\hbox {d}\xi \nonumber \\&+\frac{a}{4}\int _{0}^{\varPhi }\left[ r_1\left( 1-\frac{1}{1-\text{ sn }^2(\omega _7\xi ,k)}\right) \right] ^\frac{3}{2}\hbox {d}\xi .\nonumber \\ \end{aligned}$$

  (81)

  Hence, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left[ r_1\left( 1-\frac{1}{1-\text{ sn }^2(\omega _7\xi ,k)}\right) \right] ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

  (82)

  where $\varTheta (\xi )$ is given by (81) and $\varPhi _3\lnot 0$.

• (iii) For $h=0$, the level curves defined by $H_2(\varPhi ,y)=h$, Eq. (18) has a homoclinic orbit to the origin (cusp point) $E_{0}(0,0)$ enclosing an equilibrium point of $E_{1}(\varPhi _1,0)$. In this case we have $F_5=(\varPhi _m-\varPhi )\varPhi ^3(\varPhi _L-\varPhi )$, where $\varPhi _m<\varPhi _1<0<\varPhi _L$. Now, (73) can be written as

  $$\begin{aligned} \omega _8\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi _m-\varPhi )\varPhi (\varPhi _L-\varPhi )}}. \end{aligned}$$

  Completing the above integral, we can get a periodic orbit of Eq. (1):

  $$\begin{aligned} \varPhi (\xi )=\left( \varPhi _m(1-\text{ sn }^2(\omega _8\xi ,k))\right) ^\frac{1}{2}, \end{aligned}$$

  (83)

  we obtain from Eq. (10) that

  $$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi +\frac{\delta }{2}\sqrt{\varPhi _m} \int _{\varPhi _m}^{\varPhi }\nonumber \\&\left( (1-\text{ sn }^2(\omega _8\xi ,k))\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+ \frac{{\hat{h}}_2}{\sqrt{\varPhi _m}}\int _{\varPhi _m}^{\varPhi }\left( \frac{1}{(1-\text{ sn }^2(\omega _8\xi ,k))}\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+\frac{a}{4}\int _{\varPhi _m}^{\varPhi }\left( \varPhi _m(1-\text{ sn }^2(\omega _8\xi ,k))\right) ^\frac{3}{2}\hbox {d}\xi .\nonumber \\ \end{aligned}$$

  (84)

  Hence, we have the following solution of Eq. (1):

  $$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m(1-\text{ sn }^2(\omega _8\xi ,k))\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] \end{aligned}$$

  (85)

  where $\varTheta (\xi )$ is given by (84) and $\ k^2=\frac{|\varPhi _m}{\varPhi _L-\varPhi _m}, \ \omega _8=\frac{4\sqrt{{b_7}}}{\sqrt{3(\varPhi _L-\varPhi _m)}}$.

5.3 The case of $b_3<0<b_5, \ h_1<0<h_2$ (see Fig. 8d)

For $h=0$, the level curve defined by $H_3(\varPhi ,y)=h$, there exist two homoclinic orbit to the origin $E_0(0,0)$ with “figure eight” of (18). In this case we have $F_5=(\varPhi -\varPhi _m)\varPhi ^2(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )$, where $\varPhi _m<\varPhi _1<0<\varPhi _2<\varPhi _M<\varPhi _L$. Now, (73) has the form

$$\begin{aligned} \omega _8\xi =\int _{\varPhi }^{\varPhi _M}\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}} \end{aligned}$$

and

$$\begin{aligned} \omega _8\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{\varPhi \sqrt{(\varPhi -\varPhi _m)(\varPhi _M-\varPhi )(\varPhi _L-\varPhi )}}. \end{aligned}$$

Using the above integrals, corresponding to the right homoclinic orbit we have the following parametric representation of Eq. (1):

$$\begin{aligned} \varPhi (\xi )=\left( \varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{1}{2}, \end{aligned}$$

(86)

we obtain from Eq. (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi \nonumber \\&+\frac{1}{2}\delta \int _{\varPhi }^{\varPhi _M}\left( \varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+ {\hat{h}}_2\int _{\varPhi }^{\varPhi _M}\left( \varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^{-\frac{1}{2}}\hbox {d}\xi \nonumber \\&+\frac{a}{4}\int _{\varPhi }^{\varPhi _M}\left( \varPhi _L+\frac{\varPhi _M-\varPhi _L}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{3}{2}\hbox {d}\xi . \end{aligned}$$

(87)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _L+\frac{(\varPhi _M-\varPhi _L)}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(88)

where $\varTheta (\xi )$ is given by (87) and corresponding to the left homoclinic orbit, we have the following parametric representation of Eq. (1):

$$\begin{aligned} \varPhi (\xi )=\left( \varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _8\xi ,k)\right) ^\frac{1}{2}. \end{aligned}$$

(89)

Thus we obtain from Eq. (10) that

$$\begin{aligned}&\varTheta (\xi )=\frac{1}{2}(2\nu -\mu )\xi \nonumber \\&\quad +\frac{1}{2}\delta \int _{\varPhi _m}^{\varPhi }\left( \varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _8\xi ,k)\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&\quad + {\hat{h}}_2\int _{\varPhi _m}^{\varPhi }\left( \varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _8\xi ,k)\right) ^{-\frac{1}{2}}\hbox {d}\xi \nonumber \\&\quad +\frac{a}{4}\int _{\varPhi _m}^{\varPhi }\left( \varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _8\xi ,k)\right) ^\frac{3}{2}\hbox {d}\xi .\nonumber \\ \end{aligned}$$

(90)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m+(\varPhi _M-\varPhi _m)\text{ sn }^2(\omega _8\xi ,k)\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(91)

where $\varTheta (\xi )$ is given by (90) and $k^2=\frac{\varPhi _M-\varPhi _m}{\varPhi _L-\varPhi _m}$.

5.4 The case of $-\frac{4}{3}\sqrt{-3b_3b_7}<b3< 3b_1^\frac{3}{2}, h_2<0<h_1$ (see Fig. 8g)

For $h=h_1$ the curves defined by $H_2(\varPhi ,y)=h$, we have $y^2=\left( \frac{8b_5}{15}\right) (\varPhi -r_1)\varPhi (\varPhi _2-\varPhi )^3$, where $r_1<\varPhi _1<0<\varPhi _2=\varPhi _3$. Equation (18) has a periodic orbit enclosing $E_1(\varPhi _1,0)$, which gives rise to a periodic orbit of system (18). Now, (23) has the form

$$\begin{aligned} \omega _4\xi =\int _{r_1}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _2-\varPhi )\sqrt{(\varPhi -r_1)\varPhi (\varPhi _2-\varPhi )}}. \end{aligned}$$

Therefore, we obtain the following parametric representation:

$$\begin{aligned} \varPhi (\xi )=\sqrt{r_1}\left( 1-\text{ sn }^2(\omega _4\xi ,k)\right) ^\frac{1}{2}, \end{aligned}$$

(92)

Thus we obtain from (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi +\frac{\delta }{2} \sqrt{r_1}\nonumber \\&\int _{r_1}^{\varPhi }\sqrt{1-\text{ sn }^2(\omega _4\xi ,k)}\hbox {d}\xi \nonumber \\&+\frac{{\hat{h}}_2}{\sqrt{r_1}}\int _{r_1}^{\varPhi }\frac{\hbox {d}\xi }{\sqrt{1-\text{ sn }^2(\omega _4\xi ,k)} }\nonumber \\&+\frac{a}{4}\int _{r_1}^{\varPhi }\left( r_1(1-\text{ sn }^2(\omega _4\xi ,k))\right) ^\frac{3}{2}\hbox {d}\xi . \end{aligned}$$

(93)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\sqrt{r_1\left( 1-\text{ sn }^2(\omega _4\xi ,k)\right) }\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(94)

where $\varTheta (\xi )$ is given by (93) and $k^2=\frac{|r_1|}{\varPhi _2-r_2}$.

5.5 The case of $3b_1^\frac{3}{2}<2\sqrt{b_3b_7}, \ 0<h_1<h_2$ (see Fig. 8h)

For $h \in (h_1,0)$, the level curve defined by $H_3(\varPhi ,y)=h$, there exist two homoclinic orbit to the equilibrium point $E_2(\varPhi _2,0)$ of system (18). Here, the origin is $E_0(0,0)$ is a high order equilibrium point. In this case we have $F_5=(\varPhi -\varPhi _m)(\varPhi -\varPhi _2)^2\varPhi (\varPhi _L-\varPhi )$, where $\varPhi _m<\varPhi _1<\varPhi _2<\varPhi _3<0<\varPhi _L$. Now, (73) has the form

$$\begin{aligned} \omega _8\xi =\int _{\varPhi _m}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _2-\varPhi )\sqrt{(\varPhi -\varPhi _m)\varPhi (\varPhi _L-\varPhi )}} \end{aligned}$$

and

$$\begin{aligned} \omega _8\xi =\int _{\varPhi }^{0}\frac{\hbox {d}\varPhi }{(\varPhi -\varPhi _2)\sqrt{(\varPhi -\varPhi _m)\varPhi (\varPhi _L-\varPhi )}}. \end{aligned}$$

Using the above integrals, corresponding to the left homoclinic orbit we have the following parametric representation of Eq. (1):

$$\begin{aligned} \varPhi (\xi )=\left( \varPhi _m(1-\text{ sn }^2(\omega _8\xi ,k))\right) ^{\frac{1}{2}}, \end{aligned}$$

(95)

we obtain from Eq. (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi \nonumber \\&+\frac{\delta }{2} \sqrt{|\varPhi _m|}\int _{\varPhi _m}^{\varPhi }\left( 1-\text{ sn }^2(\omega _8\xi ,k)\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+ \frac{{\hat{h}}_2}{\sqrt{|\varPhi _m|}}\int _{\varPhi _m}^{\varPhi }\left( 1-\text{ sn }^2(\omega _8\xi ,k)\right) ^{-\frac{1}{2}}\hbox {d}\xi \nonumber \\&+\frac{a}{4}\int _{\varPhi _m}^{\varPhi }\left( \varPhi _m(1-\text{ sn }^2(\omega _8\xi ,k))\right) ^\frac{3}{2}\hbox {d}\xi . \end{aligned}$$

(96)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _m(1-\text{ sn }^2(\omega _8\xi ,k))\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(97)

where $\varTheta (\xi )$ is given by (96) and $k^2=\frac{|\varPhi _m|}{\varPhi _L-\varPhi _m}, \varPhi _m\ne \varPhi _2$.

and corresponding to the right homoclinic orbit, we have the following parametric representation of Eq. (1):

$$\begin{aligned} \varPhi (\xi )=\left( \varPhi _L(1-\frac{1}{\text{ dn }^2(\omega _8\xi ,k)})\right) ^\frac{1}{2}, \end{aligned}$$

(98)

we obtain from Eq. (10) that

$$\begin{aligned} \varTheta (\xi )= & {} \frac{1}{2}(2\nu -\mu )\xi \nonumber \\&+\frac{\delta }{2} \sqrt{\varPhi _L}\int _{\varPhi }^{0}\left( 1-\frac{1}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{1}{2}\hbox {d}\xi \nonumber \\&+ \frac{{\hat{h}}_2}{\sqrt{\varPhi _L}}\int _{\varPhi }^{\varPhi _M}\left( 1-\frac{1}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^{-\frac{1}{2}}\hbox {d}\xi \nonumber \\&+\frac{a}{4}\int _{\varPhi }^{\varPhi _M}\left( \varPhi _L(1-\frac{1}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{3}{2}\hbox {d}\xi . \end{aligned}$$

(99)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _L(1-\frac{1}{\text{ dn }^2(\omega _8\xi ,k)}\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(100)

where $\varTheta (\xi )$ is given by (99).

5.6 The case of $3b_1^\frac{3}{2}<2\sqrt{b_3b_7}, 0<h_1<h_2$ (see Fig. 8k).

For $h=h_2$, the level curves defined by $H(\varPhi ,y)=h$, in (72), we have a homoclinic orbit at a cusp point $E_{12}(\varPhi _1,0)$ of system (18) enclosing $E_{3}(\varPhi _3,0)$. In this case we have $F_5=(\varPhi -\varPhi _1)^3(\varPhi _M-\varPhi )\varPhi$, where $\varPhi _1<\varPhi _3<\varPhi _M<0$. Now, (73) can be written as

$$\begin{aligned} \omega _6\xi =\int _{\varPhi }^{\varPhi _M}\frac{\hbox {d}\varPhi }{(\varPhi -\varPhi _1)\sqrt{(\varPhi -\varPhi _1)(\varPhi _M-\varPhi )\varPhi }}. \end{aligned}$$

It gives rise to the parametric representations of Eq. (1) as follows:

$$\begin{aligned} \varPhi (\xi )=\sqrt{\varPhi _M}\text{ nd }(\omega _6\xi ,k), \end{aligned}$$

(101)

Thus we obtain from (10) that

$$\begin{aligned}&\varTheta (\xi )=\frac{1}{2}(2\nu -\mu )\xi \nonumber \\&\quad +\frac{\delta \sqrt{\varPhi _M}}{2k^{\prime }} \text{ tanh }^{-1}\left[ \frac{k^{\prime }\text{ sn }(\omega _6\xi ,k)- \text{ cn }(\omega _2\xi ,k)}{k^{\prime }\text{ sn }(\omega _6\xi ,k)+\text{ cn }(\omega _2\xi ,k)}\right] \nonumber \\&\quad + \frac{{\hat{h}}_2}{\sqrt{\varPhi _M}}\text{ sinh }^{-1}(\omega _2\xi ,k)+\frac{a}{4k^{\prime 3}}\varPhi _M^\frac{3}{2}\nonumber \\&\quad \left( (2-k^2)\text{ tanh }^{-1}\left( \frac{k^{\prime }\text{ sn }(\omega _6\xi ,k)- \text{ cn }(\omega _2\xi ,k)}{k^{\prime }\text{ sn }(\omega _6\xi ,k)+\text{ cn }(\omega _2\xi ,k)}\right) \right) \nonumber \\&\quad -\frac{a}{4k^{\prime 3}}\varPhi _M^\frac{3}{2} k^{\prime }k^2\text{ sn }(\omega _2\xi ,k)\text{ cn }(\omega _2\xi ,k)\text{ nd }^2(\omega _2\xi ,k).\nonumber \\ \end{aligned}$$

(102)

Therefore, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _M\text{ nd }^2(\omega _6\xi ,k)\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(103)

where $\varTheta (\xi )$ is given by (102).

5.7 The case of $b_3<0<b_5,\ \Delta =0, \ h_1=h_2$ (see Fig. 8l).

For $h=h_1, \$ the level curves defined by $H(\varPhi ,y)=h$, in (72), we have a homoclinic orbit to the multiple equilibrium point (cusp point) $(\varPhi _1,0)$ and $\varPhi _1=\varPhi _2$of system (18) enclosing $E_{3}(\varPhi _3,0)$. In this case we have $F_5=(\varPhi -\varPhi _1)^4\varPhi$, where $b_2=3(b_1)^\frac{3}{2}, \ b_3=-3(b_1)^\frac{3}{2}$ and $\varPhi _1<\varPhi _3<\varPhi _M<0$. Now, (73) can be written as

$$\begin{aligned} \omega _6\xi =\int _{0}^{\varPhi }\frac{\hbox {d}\varPhi }{(\varPhi _2-\varPhi )^2\sqrt{\varPhi }}. \end{aligned}$$

It gives rise to the parametric representations of Eq. (1) as follows:

$$\begin{aligned} \varPhi ({\hat{\xi }})=\sqrt{\varPhi _2}\text{ tanh }({\hat{\xi }}), \end{aligned}$$

(104)

and by using Eq. (10) we have

$$\begin{aligned} \varTheta (({\hat{\xi }}))= & {} \frac{1}{2}(2\nu -\mu ){\hat{\xi }}+\frac{\delta }{2} \sqrt{\varPhi _2}\text{ ln }\left( \text{ cosh }({\hat{\xi }})\right) \nonumber \\&+\frac{1}{2}{\hat{h}}_2\text{ ln }\left( \frac{1+\text{ sinh }({\hat{\xi }})}{1-\text{ sinh }({\hat{\xi }})}\right) +\frac{a}{4}\varPhi _2^\frac{3}{2}\frac{\text{ sinh }({\hat{\xi }})}{2\text{ cosh }^2({\hat{\xi }})}\nonumber \\&+\frac{1}{4}\text{ ln }\left( \frac{1+\text{ sinh } ({\hat{\xi }})}{1-\text{ sinh }({\hat{\xi }})}\right) . \end{aligned}$$

(105)

Hence, we have the following solution of Eq. (1):

$$\begin{aligned} A(x,t)= & {} i\left( \varPhi _2\text{ tanh }^2({\hat{\xi }})\right) ^\frac{1}{4}\nonumber \\&\times \text{ exp }\left[ -i\varTheta +i(\nu x-\lambda t)\right] , \end{aligned}$$

(106)

where $\varTheta ({\hat{\xi }})$ is given by (105).

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 $(b_2,b_3,b_4)$ for $b_1<0$ and $b_1>0$, 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 $\varPhi (\xi )=\sqrt{\varPhi }\text{ sin }\varTheta , \text{ and } \psi (\xi )=\sqrt{\varPhi }\text{ cos }\varTheta$, where $\varPhi (\xi )$ and $\psi (\xi )$ are given in Sect. 3.

Theorem 2

• (i) Depending on change of parameters regions of $(b_2,b_3, b_4)$ for $b_1<0$ and $b_1>0$, we have found 11 solutions corresponding to the periodic, homoclinic and heteroclinic orbits of system (18) with $p=3$. The septic derivative nonlinear Schrödinger equation (1) has 11 exact solutions given by (76), (79), (82), (85), (88), (91), (94), (97), (100), (103) and (106).

• (ii) System (3) has 11 exact explicit solutions $\varPhi (\xi )=\sqrt{\varPhi }\text{ sin }\varTheta , \text{ and } \psi (\xi )=\sqrt{\varPhi }\text{ cos }\varTheta$, where $\varPhi (\xi )$ and $\psi (\xi )$ are given in Sect. 4.