Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative

Abstract

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

1. Introduction

Fractional differential equations is a a branch of mathematics with many applications. It provides more flexibility in modeling physical phenomena [1,2,3,4,5]. In this paper, the fractional nonlinear Schrödinger differential equation is used model rogue waves. These are defined as high amplitude waves with height greater than double the significant wave height of the background sea. Rogue waves appear in the deep ocean, in the atmosphere, in optics, in plasma, and in other settings [6]. The study of these waves falls within an active area of research, namely the study of nonlinear partial deferential Equations (NLPDEs), e.g., [7,8,9,10,11,12,13,14,15,16,17,18]. Within this wider field, one finds studies addressing the physical properties of waves in oceans. These studies include the study of surface tide, internal tides, and deep-water waves [19,20,21,22]. Within the latter studies, rogue waves have become a topic of intense research. These waves were examined as solutions of the nonlinear Schrödinger equation in (1 + 1) and (2 + 1) dimensions [23]. The nonlinear rogue wave dynamics in the case of unidirectional wave propagation was studied in [24]. The phase-resolved numerical simulations based on a high-order spectral (HOS) method was used to study the occurrence and dynamics of rogue waves in three-dimensional deep water [25]. For water waves, the nonlinear Schrödinger Equation (NLS) and its modifications were used to model the behavior of the waves in deep water condition [26,27,28,29,30,31]. Exact solutions of the Schrödinger equation were constructed using the bifurcation theory of dynamical systems [32,33].

Deep-water irrational gravity waves propagating at the surface of an inviscid incompressible fluid are governed to the third order in amplitude by an equation first introduced by Zakharov [26]

$$\begin{array}{cc}\hfill i\frac{\partial B(k,t)}{\partial t}& =\int \int {\int }_{-\infty }^{\infty }T(k,k_1,k_2,k_3)B^{*}(k_1,t)B(k_2,t)B(k_3,t)\hfill \\ \hfill & \times \delta (k+k_1-k_2-k_3)e^{it[\omega \left(k\right)+\omega \left(k_1\right)-\omega \left(k_2\right)-\omega \left(k_3\right)]}d{k}_{1}d{k}_{2}d{k}_3,\hfill \end{array}$$

(1)

where ${}^{*}$ denotes the complex conjugate, $T(k,k_1,k_2,k_3)$ is Krasitskii’s kernel [34], $\omega \left(k\right)=\sqrt{g\left|k\right|}$ is the dispersion relation for gravitational waves, g is the acceleration due to the gravity, and $k_1,k_2,k_3$ are phase frequencies. The modified nonlinear Schrödinger equation was derived in detail in [28] and has the form

$$\begin{array}{cc}\hfill & i\left(a_{x_1}+\frac{2k_0}{{\omega }_0}{a}_t\right)-\frac{1}{4k_0}{a}_{x_1{x}_1}+\frac{1}{2k_0}{a}_{x_2{x}_2}-\frac{i}{8k_0^2}{a}_{x_1{x}_1{x}_1}+\frac{3i}{4k_0^2}{a}_{x_1{x}_2{x}_2}\hfill \\ \hfill & =k_0^3{\left|a\right|}^{2}a+\frac{i{k}_0^2}{2}{a}^2{a}_{x_1}^{*}-3i{k}_0^2{\left|a\right|}^2{a}_{x_1}+\frac{2k_0^2}{\omega \left(k_0\right)}a{\mathsf{\Phi }}_{x_1}(x_3=0),\hfill \end{array}$$

(2)

where $\mathsf{\Phi }(x_1,x_2,x_3,t)$ is the hydrodynamic potential, $a(x_1,x_2,t)$ is a complex amplitude and ${\omega }_0,k_0$ are the frequency and wave number of the carrier wave, respectively. Introducing new variable x in the form

$$x=\left(x_1-\frac{{\omega }_{0}t}{2k_0}\right)cos\theta +x_{2}sin\theta ,$$

(3)

where $\theta$ is the coefficient of the surface tension. Inserting the transformation (3) into Equation (2), and assuming $a(x_1,x_2,t)=P(x,t)$, we obtain

$$i{P}_t+{\alpha }_1{P}_{xx}+{\alpha }_2{\left|P\right|}^{2}P=i{r}_1{P}_{xxx}+i{r}_2{P}^2{P}_x^{*}-i{r}_3{\left|P\right|}^2{P}_x+r_{4}P,$$

(4)

where ${\alpha }_1,{\alpha }_2,r_1,r_2,r_3,r_4$ are constants introduced for notational convenience and given by

$$\begin{array}{cc}\hfill {\alpha }_1& =\frac{{\omega }_0}{8k_0^2}(2-3{cos}^2\theta ),{\alpha }_2=\frac{-{\omega }_0{k}_0^2}{2},r_1=\frac{{\omega }_{0}cos\theta }{16k_0^3}(7{cos}^2\theta -6),\hfill \\ \hfill r_2& =\frac{{\omega }_0{k}_0}{4}cos\theta ,r_3=\frac{3{\omega }_0{k}_0}{2}cos\theta ,r_4=\frac{k_0}{2}{\left(\right|P|}^2{{)}_x|}_{x_3=0}.\hfill \end{array}$$

Previously, Equation (4) appeared frequently in the literature. In [29], the authors used the $G^{\prime }/G$ method to obtain analytical solutions and studied the rogue wave’s amplitude to highlight their destructive power. In [30], the authors utilized both the $G^{\prime }/G^2$ method and extended Sinh-Gordon equation expansion to construct analytical solutions that are expressed using trigonometric and hyperbolic functions and solitary solutions.

In the current work, we study Equation (4) with time M-fractional fractional order $\alpha$ taking the form

$$i{D}_{M,t}^{\alpha ,\beta }P+{\alpha }_1{P}_{xx}+{\alpha }_2{\left|P\right|}^{2}P=i{r}_1{P}_{xxx}+i{r}_2{P}^2{P}_x^{*}-i{r}_3{\left|P\right|}^2{P}_x+r_{4}P,$$

(5)

where the operator $D_{M,t}^{\alpha ,\beta }$ is the time M-fractional operator of order $\alpha$ defined in Appendix A. Equation (5) is a generalization for Equation (4) since the latter can be obtained as a special case when $\alpha =1,\beta =0$. Equation (4) has many uses in ocean engineering [26,28,34].

To the extent of the authors’ knowledge, the study of the M-time fractional modified nonlinear Schrödinger equation in (5) has not been considered in the literature. This motivated the authors to study the dynamical behaviour of Equation (5) using bifurcation theory and to construct some wave solutions to this equation. We also studied the influence of the fractional order $\alpha$ on the obtained solutions and examined the dependence of the solutions on the initial conditions.

This paper is structured as follows: Section 2 contains the mathematical analysis covering the conversion of Equation (5) into a traveling wave system using some wave transformation. Section 3 is devoted to bifurcation and phase portrait analysis of the solutions. In Section 4, we discuss the obtained solution in some details. Finally, in Section 5, we provide 3D- graphical representations and illustrate the effect of the fractional order $\alpha$ on the solutions. Finally, Section 6 gives a summary of the obtained result.

2. Traveling Wave System

We look for wave solutions for Equation (5) in the form

$$P(x,t)=R\left(\rho \right)e^{i\theta },$$

(6)

where

$$\rho =\zeta \left(x-\frac{\nu }{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha }\right),\theta =-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha },$$

(7)

where $\zeta ,\nu ,w,k$ are free constants characterizing the soliton amplitude, soliton velocity, phase constant, frequency, and wave number, respectively, while $R\left(\rho \right)$ is a real valued function. Direct calculations give

$$\begin{array}{cc}\hfill D_{M,t}^{\alpha ,\beta }P& =e^{i\theta }\left(-\zeta \nu R^{\prime }+iwR\right),P_x=e^{i\theta }\left(\zeta R^{\prime }-ikR\right),\hfill \\ \hfill P_{xx}& =e^{i\theta }\left(-k^{2}R+{\zeta }^2{R}^{″}-2i\zeta k{R}^{\prime }\right),P_x^{*}=e^{-i\theta }\left(\zeta R^{\prime }+ikR\right),\hfill \\ \hfill P_{xxx}& =e^{i\theta }[{\zeta }^3{R}^{\prime \prime \prime }-3\zeta k^2{R}^{\prime }-i(3{\zeta }^{2}k{R}^{″}-k^{3}R)].\hfill \end{array}$$

(8)

Inserting the expressions (8) into Equation (5) and dividing by $e^{i\theta }$ then taking real and imaginary parts, we obtain:

From the real part

$${\zeta }^2\left({\alpha }_1-3k{r}_1\right)R^{″}+\left({\alpha }_2+k(r_2+r_3)\right)R^3+\left(k^3{r}_1-{\alpha }_1^2{k}^2-w-r_4\right)R=0.$$

(9)

From the imaginary part, after dividing by $\zeta$

$${\zeta }^2{r}_1{R}^{\prime \prime \prime }-\left(3k^2{r}_1-2{\alpha }_{1}k-\nu \right)R^{\prime }+\left(r_2-r_3\right)R^2{R}^{\prime }=0.$$

(10)

Integrating Equation (10) with respect to $\rho$, and setting the integration constant to zero, we obtain

$${\zeta }^2{r}_1{R}^{″}-\left(3k^2{r}_1-2{\alpha }_{1}k-\nu \right)R+\left(\frac{r_2-r_3}{3}\right)R^3=0.$$

(11)

The two Equations (9) and (11) are identical if

$$\frac{(r_2-r_3)}{3\left[k(r_2+r_3)+{\alpha }_2\right]}=\frac{r_1}{{\alpha }_1-3k{r}_1}=\frac{\left(3k^2{r}_1-2{\alpha }_{1}k-\nu \right)}{k^3{r}_1-{\alpha }_1{k}^2-w-r_4}$$

which is equivalent to

$$k=\frac{{\alpha }_1\left(r_2-r_3\right)-3{\alpha }_2{r}_1}{6r_1{r}_2},\nu =\frac{10k^3{r}_1^2+r_1\left(w+r_4-10{\alpha }_1{k}^2\right)+2{\alpha }_1^{2}k}{3k{r}_1-{\alpha }_1}.$$

(12)

Thus, with the values given in (12), both Equations (9) and (11) reduce to

$$R^{″}\left(\rho \right)+f_{1}R\left(\rho \right)-f_2{R}^3\left(\rho \right)=0,$$

(13)

where $f_1,f_2$ are two constants given by

$$\begin{array}{cc}\hfill f_1& =\frac{1}{108r_1^2{r}_2^2{\zeta }^2[{\alpha }_1(r_2+r_3)+3{\alpha }_2{r}_1]}[-{(r_2-r_3)}^2(5r_2+r_3){\alpha }_1^3+9{\alpha }_2{\alpha }_1^2{r}_1(r_2-r_3)(3r_2+r_3)\hfill \\ \hfill & -27r_1^2{\alpha }_2^2{\alpha }_1(r_2+r_3)-27r_1^2[r_1{\alpha }_2^3+8r_2^3(\omega +r_4)]],\hfill \\ \hfill f_2& =-\frac{r_2-r_3}{3{\zeta }^2{r}_1}.\hfill \end{array}$$

(14)

Equation (13) is equivalent to the following planar system, referred to in the literature as traveling wave system.

$$\left\{\begin{array}{c}{R}^{\prime }\left(\rho \right)=S\left(\rho \right)\hfill \\ S^{\prime }\left(\rho \right)=R\left(\rho \right)[f_2{R}^2\left(\rho \right)-f_1].\hfill \end{array}\right.$$

(15)

The system (15) is a conservative Hamiltonian system since $\mathrm{div}(R^{\prime },S^{\prime })=0$ and can be obtained form the Hamiltonian

$$H=\frac{1}{2}{S}^2\left(\rho \right)+\frac{f_1}{2}{R}^2\left(\rho \right)-\frac{f_2}{4}{R}^4\left(\rho \right),$$

(16)

through Hamilton’s canonical equations. Since $\frac{\partial H}{\partial \rho }=0$, the Hamiltonian (16) is a first integral (total energy), for the system (15), i.e.,

$$\frac{1}{2}{S}^2\left(\rho \right)+\frac{f_1}{2}{R}^2\left(\rho \right)-\frac{f_2}{4}{R}^4\left(\rho \right)=\delta ,$$

(17)

where $\delta$ is a constant, determined from the initial conditions. The constant $\delta$ plays a significant role in the the remainder of this study. Since H is constant, the Hamiltonian system (15) is a one dimensional integrable system. Thus, its solution can be obtained by quadrature utilizing the first integral (17), and the problem of finding the solution for Equation (5) is equivalent to the problem of finding the solution for the particle motion described by the Hamiltonian (16). Inserting the first expression in Equation (15) in the first integral (17) and separating the variables, we obtain

$$\frac{dR}{\sqrt{M_4\left(R\right)}}=\sqrt{2}d\rho ,$$

(18)

where

$$M_4\left(R\right)=f_2(R^4-\frac{2f_1}{f_2}{R}^2+\frac{4\delta }{f_2}).$$

(19)

3. Bifurcation and Phase Portrait Analysis

To integrate both sides of Equation (18), we need to specify the range of the three parameters $f_1,f_2$, and $\delta$. Usually, one of two methods can be employed to find this range. The first is the complete discriminate system for the polynomial $M_4\left(R\right)$ [35], which has been applied in previous research, e.g., [36]. The second is bifurcation analysis [37,38]. Bifurcation analysis has been applied successfully by various researchers, e.g., [39,40,41,42,43,44,45,46,47]. Its use gives all possible solutions and classifies these solutions by the type of orbit. It also clarifies the dependence of the solutions on the initial conditions. With these advantages in mind we chose to study the system (15) using bifurcation analysis.

To study the bifurcation and investigate the phase portrait for the system (15), we start by finding the fixed points. These are $(R_1,0)$, where $R_1$ is a zero of the function $g\left(R_1\right)=R_1[f_2{R}_1^2-f_1]$. If $f_1{f}_2<0$, the system (15) has $F_0=(0,0)$ as the only fixed point. If $f_1{f}_2>0$ it has three fixed points, $F_0=(0,0),F_{1,2}=\left(\pm \sqrt{\frac{f_1}{f_2}},0\right)$. The determinant of the Jacobian matrix for the Hamiltonian system (15) at the fixed point $(R_1,0)$ has the form

$$det\left(J(R_1,0)\right)=f_1-3f_2{R}_1^2.$$

(20)

Exploring the qualitative theory for the planar integrable system [37,38], the fixed point $(R_1,0)$ is either a center if $det\left(J(R_1,0)\right)>0$, a saddle if $det\left(J(R_1,0)\right)<0$, or a cusp if $det\left(J(R_1,0)\right)=0$ and its Poincaré index is zero. Evaluating the determinant of the Jacobi matrix (20) at the fixed points $F_i,i=0,1,2$, we obtain

$$det\left(J\left(F_0\right)\right)=f_1,det\left(J\left(F_{1,2}\right)\right)=-2f_1.$$

(21)

Furthermore, the values of the parameter $\delta$ at the fixed points $F_i,i=0,1,2$ are

$${\delta }_0=H\left(F_0\right)=0,{\delta }_1=H\left(F_{1,2}\right)=\frac{f_1^2}{4f_2}.$$

(22)

In analyzing the behavior of the solution, we consider the two following cases:

(a)

If $f_1{f}_2<0$, the Hamiltonian system (15) has a single fixed point which is either a saddle when $f_1<0$ and $f_2>0$, or a center if $f_1>0$ and $f_2<0$. The phase portraits for both cases are shown in Figure 1. For $f_1<0,f_2>0$, Figure 1a shows the phase plane containing two unbounded families of orbits in green for $\delta >0$ and in blue for $\delta <0$ with the two families being separated by unbounded orbits corresponding $\delta =0$ in red. The phase portrait for $f_1>0,f_2<0$, is shown in Figure 1b. There is only one bounded family of periodic orbits, in green, for $\delta >0$ about the center point $F_0$. Note that when $\delta \to 0$, the family of periodic orbits approaches the fixed point $F_0$.

Figure 1. Phase portrait of system (15) in the case that $f_1{f}_2<0$ for different values of $\delta$. (a) $f_1=-1,f_2=1$; (b) $f_1=1,f_2=-1$.

(b)

If $f_1{f}_2>0$, the Hamilton system (15) has three fixed points $F_i,i=0,1,2$. If $f_1<0$ and $f_2<0$, the fixed point $F_0$ is a saddle while $F_{1,2}$ are centers. In this case we have three kinds of level sets for H. For $\delta =0$ there are two homoclinic orbit connecting the saddle point $F_0$ to itself, shown in blue in Figure 2a. For $\delta \in ]{\delta }_1,0[$ we obtain two periodic orbits each containing one of the centers $F_1$ and $F_2$ and contained in an oval of the $\delta =0$ homoclinic orbits, as shown in red in Figure 2a. Finally, for $\delta >0$ we obtain a super periodic orbit surrounding the $\delta =0$ level set as shown in green. Notice, when $\delta \to 0$, the super-periodic family of orbits in green approaches the homoclinic orbit in blue, while the two periodic families of red orbits either approaches the homoclinic orbit in blue when $\delta \to 0$, or shrinks to the two fixed points when $\delta \to {\delta }_1$.

Figure 2. Phase portrait of system (15) in the case that $f_1{f}_2>0$ for different values of $\delta$. (a) $f_1=-1,f_2=-1$; (b) $f_1=1,f_2=1$.

If, on the other hand, $f_1>0$ and $f_2>0$, then $F_0$ is a center while $F_{1,2}$ are saddle points. In this case, we have five families of curves. For $\delta >{\delta }_1$, we have two families of unbounded orbits shown in red in Figure 2b. For $\delta ={\delta }_1$, we have a heteroclinic orbit connecting the two saddle points $F_1$ and $F_2$ as shown in blue. For $0<\delta <{\delta }_1$, we obtain two unbounded orbits and a periodic orbit surrounding the center $F_0$ and contained in the heteroclinic orbits of the $\delta ={\delta }_1$ case as shown in green. The periodic orbits for $0<\delta <{\delta }_1$ shrink to the center $F_0$ as $\delta \to 0$. For $\delta =0$, we obtain two unbounded orbits and the stationary point $F_0$, shown in black. Finally for $\delta <0$, we obtain two unbounded orbits as shown in purple. It is should be noted that as $\delta \to 0$, the family of periodic orbits in green shrinks to the fixed point $F_0$ and approach the heteroclinic orbit in blue when $\delta$ approaches ${\delta }_1$. Furthermore, when $\delta \to {\delta }_1$, the unbounded orbits in red will approach the heteroclinic orbit in blue.

The study of the transition between the phase orbits discussed above has two significant uses. The first is that it enables us to examine the dependence of the solutions on the initial conditions through the parameter $\delta$ which is determined by the initial conditions. The second is that it allows the construction the solution of the limiting phase orbit by a limiting process.

4. Solutions

In this section, we construct some wave solution for Equation (5) in the form Equation (6) by integrating both sides of the differential form (18) using the bifurcation restrictions on the parameters. We restrict ourselves to finding wave solutions corresponding to bounded orbits.

4.1. Periodic Solutions

The periodic wave solutions for Equation (5) are constructed by integrating the one differential forms along periodic phase orbits. Thus, we have the following cases:

(a)

If $(f_1,f_2,\delta )\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{-}\times {\mathbb{R}}^{+}$ then the system (15) has a periodic orbit for fixed values of $f_1,f_2,\delta$ intersecting $R-$ axis in two points and consequently, the polynomial $M_4\left(R\right)$ has two real zeros, $\pm z_1$, where $z_1>0$, and two purely imaginary roots $i{z}_2,-i{z}_2$. Thus, we can write $M_4\left(R\right)=-f_2(z_1^2-R^2)(z_2^2+R^2)$. Assuming $R\left(0\right)=0$, $0<R<z_1$ and integrating both sides of Equation (18), we obtain

$$R\left(\rho \right)=\frac{z_1{z}_2}{\sqrt{z_1^2+z_2^2}}\mathrm{sd}(\sqrt{-2f_2(z_1^2+z_2^2)}\rho ,\frac{z_1}{\sqrt{z_1^2+z_2^2}}).$$

(23)

Using the last expression for $R\left(\rho \right)$ and Equation (6), we obtain a novel periodic solution for Equation (5) in the form

$$P(x,t)=\frac{z_1{z}_2}{\sqrt{z_1^2+z_2^2}}\mathrm{sd}(\sqrt{-2f_2(z_1^2+z_2^2)}\zeta (x-\frac{\nu }{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha }),\frac{z_1}{\sqrt{z_1^2+z_2^2}})e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

(24)

(b)

For fixed values of $(f_1,f_2,\delta )\in {\mathbb{R}}^{-}\times {\mathbb{R}}^{-}\times {\mathbb{R}}^{+}$, the system (15) has a super periodic orbit crossing $R-$ axis in exactly two points. Therefore, we write $M_4\left(R\right)=-f_2(z_3^2-R^2)(z_4^2+R^2)$, where $z_3>0$. Assuming $R\left(0\right)=z_3$ and integrating both sides of Equation (18), we obtain

$$R\left(\rho \right)=z_3\mathrm{cn}(\sqrt{2f_2(z_3^2+z_4^2)}\rho ,\frac{z_3}{\sqrt{z_3^2+z_4^2}}).$$

(25)

Inserting the last expression for $R\left(\rho \right)$ in (6), we obtain

$$P(x,t)=z_3\mathrm{cn}(\sqrt{2f_2(z_3^2+z_4^2)}\rho ,\frac{z_3}{\sqrt{z_3^2+z_4^2}})e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

(26)

(c)

If $f_1<0,f_2<0$, and $\delta \in ]{\delta }_1,0[$, then system (15) has two families of periodic orbits shown in red in Figure 2a. This shows that the polynomial $M_4\left(R\right)$ has the form $M_4\left(R\right)=-f_2(z_5^2-R^2)(R^2-z_6^2)$, where $0<z_5<z_6$. There are two possible intervals giving real values for $R\left(\rho \right)$. They are $R\in ]-z_6,-z_5[\cup ]z_5,z_6[$. We compute the solution in each interval separately.

• If $R\in ]z_5,z_6[$ and $R\left(0\right)=z_6$, then integrating of both sides of Equation (18) yields

  $$R\left(\rho \right)=z_6\mathrm{dn}(z_6\sqrt{-2f_2}\rho ,\sqrt{1-\frac{z_5^2}{z_6^2}}).$$

  (27)
  Taking into account the last expression and Equation (6), we obtain a new solution for Equation (5) in the form

  $$P(x,t)=z_6\mathrm{dn}(z_6\sqrt{-2f_2}\rho ,\sqrt{1-\frac{z_5^2}{z_6^2}})e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

  (28)
• If $R\in ]-z_6,-z_5[$ and $R\left(0\right)=-z_6$, then using the same procedures as in the previous case we obtain a solution for Equation (5) in the form

  $$P(x,t)=-z_6\mathrm{dn}(z_6\sqrt{-2f_2}\rho ,\sqrt{1-\frac{z_5^2}{z_6^2}})e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

  (29)
  Note that the two solutions (28) and (29) correspond to the periodic families of orbits contained in the right and left ovals of the homoclinic orbit, respectively.

(d)

For values of $(f_1,f_2,\delta )\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\times ]0,{\delta }_1[$ the system (5) has a periodic orbit in addition to unbounded orbit as shown in green in Figure 2b. This shows that the polynomial $M_4\left(R\right)$ has four real zeros, that we denote $\pm z_7,\pm z_8$, where $z_7<z_8$. Thus, we can write $M_4\left(R\right)=f_2(z_7^2-R^2)(z_8^2-R^2)$. Assuming $R\left(0\right)=0$ and $R\in ]-z_7,z_7[$ which corresponds to the periodic orbit contained in the heteroclinic orbit and integrating both sides of Equation (18) we obtain

$$R\left(\rho \right)=z_7\mathrm{sn}(z_8\sqrt{2f_2}\rho ,\frac{z_7}{z_8}).$$

(30)

Inserting Equation (30) in (6), we obtain a new periodic wave solution for Equation (5) in the form

$$P(x,t)=z_7\mathrm{sn}(z_8\sqrt{2f_2}\zeta (x-\frac{\nu }{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha }),\frac{z_7}{z_8})e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

(31)

4.2. Soliton Solution

The existence of a homoclinic phase orbit for the system (15) shows the presence of solitary wave solutions for Equation (5). System (15) has a homoclinic orbit if $f_1<0,f_2<0$, and $\delta =0$ as shown in Figure 2a in blue. This orbits intersect $R-$ axis in three points and thus the polynomial $M_4\left(R\right)$ has two simple roots, that we denote $\pm z_9$, where $z_9>0$ and one double root at the origin. Thus, $M_4\left(R\right)=-f_2{R}^2(z_9^2-R^2)$. Assuming $R\left(0\right)=z_9$, and that $R\in ]0,z_9[$, and integrating both sides of Equation (18), we obtain

$$R\left(\rho \right)=z_9\sec \mathrm{h}\left(z_9\sqrt{-2f_2}\rho \right).$$

(32)

Using the transformation (6), we obtain a one-soliton solution for Equation (5) in the form

$$P(x,t)=z_9\sec \mathrm{h}\left(z_9\sqrt{-2f_2}\zeta (x-\frac{\nu }{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })\right)e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

(33)

This solution corresponds to the right oval of the homoclinic orbit. With a similar procedure, we can obtain the solitary solution corresponding to the left oval.

4.3. Kink Solution

If $f_1>0,f_2>0,\delta ={\delta }_1$, the system (15) has a heteroclinic orbit shown in blue in Figure 2b. The existence of such orbits indicates the presence of kink (or anti-kink) solutions. This orbit crosses $R-$ axis twice, and so the polynomial $M_4\left(R\right)$ has two double roots which are the $R-$ coordinate of the two saddle points $F_{1,2}$. Therefore, $M_4\left(R\right)=f_2{(R^2-\frac{f_1}{f_2})}^2$. Assuming $R\left(0\right)=0$, and considering the values $R\in ]-\sqrt{\frac{f_1}{f_2}},\sqrt{\frac{f_1}{f_2}}[$, we obtain on integrating both sides of Equation (18) that

$$R\left(\rho \right)=\sqrt{\frac{f_1}{f_2}}tanh\left(\sqrt{2f_1}\rho \right).$$

(34)

Utilizing the transformation (6) and the above expression, we obtain a kink solution for Equation (5) in the form

$$P(x,t)=\sqrt{\frac{f_1}{f_2}}tanh\left(\sqrt{2f_1}\zeta (x-\frac{\nu }{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })\right)e^{i(-kx+\frac{w}{\alpha }\mathsf{\Gamma }(\beta +1)t^{\alpha })}.$$

(35)

4.4. Degeneracy Property

The present subsection aims to study the degeneracy of the solutions using the transition between the phase orbits. Clearly, each orbit in the phase space corresponds to a certain initial condition which determines the constant $\delta$. Hence, the study of the degeneracy of the solutions through the parameter $\delta$ is equivalent to studying the dependence of the solutions on the initial conditions. We consider the the following cases:

(1)

The family of orbits shown in Figure 1b in green degenerates into the fixed point O when $\delta$ tends to zero. Consequently, $z_1$ tends to zero. Hence, $R\left(\rho \right)$ tends to zero which is $R-$ coordinate of the fixed point O. Therefore, the solution (24) goes to zero.

(2)

The family of super periodic orbits shown in Figure 2a in green is transformed into the homoclinic orbit in blue when $\delta$ approaches zero. If $\delta \to 0$, then $z_3\to z_9$ and $z_4\to 0$. Thus, the super-periodic solution degenerates into the solitary solution (33).

(3)

If $\delta \to 0$, the two families of periodic orbits shown in Figure 2a in red degenerate to the homoclinic orbits, with each family degenerating into the homoclinic orbit containing it. As a result when $\delta \to 0$, we have $z_5\to 0$, and $z_6\to z_9$. Hence, the solution (28) changes to the solution (33).

(4)

If $\delta \to {\delta }_1$, the two periodic families of orbits shown in Figure 2a in red degenerate into the two fixed points $F_{1,2}$. The periodic family in the right oval of the homoclinic orbit is converted into $F_1=(\frac{f_1}{f_2},0)$ and consequently, $z_{5,6}\to \sqrt{\frac{f_1}{f_2}}$. Hence, the solution (28) turns into $R-$ coordinate of the fixed point $F_1$.

(5)

If $\delta \to {\delta }_1$, the periodic family of orbits shown in Figure 2b in green goes to the heteroclinic orbit in blue and $z_{7,8}\to \sqrt{\frac{f_1}{f_2}}$. Hence, the solution (31) is degenerated into the solution (35).

It is worth mentioning that the study of the solution degeneracy through the transition of the phase orbits is significant because it shows the consistency of the obtained solution. It is also clarifies the dependence of the solutions on the initial conditions.

5. Graphical Representations

This section aims to give graphical representations of some obtained solutions and illustrate the effect of the fractional order $\alpha$ on these solutions. In all the following figures, we take $\beta =1,\nu =0.4,\omega =0.75,k=-0.5$. We investigate the effect of the fractional order $\alpha$ on the periodic, solitary, and kink solutions separately:

(a)

We take $f_1=1,f_2=-1,\delta =0.5$. For these values, the polynomial $M_4\left(R\right)$ (19) has two real roots $z_1=0.8555996772,z_2=1.652891650$. Based on the bifurcation analysis, Equation (5) has a solution (24) and hence $\left|P\right(x,t\left)\right|$ is given by

$$|P(x,t)|=0.7598356855|\mathrm{sd}(2.632148026x-1.052859210\frac{t^{\alpha }}{\alpha },0.4597008434)|.$$

(36)

Figure 3 illustrates graphically $\left|P\right(x,t\left)\right|$ of the solution (24) with different values of the fractional derivative order $\alpha$. For $\alpha =1$ corresponding to the classical case, $\left|P\right(x,t\left)\right|$ is periodic as shown in Figure 3a. As the values of $\alpha$ decreases from one, it loses its periodicity as shown in Figure 3b,c. Figure 3d gives the 2D representation of $\left|P\right(x,t\left)\right|$ when $x=1$ showing that the amplitude of the solution remains approximately unchanged while the width of the solution grows as the fractional order $\alpha$ decreases from one.

Figure 3. Graphic representation of real part of the solution (33) $f_1=1,f_2=-1,\delta =0.5,\beta =1$, $\nu =0.4$, $\omega =0.75,k=-0.5$ with different values of $\alpha$. (a) 3D representation with $\alpha =1$; (b) The 3D representation with $\alpha =0.7$; (c) 3D representation with $\alpha =0.3$; and (d) $2D$ with different values of $\alpha$.

(b)

For $f_1=-1,f_2=-1,\delta =0$, the polynomial $M_4\left(R\right)$ (19) has three roots: one is double at the origin and the other are $\pm z_9$, where $z_9=1.414213562$. According to the bifurcation analysis, Equation (5) has a solution in the form (33) and consequently, $\left|P\right(x,t\left)\right|$ is given by

$$\left|P(x,t)\right|=1.414213562|\sec \mathrm{h}(1.999999999x-\frac{0.7999999996}{\alpha }{t}^{\alpha })|.$$

(37)

Figure 4a–c show the 3D representations of $\left|P\right(x,t\left)\right|$ for the solution (33) for distinct values of the fractional order $\alpha$. Figure 4d shows the amplitude of $\left|P\right(x,t\left)\right|$ is approximately unaffected as $\alpha$ decreases from one.

Figure 4. Graphic representation of $\left|P\right(x,t\left)\right|$ for the solution (33) when $f_1=f_2=-1,\delta =0$, $\beta =1$, $\nu =0.4,\omega -0.75,k=-0.5$ with different values of $\alpha$. (a) The 3D representation with $\alpha =1$; (b) 3D representation with $\alpha =0.7$; (c) 3D representation with $\alpha =0.3$; and (d) $2D$ with different values of $\alpha$.

(c)

For $f_1=f_2=1,\delta =0.25$, Equation (5) has a solution in the form (35) and therefore, we have

$$\left|P(x,t)\right|=\left|\tan \mathrm{h}\right(\sqrt{2}(x-\frac{0.4}{\alpha }{t}^{\alpha }))|.$$

(38)

Figure 5a–c show $\left|P\right(x,t\left)\right|$ for the solution (35) for several values of the fractional order derivative. Figure 5d shows the amplitude of $\left|P\right(x,t\left)\right|$ is slowly increased with decreasing the order of the fractional order $\alpha$.

Figure 5. Graphic representation of $\left|P\right(x,t\left)\right|$ of the solution (35) $f_1=f_2=1,\delta =0.25,\beta =1$, $\nu =0.4$, $\omega =0.75,k=-0.5$ with different values of $\alpha$. (a) The 3D representation with $\alpha =1$; (b) 3D representation with $\alpha =0.7$; (c) 3D representation with $\alpha =0.3$; and (d) $2D$ with different values of $\alpha$.

6. Conclusions

The present study investigated the time M- fractional modified nonlinear Schrödinger equation that characterizes the propagation of rogue waves in deep waters. A wave transformation was applied to Equation (5) to transform it into a traveling wave system which was shown to be a one-dimensional Hamiltonian system. The qualitative theory for the planar integrable systems was utilized to perform the bifurcation analysis and inspect the phase portrait. Based on the bifurcation constraints on the system parameters, we have utilized the first integral for the traveling wave system to construct some new wave solutions for Equation (5) which were categorized into periodic, super-periodic, solitary, and kink wave solutions. We studied the degeneracy of the obtained solutions using the transition between the phase orbits. We investigated the dependence of the solutions on the initial conditions. These solutions were shown graphically for different values of the fractional order $\alpha$. Figure 3 gives the 2D representation of $\left|P\right(x,t\left)\right|$ given by (24) when $x=1$ and it shows that the amplitude of the solution remains approximately unchanged while the width of the solution grows as the fractional order $\alpha$ decreases from one. Figure 4 shows the amplitude of $\left|P\right(x,t\left)\right|$ given by (33) is approximately unaffected as $\alpha$ decreases from one. Figure 5 shows the amplitude of $\left|P\right(x,t\left)\right|$ given by (35) is slowly increased with decreasing order of the fractional order $\alpha$.