Bifurcation analysis and exact solutions for a class of generalized time-space fractional nonlinear Schrödinger equations

1.   Introduction

In recent years, due to the rapid development and wide applications in nonlinear science of fractional calculus theory, many problems of mathematical physics and engineering have been successfully modeled by fractional differential equations (FDEs), such as materials ^[1], plasma physics ^[2], chaotic oscillations ^[3], chemistry and biochemistry ^[4], hydrology ^[5] and so on ^[6,7,8,9]. To better understand the physical meanings of these models, people have constructed many efficient methods for finding the exact explicit solutions of these FDEs, including the Bäcklund transformation ^[10], Darboux transformation ^[11] and Hirota bilinear method ^[12], which can be used to find N-soliton solutions. Furthermore, the general algebraic method ^[13], projective Riccati equations method ^[14], Jacobi elliptic function expansion method ^[15], $G'/G$-expansion method ^[16], sine-Gordon method ^[17], new Kudryashov method ^[18], fractional sub-equation method ^[19], fractional Hirota bilinear method ^[20], Riemann-Hilbert method ^[21], complex method ^[22], Bernoulli $G'/G$-expansion method ^[23], etc. ^{[24,25,26,27,28,29]} can be used to find doubly periodic solutions, solitary wave solutions and trigonometric solutions of these models.

As we all know, the nonlinear Schrödinger equation is highly focused in nonlinear science, and it describes many phenomena, including plasma ^[30], electromagnetic wave propagation ^[31], quantum mechanics ^[32], optics of nonlinear media ^[33], underwater acoustics ^[34], etc. ^[35]. Hence, solving this equation is highly important for researchers.

In this article, let us consider the following generalized time-space fractional nonlinear Schrödinger equation (GTSFNLS) mentioned in ^{[36,37,38,39,40,41,42,43,44,45,46]}:

$D_x^{2\beta }u = D_x^\beta (D_x^\beta u), u = u(x, t), i = \sqrt { - 1}$, and $a, \gamma, v$ are real parameters. When $a = 1, s = 1,$ $v = \gamma = 2\lambda$, Eq (1) turns into an unstable nonlinear Schrödinger equation describing the bilayer baroclinic instability of some long waves ^[36]. When $a = 1, s = 1, v = 0$, Eq (1) occurs in various fields of physics, including optical fiber communication, quantum mechanics, fluid mechanics, superconductivity, plasma physics, etc. ^{[37,38,39,40,41]}. The authors studied its approximate solution by Adomian expansion in ^[37]. When $a = 1, \gamma = \frac{1}{2}\mu,$ $s = 1, v = 0$, some exact solutions were obtained by direct method in ^[38]. When $\alpha = \beta = 1$, $a = 1, s = 1, v = 0$, it translates into the classical nonlinear Schrödinger equation ^[39,40,41]. In addition, Eq (1) has many special cases, and related studies can be found in ^{[42,43,44,45,46,47,48]}. The main purpose of this paper is to find the new exact solution of Eq (1) under the famous Caputo fractional derivative definition by using the general mapping deformation method and to study the structure of these solutions by using the theory of planar dynamical systems. Next, let us review some definitions about classical fractional calculus ^[49,50,51].

Definition 1. For a function $f(t):[0, \infty) \to R$, we define the Riemann-Liouville fractional integral operator of order $\alpha > 0$ as ^[49,50,51]

It admits the following properties:

Definition 2. For $\alpha > 0,$ the Caputo fractional derivative operator of order $\alpha$ is defined as ^[49,50,51]

Moreover, we have the following properties:

This article is organized as follows: The general mapping deformation method is described in Section 2. In Section 3, some new exact solutions and bifurcation structures of the GTSFNLS are found by utilizing the proposed method and the planar dynamic system theory method. Finally, the conclusion is presented in Section 4.

2.   The general mapping deformation method (GMDM)

Consider the following partial differential equation:

We assume Eq (2) has solutions as follows:

$n$ is a balance number, and the coefficients ${A_i}$, ${A_{ - i}}$ and variable function $\eta = \eta (x, t)$ are determined later. $F(\eta)$ satisfies the following auxiliary equation:

Substituting Eqs (3) and (4) into Eq (2), collecting the coefficients of ${F^j}(\eta)\sqrt {\sum\limits_{i = 0}^4 {{a_i}} {F^i}(\eta)}$ and ${F^i}(\eta)(i, j = 0, \pm 1, \pm 2, \cdots)$ to zero yields algebraic equations (AEs) for ${a_0}, {a_1}, {a_2}, {a_3}, {a_4},$ ${A_i}, {A_{ - i}}$ and $\eta$. Utilizing mathematical software to solve the AEs and substituting each $F(\eta)$ into Eq (3), we obtain the solutions of Eq (2). For finding some new general solutions of Eq (4), we assume

${b_i}(i = 0, \pm 1, \cdots, \pm 4)$ are undetermined coefficients, and the functions $e = e(\eta), f = f(\eta),$ $g = g(\eta), h = h(\eta)$ are constructed as below ^[15,52,53]:

$p, q, r, l$ are undetermined coefficients, and $e, f, g, h$ satisfy the nexus (4) and (5a–5d) mentioned in ^[15,52,53].

Remark 1. Our method proposed here can be used to extend many other traditional methods such as the generalized Jacobi elliptic functions expansion method ^[15,53], the extended projective Riccati equations method ^[14], many other algebra expansion methods ^{[18,19,38,46]}, etc.

3.   Bifurcation and exact solutions for the GTSFNLS

3.1.   Exact solutions

If we let $D_t^\alpha u = \frac{{{\partial ^\alpha }u}}{{\partial {t^\alpha }}}, D_x^{2\beta }u = \frac{{{\partial ^{2\beta }}u}}{{\partial {x^{2\beta }}}}$, Eq (1) can be rewritten as follows:

Let us give a functional transformation ^[46,54,55]

${k_1}, {k_2}, {c_1}, {c_2}$ are parameters to be determined later. Substituting Eqs (7) and (8) into Eq (6), separating the real part and the imaginary part, we obtain

Here, ${\varphi _{\eta \eta }}(\eta) = \frac{{{d^2}\varphi (\eta)}}{{d{\eta ^2}}}, {\varphi _\eta }(\xi) = \frac{{d\varphi (\eta)}}{{d\eta }}$.

Using the transformation $\psi = \psi (\eta) = {\varphi ^s}(\eta) = {\varphi ^s}$ for (9.1) yields

Clearly, the balance number $n = 1$, and we assume solutions of Eq (10) have the form

where $F{'^2} = \sum\limits_{i = 0}^4 {{a_i}{F^i}}, F = F(\eta)$ and $F' = \frac{{dF(\eta)}}{{d\eta }}$.

Substituting Eq (4) and (11) into Eq (10) and by utilizing the GMDM with the aid of mathematical software, we get the following solutions:

Family 1. $\varphi (\eta) = C.$

We find the trivial solution of Eq (6)

Remark 2. If we select ${c_1} = c_0^{2s}a + v, {k_1} = c_0^s, \gamma = 2a, \alpha = \beta = 1$ in ${u_0}$, we have the solution ${u_{01}} = {c_0}{e^{i({k_1}x + {c_1}t)}},$ which can be obtained by many authors by using approximate methods such as VIM or HAM ^[37,39].

Family 2. ${c_2} + 2a{k_1}{k_2} = 0.$

Case 1

Case 2

We obtain two types of solutions for Eq (6):

${F_i}$ is an arbitrary solution of the auxiliary equation ${F_i}{'^2} = {a_0} + {a_2}{F_i}^2 + {a_4}{F_i}^4$ in ${u_1}, {u_2}$, and the coefficients ${a_0}, {a_2}, {a_4}$ are arbitrary constants. Many types of ${F_i}$ have been found in a large number of papers, such as ^[53,56,57]. Let us choose ${a_0} = 1 - {m^2}, {a_2} = 2{m^2} - 1,$ ${a_4} = - {m^2}, {F_1} = cn\xi$. Thus,

and the solution ${u_{1.1}}$ is translated into a bell-soliton solution when $\alpha = \beta = 1, m = 1$.

If we let ${a_0} = 1, {a_2} = - {m^2} - 1, {a_4} = {m^2},$${F_0} = sn\xi$, then

Case 3

Case 4

Case 5

We find the following solutions of Eq (6), respectively:

If we let $m = 1$ or $m = 0$, solution ${u_5}$ is degenerated to the following form:

Case 6

Case 7

Case 8

With the same process, we obtain

If selecting ${k_1} = 0, v = 0$ in ${u_6}$, we get

Remark 3. If we select ${k_1} = 0, v = 0$, $\alpha = \beta = 1, a = 1, \gamma = 1, {k_2} = 1, {c_1} = \omega, s = \frac{{p - 1}}{2}$, ${u_6}$ turns into the following solution mentioned in ^[58].

We simulate some structures of the periodic and solitary solutions for Eq (1) below. Some optical waves of the GTSFNLS are propagated by a periodic wave pattern in Figures 1 and 2 or a bright-soliton wave pattern in Figure 3 and blow-up wave pattern in Figure 4 with the fractional order $\alpha = 0.4, \beta = 0.8$. The density plots of ${Re} {u_4}, {Im} {u_5}, \left| {{u_6}} \right|$ and $\left| {{u_{8.1}}} \right|$ are shown in Figure 5.

3.2.   Bifurcation analysis

Below, let us discuss the plane phase portrait structure of Eq (9) by using the plane dynamic system theory ^[59,60]. Without loss of generality, select ${k_1} = {k_2} = 1,$ $a = 0.5, {c_2} = - 1$. Thus,

Let $\frac{{d\varphi }}{{d\eta }} = y$, and clearly, Eq (12) is equivalent to the following regular system:

We can get the following Hamiltonian of Eq (13):

Indeed, system (13) has three equilibrium points $P(\varphi, y) = P(\varphi, \varphi ')$ on the $\varphi$-axis:

The coefficient matrix of system (13) is defined as $M(\varphi, y)$, and $J({P_i}) = \det M({\varphi _i}, {y_i}), i = 0, 1, 2$ is the determinant of $M(\varphi, y)$ about ${P_i}$.

By the dynamical bifurcation theory of planar systems, as we all know, the equilibrium point ${P_i}$ of system (13) is a center if $J({P_i}) > 0$, it is a saddle if $J({P_i}) < 0$, and it is a cusp if $J({P_i}) = 0$. Let us analyze the bifurcation structures of system (13) by using mathematical software and the above facts.

Case 1. ${c_1} - v = 1, \gamma = \frac{1}{2}, s \in {Z^ + }$.

Notice that $J({P_0}) = - 3 < 0$, $J({P_1}) = J({P_2}) = 6s > 0$. Hence, ${P_1}, {P_2}$ are center points, and the origin ${P_0}$ is a saddle point. Let us discuss three situations of Eq (14) with $s = 1$ and

(i) When $h < 0, h \in (- 4.5, 0)$, we can find two clusters of doubly periodic solutions for the periodic orbits of Eq (13) defined by the following integral equation (see Figure 6(a)):

Integrate this equation along the periodic orbits thus:

We get the following smooth doubly periodic solutions:

where ${\varphi _1} = \sqrt {3 - \sqrt {9 + 2h} }, {\varphi _2} = \sqrt {3 + \sqrt {9 + 2h} }$.

(ii) When $h = 0$, there exist two clusters of bell-soliton wave solutions for the symmetric homoclinic orbits of system (13) with the following form:

(iii) When $h > 0$, we obtain the Jacobi cosine wave solutions for the periodic orbit of system (13) with the following form:

where $\varphi _1^2 = \sqrt {9 + 2h} - 3, \varphi _2^2 = \sqrt {9 + 2h} + 3$.

Case 2. ${c_1} - v = - 1, \gamma = - \frac{1}{2}, s \in {Z^ + }$.

Clearly, $J({P_0}) = 1 > 0$, $J({P_1}) = J({P_2}) = - 2s < 0$. Hence, ${P_1}, {P_2}$ are saddle points, and ${P_0}$ is a center point (see Figure 6(b)). Let $s = 1, {({\varphi _\eta })^2} = \frac{1}{2}{\varphi ^4} - {\varphi ^2} + h$, and we can discuss the solutions of Eq (12) with the same process. Due to space limitations, we only give the solutions. When $h \in (0, 0.5)$, we can obtain two clusters of Jacobi sine wave solutions for the periodic orbits. When $h = 0.5$, there exist two kink and antikink solutions corresponding to the solutions of two symmetric heteroclinic orbits.

When $h < 0$, there exist two blow-up wave solutions for the corresponding orbits.

where ${\varphi _1} = \sqrt {1 - \sqrt {1 - 2h} }, {\varphi _2} = \sqrt {1 + \sqrt {1 - 2h} }$.

where $\varphi _1^2 = \sqrt {1 - 2h} - 1, \varphi _2^2 = \sqrt {1 - 2h} + 1$.

Some planar phase graphs of system (13) with the parameter $s \ne 1$ are shown in Figure 7. From Figure 7(a), ($\gamma = 1.25, {c_1} - v = - 4, s = 1.5$), and we can find that there exist a periodic solution and a bell-soliton solution, while there are not these two types of solutions in Figure 7(b) ($\gamma = - 2.5, {c_1} - v = 0.5, s = 4$).

Remark 4. It is notable that with the increase of $h$, the periodic wave solution is degenerated into a solitary wave solution and then into another periodic solution in Figure 6(a). The periodic wave solution is transformed into a kink or antikink wave solution and then into an unbounded solution in Figure 6(b).

4.   Conclusions

In brief, many types of new exact solutions for the GTSFNLS have been found after utilizing the GMDM. Some dynamic behaviors of these solutions are discussed using bifurcation theory. We simulate the 3D plots, 2D plots, density plots and phase graphs of the partial solutions in Figures 1–7, which show that these doubly periodic wave solutions, solitary wave solutions and single periodic solutions can be mutually transformed along with the concomitant energy constant and its corresponding orbits. These efficient and significant two methods can be used for many other nonlinear models, such as the Korteweg–de Vries (KdV) equation, Ginzburg-Landau equation, Burgers-BBM (Benjamin-Bona-Mahony) equation, etc.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the practical innovation training program projects for the university students of Jiangsu Province (Grant No. 202311276081Y; 202211276054Y)).

Conflicts of Interest

The author declare that he has no competing interests in this paper.