Bifurcation analysis and investigations of optical soliton solutions to fractional generalized third-order nonlinear Schrödinger equation

1 Introduction

Fractional partial differential equations (FPDEs) are key to representing systems that involve non-local interactions [1], irregular diffusion patterns [2], and long-range correlations [3]. These equations are extensively applied in data analysis, processing of images and signals and advance material. FPDEs are increasingly attracting attention due to their ability to offer more precise system modeling. The solution of partial differential equations (PDEs) and FPDEs can be extremely difficult due to their basic non-local and non-linear characteristics. Although frequently utilized, methods are like the finite difference method [4], finite element method [5], and others can be resource-intensive for complex FPDEs.

They have been applied in physics to explain heat flow in materials with exceptional properties, wave propagation in viscoelastic materials, and diffusion processes in fractal media. Fractional nonlinear evolution equations are used in chemistry to describe reactions influenced by past events and to analyze reaction-diffusion phenomena, and in biology to comprehend processes such as population growth, growth of tumors, and the spread of diseases that are infectious. Additionally, in both finance and economics, fractional nonlinear evolution equations have been used. They have been applied to risk management plans, pricing of option models, and the estimation and evaluation of financial time series data. A more realistic representation of the volatility clustering and long-term memory effects seen in financial markets is made possible by the use of fractional derivatives [6]. Various methods have been utilized to find the analytical solutions for fractional non-linear equations such as Kudryashov method [7], sine-Gordon expansion technique [8], F-expansion method [9] and auxiliary equation method [10].

In this article, two approaches new mapping method and bifurcation theory are used to solve the fractional generalized third-order nonlinear Schrödinger model with the Caputo fractional derivative. The nonlinear Schrödinger equation is a kind of PDE, which is applicable in different branches of physics, including hydrodynamics, superconductivity, optics, optical fibers, quantum mechanics and magneto-static rotating waves [11]. Soliton theory has been explored and used in science, engineering, and many analytical techniques thoroughly analyzing these equations. In nonlinear optical systems, the nonlinear Schrödinger equation is frequently utilized to describe soliton behavior in optical fibers, ultrashort pulse propagation, and other nonlinear wave phenomena. The development of numerous techniques, including bilinear, analytical, and numerical approaches, has resulted from advancements in the solution of complicated problems, and each approach has produced significant outcomes. The well-known mapping method is useful to derive the exact solutions efficiently, making them in studying solitons, wave dynamics and integrable functions. The new mapping method [12], is utilized to determine the exact solutions and the bifurcation, phase portrait and traveling wave solution of generalized third-order nonlinear Schrödinger model with Caputo fractional derivative of the model under consideration. While the proposed model is space-time-fractional, this work extends it by replacing the space derivative with Caputo fractional derivative. A mathematical equation that explains the movement of wave packets in nonlinear media is the generalized third-order nonlinear Schrödinger model. It is a modification of the traditional nonlinear Schrödinger equation, which explains the wave propagation of waves in linear media. The fractional derivatives presented in (4.1) is described by the Caputo derivative operator [12] for analytical solutions of proposed model.

The structure of the paper is as follows. In Section 2 the preliminaries is defined. In Section 3 description of the new mapping method is explained. In Section 4 mathematical analysis of the model and in Section 5 graphical explanations is discussed. In Section 6, Section 7 and Section 8 bifurcation theory, chaotic behaviour and sensitivity analysis is discussed respectively. The novelty and comparison of model is described in Section 9, and physical interpretation of the model is explained in 10. The conclusion is discussed in Section 11.

2 Preliminaries

This section provides the overview of Caputo fractional derivative (FD) and some of its basic properties. For a function u(x, t), Caputo FD of order α is described as:

where d a α C u ( x , t ) , shows the Caputo FD of order α. The function u(x, t), is suitably smooth. The following characteristics of this operator are applied in order to convert FPDEs into NODEs.

1. d y α x p = Γ ( 1 + p ) Γ ( 1 + p − α ) x p − α .

2. d y α a [ b ( z ) ] = a b ′ ( b ( z ) ) d y α b ( z ) = d h α a ( b ( z ) ) [ b ′ ( y ) ] α , 0 < α ≤ 1 ,

3 Description of the new mapping method

Consider the following nonlinear PDEs:

where R is a polynomial in v with its partial derivatives, including nonlinear terms and the highest order derivatives, and v = v(x, t) is an unknown function. The following is an explanation of the main steps of the new mapping method [13].

Step 1:

Traveling wave transformation.

where μ is constant, Eq. (3.1) reduces to the nonlinear ordinary differential equation (ODE) given below:

where G symbolizes the polynomial v(ξ) and its total derivatives with respect to ξ, where  ′ = ( d d ξ ) .

Step 2:

Assume the subsequent structure is the solution to Eq. (3.3):

where H(ξ) satisfies the conditions of the first-order nonlinear ODE:

where α _j  (j = 0, 1, …2N), f, g, c and d all are constant that must be determined to ensure α _2N ≠ 0.

Step 3:

The balancing parameter N of Eq. (3.4) is obtained by comparing the highest-order derivatives with the highest nonlinear terms in Eq. (3.3).

Step 4:

By putting Eq. (3.4) and Eq. (3.5) into Eq. (3.3), and all the coefficients of (H ^j ) are collected.

Step 5:

By setting the coefficients to zero, a set of algebraic equations is derived, that can be solved using mathematica to calculate the values of α _j (j = 0,1,2…2N) and f, g, c and d.

Step 6:

Novel mapping method [14] has different types of families of solutions which have been described in Appendix A.

4 Mathematical analysis

In the present section, the suggested method is used to solve the fractional generalized third-order nonlinear Schrödinger equation by applying Caputo fractional derivative which is given as below:

where 0 < γ, δ ≤ 1, Q = Q(x, t) is denotes the complex function and β ₁, β ₂ and β ₃ are arbitrary constants.

The following transformation of variables is used:

where the phase component of soliton is represented by θ(x, t), while its wave number and velocity are denoted as B ₁ and A ₁ respectively. After substituting Eq. (4.2) into Eq. (4.1), the expression takes the form:

For the real part.

and for imaginary part

Eq. (4.4) is obtained by integrating it once with respect to ξ.

Therefore, Eq. (4.5) and Eq. (4.3) are equivalent under the given constraint conditions

Applying the homogeneous balancing principle in equation Eq. (4.4) gives always to the balance number N = 1. Thus Eq. (4.5) admits the following formal solutions:

where Q(ξ) satisfies the auxiliary equation:

Substituting Eq. (4.8) along with Eq. (4.7) into Eq. (4.5), one arrives

Type-1: By using f = 16 g 2 27 c and d = 3 c 2 16 g into the algebraic equations Eq. (4.9), after solving these equations using mathematica, the following results are obtained:

By putting Eq. (4.10), along with Eq. (11.1), into Eq. (4.7). The resulting analytical solutions are:

By putting Eq. (4.10), together with Eq. (11.2), into Eq. (4.7).The resulting analytical solutions are:

By putting Eq. (4.10), along with Eq. (11.3), into Eq. (4.7). The resulting analytical solutions are:

Type-2: By using f = 0 and d = 3 c 2 16 g into the algebraic equations Eq. (4.8), after solving these equations using mathematica, the following results are obtained:

By putting Eq. (4.14) with the help of Eq. (11.4) into Eq. (4.7). The resulting analytical solutions are:

Type-3: By using f = 0 into the algebraic equations Eq. (4.9), after solving these equations using mathematica, the following results are obtained:

To enhance clarity, we explicitly state the analytical framework used in solving the fractional generalized third-order nonlinear Schrödinger equation. A traveling wave transformation is applied to reduce the space-time fractional PDE into a nonlinear ODE, assuming a separable solution where the wave profile and phase are treated independently. The Caputo fractional derivative is used due to its physical relevance in initial value problems. The new mapping method is then applied, where a polynomial ansatz in terms of an auxiliary function is introduced. The structure of H(ξ) is governed by a nonlinear first-order ODE involving arbitrary constants, which are determined by the homogeneous balancing principle. These constants are computed by substituting the ansatz into the reduced ODE and equating coefficients of like powers of H(ξ), resulting in a solvable algebraic system. This rigorous method ensures the exactness and physical relevance of the soliton solutions obtained.

5 Graphical explanation

This section explores the visual representations of the fractional generalized third-order nonlinear Schrödinger equation with variable refraction. The 3D, 2D and contour diagrams of selected solutions are presented to illustrate the physical behaviour of derived results.

Figure 1 shows the bell shape soliton solution of Eq. (4.10), by taking the parametric values A ₁ = 0.5,  B ₁ = 0.5 γ = 1,  δ = 1,  β ₂ = 0.1,  β ₃ = 2,  c = 1,  g = −2,  ϵ = −1,  and a ₁ = −1. Nonlinear optics makes extensive use of these solitons. A bell-shaped soliton is an isolated wave that moves through a medium with a single, smooth hump. In terms of physics, this form is a steady concentration of energy or disturbance that, in contrast to regular waves, doesn’t spread over time. Its stability results from the balance between two opposing effects: nonlinearity, which attempts to compress the wave, and dispersion, which tends to make it spread. A stable, self-reinforcing wave is created when these two forces perfectly balance one another out. The bell-shaped soliton is observed in systems such as quantum fluids, optical fibers, and shallow water waves. Its energy is most powerful in the center and tapers off smoothly toward the edges.

Figure 1:

The solitary wave solution H ₁(x, t) generated by the Eq. (4.10): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

Figure 2 shows the w-shape soliton solution of Eq. (4.11) with the parametric values A ₁ = 0.5,  B ₁ = 0.5 γ = 1,  δ = 1,  β ₂ = 1,  β ₃ = 2,  c = 1,  g = −2,  ϵ = −1,  and a ₁ = −1. These solutions are useful to represents complex wave patterns. A W-shaped soliton is a wave that occurs only in one place and has two peaks and a dip in the middle. It appears to resemble the letter ”W.” It depicts a more complicated distribution of energy because waves interact with each other or modify shape. It commonly shows up in systems having nonlinear effects of a higher order, like plasmas or optical fibers. The drop in the center illustrates momentary energy depletion, but the wave remains stable overall. This shape shows complex dynamics not apparent in simple bell-shaped solitons.

Figure 2:

The solitary wave solution H ₂(x, t) generated by the Eq. (4.11): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

Figure 3 illustrates anti bell shape soliton solution of Eq. (4.12) with the parametric values A ₁ = −0.5,  B ₁ = −0.5 γ = 1,  δ = 1,  β ₂ = 0.1,  β ₃ = 2,  c = −3,  g = 2,  ϵ = −1,  and a ₁ = −1. An anti-bell-shaped soliton, commonly referred to as a dark soliton, is a local depression or dip in the steady wave background’s amplitude that appears as an inverted bell curve. Physically, it is a stable ”hole” or drop in energy or particle density that travels through the medium without changing form, as opposed to a high-intensity pulse (such as in a bell-shaped or brilliant soliton).

Figure 3:

The solitary wave solution H ₃(x, t) generated by the Eq. (4.12): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

The balance between nonlinearity, which attempts to sustain the localized drop by modifying the wave’s phase and amplitude, and dispersion, which tends to spread out the wave, is what causes this dip. The anti-bell form demonstrates that energy may be preserved in both stable ”voids” inside a continuous wave and high amplitude bursts.

Figure 4 shows the kink wave soliton solution of Eq. (4.13) with the parametric values A ₁ = 1.5,  B ₁ = 1.5 γ = 2,  δ = 1,  β ₂ = 0.1,  β ₃ = 2,  c = 1,  g = 2,  ϵ = −1,  and a ₁ = −1. The kink soliton solution are powerful tool for understanding complex waves behaviour. A kink soliton is a kind of solitary wave that connects two distinct consistent states of a system and is defined by a smooth, step-like shift in the wave’s value of terms of physics, it denotes a boundary or transition (often referred to as a domain wall) between two separate but stable configurations or phases of a material. Kink solitons are a physical representation of moving interfaces or fronts that sustain the structure of the transition between various phases or configurations while propagating steadily.

Figure 4:

The solitary wave solution H ₅(x, t) generated by the Eq. (4.13): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

Figure 5 represents the periodic wave soliton solution of Eq. (4.16) by using A ₁ = 1,  B ₁ = 1 γ = 2,  δ = 1,  β ₂ = 4,  β ₃ = 5,  c = 11,  g = 0.9,  ϵ = 1,  a ₁ = 0.5, and d = −9. A sequence of steady, repeating wave pulses that go across a material without changing form is physically represented by a periodic wave soliton solution. Periodic solitons are used in plasma physics to investigate electric field structures or ion-acoustic waves that follow the same pattern over and over across an ionized gas. Additionally, periodic wave solitons point out the quantum characteristics of compressed atomic clouds under certain conditions by appearing as matter-wave lattices in Bose–Einstein condensates. As a result, a periodic wave soliton is more than simply a mathematical curiosity; it is a physically observable structure that is essential to many areas of engineering and science, particularly in systems where steady, repeatable energy or information transfer is required.

Figure 5:

The solitary wave solution H ₇(x, t) generated by the Eq. (4.16): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

Figure 6 indicates the dark wave soliton solution of Eq. (4.17) with the parametric values A ₁ = 4,  B ₁ = 2 γ = 2,  δ = 1,  β ₂ = 1,  β ₃ = 2,  c = 1,  g = 0.9,  ϵ = 1,  a ₁ = 0.5, and d = 0.1. Localized wave structures known as dark solitons show up as dips in intensity or density over a continuous backdrop. They appear in systems controlled by nonlinear wave equations, such as the Nonlinear Schrödinger Equation (NLSE). A black soliton is distinguished from other kinds of solitons, such bright solitons, by its phase leap throughout its minimum and amplitude decrease (as opposed to a localized rise).

Figure 6:

The solitary wave solution H ₆(x, t) generated by the Eq. (4.17): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

Figure 7 shows the bell shape soliton solution of Eq. (4.18) with the parametric values A ₁ = 1,  B ₁ = 1 γ = 2,  δ = 1,  β ₂ = 9,  β ₃ = 6,  c = −0.03,  g = 1,  ϵ = −1,  a ₁ = 0.14, and d = −3. A confined wave passing through a nonlinear medium and keeping its form is represented as a bell-shaped soliton. A balance between dispersion of variance which tries to spread the wave, and nonlinearity, which tends to steepen or localize it, produces this kind of solution.

Figure 7:

The solitary wave solution H ₁₃(x, t) generated by the Eq. (4.18): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

Figure 8 represents the periodic wave soliton solution of Eq. (4.19) with the parametric values A ₁ = 4,  B ₁ = 1 γ = 2,  δ = 1,  β ₂ = 1,  β ₃ = 1,  c = 1,  g = −0.24,  ϵ = −1,  a ₁ = 1, and d = 1. Waveforms that recur frequently in space or time to generate continuous, oscillating patterns are represented by periodic wave solutions. When the underlying medium or boundary conditions of a physical system exhibit a recurring structure, these waves naturally occur. From a physical perspective, they are observed in surface water waves that recur evenly over the surface, such as ocean tides and ripples. Periodic waves are used in solid-state physics to simulate phonons, which are atoms’ vibrations in a crystal lattice that transfer energy.

Figure 8:

The solitary wave solution H ₁₈(x, t) generated by the Eq. (4.19): (a) 3D bell shape wave profile, (b) 2D graph representation for different values of parameters and (c) interdependent contour representation.

6 Bifurcation analysis

In this section bifurcation theory [15] is used to analyze Eq. (4.1) to understand how changes in parameters affect the behavior of the system [16]. By applying a Galilean transformation to Eq. (4.5), becomes a planar dynamical system simplifying the study of its equilibrium points and phase space. In dynamic systems, bifurcation occurs when the system behaves differently due to minor modifications in the value of parameters [17]. It frequently results in new stability phases, periodic orbits, and unpredictable behaviors. The bifurcation hypothesis describes these unanticipated events and helps forecast how structures would behave at different phases [18]. Phase diagrams and equilibrium points for the future planner dynamical system are presented. Equilibrium points are where a system stays still, and phase portraits show its stability and response to changes. With this approach, nonlinear systems can be qualitatively analyzed, offering significant understanding into intricate behaviors and dynamics rather than depending solely on numerical data.

Physically, bifurcation analysis analyzes how a system’s behavior abruptly alters when a control parameter (such as energy input, velocity, pressure, or amplitude) arrives at a critical threshold. A bifurcation point physically defines the change from one kind of motion or wave pattern to another. For instance, a single oscillation may divide into many oscillations with distinct frequencies, or a calm fluid flow may become turbulent as speed increases. Bifurcation analysis in nonlinear wave systems illustrates important points when the behavior of the system radically changes by explaining how little changes in parameters can result in abrupt transitions, such as a single wave becoming unstable, generating periodic patterns, or becoming chaotic. Let d Q d ξ = Z , the planar system for the required model of Eq. (4.1) can be represented as follows:

where

The first integral for the planar system Eq. (6.1) is as follows:

G ( Q , Z ) = Z 2 2 + α 1 Q 2 2 + α 2 Q 4 4 .

Next, the bifurcation behaviour of the phase profile of the system in Eq. (6.1) is analyzed across the parameter space defined by α ₁ and α ₂. The system in Eq. (4.4) has three equilibrium points. The outcomes of the qualitative analysis are listed below:

Moreover, the Jacobian of Eq. (6.1) is:

Thus, G(Q, 0) represents the saddle point when G(Q, Z) < 0, G(Q, 0) represents the center point when G(Q, Z) > 0 and G(Q, 0) represents the cuspidal point when G(Q, Z) = 0. The following outcomes suggest possible variation in the parametric values.

1. Case-(1):

     If α ₁ < 0 and α ₂ > 0, Eq. (6.1) gives three equilibrium points which are given as: h ₁=(0, 0),h ₂=(2, 0) and h ₃ = (−2, 0); since h ₂ and h ₃ are center points and h ₁ is a saddle point as shown in Figure 9.

1. Case-(2):

Figure 9:

Phase portrait of Case-(1) with the given parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

     If α ₁ > 0 and α ₂ < 0, Eq. (6.1) gives three equilibrium points which are given as h ₁ = (0, 0),h ₂ = (3, 0) and h ₃ = (−3, 0); since h ₂ and h ₃ are saddle points in this case and h ₁ is a center point as shown in Figure 10.

1. Case-(3):

Figure 10:

Phase portrait of Case-(2) with the given parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

     If α ₁ < 0 and α ₂ < 0, Eq. (6.1) gives three equilibrium points which are given as h ₁ = (0, 0), h 2 = ( − 1 , 0 ) and h 3 = ( − 1 , 0 ) with h ₂ and h ₃ as being imaginary points and h ₁ as a center point as shown in Figure 11.

1. Case-(4):

Figure 11:

Phase portrait of Case-(3) with the given parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

     If α ₁ < 0 and α ₂ < 0, Eq. (6.1) gives three equilibrium points which are given as h ₁ = (0, 0), h 2 = ( − 1 , 0 ) and h 3 = ( − 1 , 0 ) with h ₂ and h ₃ as being imaginary points and h ₁ as a saddle point as shown in Figure 12.

Figure 12:

Phase portrait of Case-(4) with the given parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

7 Chaotic behaviour

The unperturbed form of the given equation is examined in the previous sections, incorporating both qualitative and quantitative analyses shows that the moving wave structure does not exhibit chaotic behavior. The analysis of chaotic behaviour in dynamics system carries important implications across engineering and scientific discipline. Unpredictable patterns in a system are found using a variety of techniques intended to expose chaos, these techniques are helpful in determining the system’s chaotic tendencies. In physical systems, motion that appears random and unpredictable yet following to precise rules is referred to as chaotic behavior. This occurs when the system becomes extremely sensitive to starting circumstances; even a small shift at the very beginning can have a profound impact on the system’s performance later on. Physically, chaos indicates when a system, albeit not really random, operates in a complicated, erratic manner devoid of recurring patterns. Numerous physical systems, such as turbulent water flow, weather systems, lasers, and plasma waves, exhibit chaotic behavior, making it hard to forecast the long-term behavior of the system. The additional perturbation term is introduced to the system in Eq. (6.1), that is L ₀Sin(ωξ), where amplitude and frequency are denoted by L ₀ and ω respectively. The following perturbed form is the final result of this modification:

Changing the perturbation term results in a variety of phase diagrams that show various chaotic behavior patterns. Graphical representation of chaotic behavior patterns in the form of 3D plot, time series and 2D plot is depicted in Figures 13–15, respectively.

Figure 13:

Graphical representation of the chaotic dynamics in system Eq. (6.1) for L ₀ = 3 and ω = 10.

Figure 14:

Graphical representation of the chaotic dynamics in system Eq. (6.1) for L ₀ = 3 and ω = 2π.

Figure 15:

Graphical representation of the chaotic dynamics in system Eq. (6.1) for L ₀ = 0.1 and ω = 0.2.

8 Sensitivity analysis

In this analysis the response of the equation to different initial values is explored. The sensitivity also analyzes that, how different kinds of input unpredictability might affect a mathematical model’s outcome [19]. Sensitivity analysis in physical systems is the study of how little modifications to beginning circumstances or parameters impact the behavior of the system as a whole. In a physical sense, it aids in our comprehension of the characteristics that most strongly impact the result of a process, wave, or dynamic system. This is particularly crucial in nonlinear systems since even little changes may have a big, unanticipated impact on stability, energy distribution, and wave patterns. To put it shortly, sensitivity analysis shows us how much control and predictability we have over a physical system and which parameters affect its behavior the most.

To examine that how the model react to different starting conditions, it is tested with three separate sets of points. The wave in red and blue colour in Figure 16 shows the initial conditions (Q, Z) = (0.7, 0.2) and (Q, Z) = (0.7, 1.5), respectively. The cure in red colour in Figure 17, indicates initial condition (Q, Z) = (0.7, 0.1), whereas the green curve represents (Q, Z) = (0.9, 0.3). Similarly, the blue curve in Figure 18 indicates the initial condition (Q, Z) = (0.6, 0.3), whereas the green curve indicates (Q, Z) = (0.9, 0.9). The red curve in Figure 19, corresponds to the initial condition (Q, Z) = (0.5, 0.3), the blue curve shows (Q, Z) = (0.7, 0.8), and the green curve represents (Q, Z) = (0.3, 0.1), as given below:

Figure 16:

The sensitivity analysis of the dynamical system’s in Eq. (6.1) for parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

Figure 17:

The sensitivity analysis of the dynamical system’s in Eq. (6.1) for parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

Figure 18:

The sensitivity analysis of the dynamical system’s in Eq. (6.1) for parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

Figure 19:

The sensitivity analysis of the dynamical system’s in Eq. (6.1) for parametric values as a ₁ = 1, k = −1, β ₂ = 1 and β ₂ = 1.

9 Novelty and comparsion

This section offers a comparison between soliton solutions of several kinds obtained of the proposed model along with other soliton structures examined by the researchers [20]. In article [20], the generalized Riccati equation method including M-truncated derivative is applied. The obtained solutions are rational, hyperbolic, trigonometry and elliptic and in article [21] soliton solutions are obtained by applying improved simple equation method and exp (-ϕ(ξ))-expansion method to the third order NLSE. In this article, the Caputo FD is used to solve the fractional generalized third-order nonlinear Schrödinger equation using a new mapping method. Various solitons solutions are obtained such as bell shape, periodic, anti bell, kink, dark and w shape. The bifurcation analysis, chaotic behaviour and sensitivity analysis of the proposed model is also investigated while these are not discussed in [20], 21]. The 3D, 2D and time series graphes are also plotted for the system.

10 Physical interpretation

In many scientific fields, the fractional generalized nonlinear Schrödinger equation has emerged as an effective tool for analyzing wave behavior. In optics, it helps researchers to understand how light pulses behave in fiber optic cables, which is crucial for improving telecommunications and laser technologies. The equation is particularly helpful for predicting extreme wave events, such rogue waves, which can appear suddenly in optical systems and oceanic environments. This is because it can represent complicated wave interactions. Researchers and biologists in the medical profession find this equation beneficial for simulating how signals pass through nerve cells and how light interacts with living tissue. In physics, it is essential for exploring quantum systems and plasma behavior. In mathematics, Figures are essential for the interpretation and visual representation of theoretical concepts and analytical results. The Figure 1, represents bell shape solitons which is the result of solution in Eq. (4.10). In fiber communication, they are used to send optical pulses across long distances without distortion. Where steady supply of energy is required, they also show up in nonlinear wave guides and plasma waves. Figure 2, represents w-shape soliton solution which is the result of solution in Eq. (4.11). The w-shape graph represents the presence of higher order soliton structure. Figure 3, represents anti bell and Figure 4, represents kink soliton derived from equations Eq. (4.12) and Eq. (4.14) respectively. Anti bell graphs represents the non-equilibrium systems where the particles preferentially accumulate at boundary points rather than center points and kink indicates the sharp and localized transformation between two different physical states. The solution obtained from Eq. (4.16) is periodic wave soliton, as shown in Figure 5. In Figure 6, the dark soliton is the solution of Eq. (4.17), bell shape soliton in Figure 7, is the solution of Eq. (4.18) and solution obtained from Eq. (4.19) is periodic wave in Figure 8.

In physics, a bell-shaped graph shows a confined, stable wave, like a soliton, with smooth decay on both sides and concentrated energy around a single peak. This shape is crucial because it illustrates how a wave can go across systems like optical fibers, water waves, and plasmas without spreading or losing its structure, which makes it perfect for energy transmission. A wave with two energy peaks and a middle drop is represented by a W-shaped graph in physics, which frequently denotes internal oscillations in nonlinear systems or relationships between many soliton particles. A dark soliton or localized energy drop is modeled by the core amplitude dip of an anti-bell-shaped graph.

The intricate wave behavior brought on by nonlinearity and parameter variations is highlighted by both forms. Kink waves are used in physics to simulate phenomena such as domain barriers, phase transitions, or shock fronts in fluids and field theories. Kink waves are a step-like transition between two distinct stable states. Periodic waves, which are frequently observed in water waves, crystal lattice vibrations, or optical systems, exhibit a repeated oscillation pattern that denotes a steady and uninterrupted energy transmission. A dark soliton, on the other hand, typically occurs in optical fibers, plasmas, or Bose–Einstein condensates and is characterized by a localized dip in amplitude on a constant background. This indicates a moving vacuum or energy depletion. These wave profiles arise in nonlinear systems and reflect different modes of energy distribution, stability, and physical behavior depending on the system parameters.

11 Conclusions

The fractional generalized third-order nonlinear Schrodinger equation has been solved by using new mapping method including Caputo fractional derivative to derive solitons solutions. By utilizing Caputo fractional derivative the various kinds of solitons solutions has been obtained. The derived solutions were bell shape, w-shape, anti bell, kink, periodic soliton and dark soliton solutions. The obtained results has been effectively shown by the 3D, 2D and contour plots by using appropriate parametric values. This work highlighted the capability and versatility of new mapping method for the proposed model. Furthermore, the bifurcation and portrait analysis for the proposed model have been performed. Figures 9–12 illustrated the equilibrium points of the system and phase diagram. The chaotic behavior and sensitivity analysis of the proposed model has also been performed. Sensitivity analysis was the examination of how modifications of the model input parameters affect its behavior or output.

While the present study offers rich analytical solutions and thorough dynamical investigations using the new mapping method applied to the fractional generalized third-order nonlinear Schrödinger equation, several limitations remain that pave the way for future exploration. First, the solutions are derived under specific assumptions about the fractional orders and parameter values; broader ranges or variable-order fractional derivatives may lead to more generalized behaviors. Second, the model considers a one-dimensional spatial domain; extending the analysis to higher-dimensional cases could uncover more complex soliton interactions and stability profiles. Additionally, the influence of noise, inhomogeneous media, and real-world boundary conditions were not accounted for, which are critical in physical applications. Future research could focus on numerical simulations that incorporate these factors, comparison with experimental data, and the development of control strategies for managing chaotic behavior in optical systems.