Exploring the chaotic, sensitivity and wave patterns to the dual-mode resonant Schrödinger equation: application in optical engineering

Abstract

The comprehension of nonlinear problems is essential for the understanding of nonlinear wave propagation in applied sciences. This work investigates the dual-mode manifestation within the nonlinear Schrodinger equation, clarifying the amplification or absorption of coupled waves. This investigation explores the dual-mode phenomenon’s simultaneous generation of two distinct waves, which are influenced by three critical parameters: phase velocity, nonlinearity, and dispersive factor. By utilizing the complex wave transformation, we derive the nonlinear ordinary differential equation of the governing model. Additionally, we employ recently developed analytical techniques, including the modified generalized exponential rational function method and the multivariate generalized exponential rational integral function method, to find a wide range of solutions such as bright-dark, bright, dark, and combined solitons for the proposed model. Moreover, the chaotic and sensitivity analysis are discussed. Different graphs are included to clarify the behavior of solutions for several parameter values.

Introduction

In light of a progressive specialization and application of mathematical models to capture increasingly complex wave dynamics¹, the interrelationship among nonlinear partial differential equations (NLPDEs)², the nonlinear Schrödinger equation (NLSE)³, the dual-mode nonlinear Schrödinger equation (DMNLSE)⁴, the resonant dual-mode nonlinear Schrödinger equation⁵, and the nonlinear evolution equations^6,7 reflects a hierarchical evolution in modeling approaches to address diverse and intricate physical phenomena. NLPDEs are essentially the mathematical tools for describing problems in which nonlinearity and spatial or temporal dynamics are essential. From quantum physics to fluid dynamics, these equations are fundamental for modeling real-world systems^8,9. Among them, the NLSE transforms into a fundamental equation for understanding wave propagation in nonlinear, dispersive systems like plasma, Bose-Einstein condensates, and optical fibers. NLSE is a specific NLPDE that effectively solves dispersion and nonlinear problems, enabling the development of soliton-stable confined wave modes. Based on the NLSE, the DMNLSE extends the model to situations in which two interacting wave modes coexist. Additional degrees of freedom introduced by this dual-mode interaction allow the dynamics of multimodal systems, including optical fibers carrying multiple polarization states or spinor Bose-Einstein condensates as well as mode coupling and energy exchange. The DMNLSE not only preserves the fundamental soliton properties of the NLSE but also considers the complicated interaction between two modes, so producing richer and more complex dynamics^10,11. By including resonance effects, the resonant DMNLSE improves this framework. Resonance results from certain frequency or phase interactions between modes amplifying their interactions to produce phenomena include synchronized energy transfer, increased energy transfer, or the development of linked solitons. Understanding systems where external or internal resonances are important, such as resonant optical waveguides, plasma waves under certain circumstances, or sophisticated photonic devices. NLPDEs essentially provide the theoretical basis; the NLSE concentrates on single-mode nonlinear wave propagation; the DMNLSE generalizes this to interacting dual-mode systems; and the resonant DMNLSE captures the extra complexity of resonance-enhanced interactions. Collectively, they provide a hierarchy of models addressing more complex nonlinear wave events in many different scientific and technical domains.

Moreover, the soliton theory is subject of extensive research due to their diverse properties and potential implications in the fields of science and technology¹². Due to its stability and localized character, soliton solutions of the DMNLSE have numerous applications in both scientific and practical domains. In optical fiber communications, they maximize dual-mode fibers for more data capacity and allow unaltered pulse transmission distances¹³. Solitons help to develop multimode laser systems and pulse shaping in nonlinear optics. They define localized particle concentrations and nonlinear interactions in dual-component and spinor condensates in Bose-Einstein condensates. While in biophysics they represent dual-mode signal transmission. Solitons can help photonic crystal and waveguide designs to comprehend mode coupling and energy transfer. Moreover, solitons provide integrable systems in mathematical physics, so advancing the study of nonlinear wave phenomena and motivating novel developments in quantum computing for stable data storage and interpretation^14–16.

Furthermore, information technology is dependent on optical fiber-based technology, which includes communications networks and internet-related phenomena¹⁷. In addition, it has a wide range of applications in the disciplines of science and biology¹⁸. Certain characteristics, including information buffering, routing, and switching, are not easily modifiable after fabrication in conventional optical fibers, in addition to the transmission of light¹⁹. Dual-core fibers have been implemented by researchers to improve fiber optic sensing techniques in response to this. The same functions can be performed with greater precision by these new dual-core optical fibers, which employ a negligible amount of mechanical pressure. Optical fiber-based technology has advanced due in part to the development of fibers containing dual particles. Their intimate proximity to one other helps these centers move mechanically. A small amount of pressure produces mechanical motion that results in optically associated behavior. The researchers might permit light to migrate from one core to another by only moving one core by a few nanometers. In contrast to traditional optical fibers, this physical phenomenon offers a multitude of novel applications for optical fiber-based technology²⁰. The internal core structure of the novel nano-mechanical fibers can also be regulated by nanometer-scale mechanical movements in the surrounding environment. In other words, these fibers are more proficient at detecting small variations in the environment and can be implemented as sensors in electronic devices.

Our comprehension of physical events is facilitated by the existence of exact solutions for numerous physical systems. It is imperative that these be incorporated, as they serve as an introductory component for further studies. A physical system must be characterized as an ODE or PDE in order to obtain an exact solution. The ability to simulate intricate natural or industrial events is facilitated by PDEs. A variety of techniques and approaches have been devised by numerous researchers to assist in the appropriate resolution of these issues. A number of methodologies have gained recognition and have been implemented frequently in recent years, including: Multiple exp-function approach²¹, Lie symmetry technique²², Hirota method^23–25, modified Sardar sub-equation method²⁶, Adomian decomposition technique²⁷, Bernoulli -expansion method²⁸, Darboux transformation²⁹, Riccati equation mapping method³⁰, bifurcation analysis^31–34, -expansion method³⁵, truncated Painlevé approach³⁶, etc.

This study aims to analyze the solitary waves of two-mode resonant nonlinear Schrödinger (RNLS) equation for two different cases such that the square root Kerr law and Kerr law. We examine the impact of the linearity-dispersive parameters on the wave propagation characteristics of the dual waves. An important phenomenon in a variety of applications, such as turbulence management and communication systems, is the convergence of left and right waves as the phase velocity increases. Due to the fact that the two-mode RNLS equation contains terms that are more highly accurate than the classical nonlinear Schrödinger equation (NLSE) in describing internal waves and multiple optical fibers in the ocean, the results of the dual-mode NLSE findings may be significant in the advancement of our understanding of wave processes. A more exhaustive comprehension of a diverse multitude of processes in science, technology, and engineering is substantially facilitated by the extensive implementation of this equation in nonlinear dispersive PDE. In order to elucidate the interacting wave dynamics in these equations, a model was developed to represent a variety of wave phenomena in plasma physics and mathematical physics. To derive the analytical solutions, novel integration techniques are implemented, such as the multivariate generalized exponential rational integral function method (MGERIFM)³⁷ and the modified generalized exponential rational function method (mGERFM)³⁸. The solutions derived from the current methods, when compared to prior approaches, illustrate their efficacy, straightforwardness, and importance in tackling these nonlinear problems, yielding innovative outcomes. The results confirm the efficacy and adaptability of the methods used, which have significant implications for the fields of engineering and applied sciences. These techniques may find significant applications that will help different technological fields to develop new soliton and wave modes to be emerged. The applied approaches provide practical recommendations and help us to understand nonlinear dynamics. Moreover, the proposed model has varied applications across many areas. In electromagnetic phenomena, it describes wave propagation in dual-band metamaterials and linked plasmonic resonances. It delineates nonlinear wave interactions in stratified flows and dual-vortex systems within fluid dynamics. In plasma physics, it regulates soliton formation and mode coupling in magnetised plasmas. In electrochemistry, it facilitates the comprehension of resonant electron transfer processes at electrode interfaces. Laser diodes use this paradigm to examine dual-wavelength emission dynamics and mode competition. Optical absorption elucidates excitonic transitions in quantum-confined systems such as quantum wells. In mathematical biology, it simulates bistable genetic oscillators and brain wave interactions among linked neuronal populations. This equation connects quantum-inspired dynamics with classical wave systems, providing insights into coherent mode interactions across many scales.

The article is sectioned as follows: Section 2 presents the governing equation, while Section 3 utilizes the mGERFM and MGERIFM to calculate soliton solutions of the proposed model. Section 4 illustrates the graphical representation of the obtained solutions. The chaotic and sensitivity are discussed in the Sections 5 and 6, respectively. Concluding remarks are delineated in Section 7.

The governing equation

Nonlinear models describing the motion of two-way waves simultaneously under the effect of contained phase velocity define the equations for the two-mode systems. In this context, the first dual-mode equation to be developed was the Korteweg-de Vries (KdV) equation³⁹. The two-mode KdV equation⁴⁰ is as follows:

1

where denotes the field function, are real numbers such that are representing nonlinearity factors and denote the phase velocities with , , characterize the interaction of dispersive components, respectively. The Hirota-Satsuma equation⁴¹ claims that two mode KdV equation regulates the interaction of waves with different dispersive components. It has been observed that the KdV equations can be employed for evaluating these waves in the absence of any interaction. Taking in Eq. (1) and integrating with respect to time with zero integration constant converts to the standard KDV equation. The general form of any NLPDE is

2

where denote the nonlinear terms and denote the linear terms. Further, Eq. (2) with dual-mode variation is expressed as:

3

In addition, in⁴², it is investigated how phase velocity affects dual-mode wave solutions in NLSEs. The resonant NLSE contributes an important function in Madelung fluids⁴³ and is written as:

4

where V stands for the complex envelope function and G for the expanded form of nonlinearity. The generalized resonant nonlinear Schrödinger equation (RNLSE) is expressed as follows:

5

The standard KdV equation is obtained by setting the term R to 0 in Eq. (1) and subsequently integrating with respect to the time derivative. In Eq. (2), we generalize any NLPDE form, and subsequently incorporate linear and nonlinear operators in Eq. (1). This process results in the derivation of the dual-mode form of Eq. (3), which is expressed in Eq. (4). Eq. (5) is derived by employing Eqs. (3) and (4). The proposed model describes quantum systems with coherently interacting states or modes, a key situation in current physics. The classic Schrödinger framework is generalised to simulate linked wavefunctions, including energy exchange between light and matter (e.g., exciton-polaritons), nonlinear wave mixing in Bose-Einstein condensates, and parity-time symmetric systems with gain and loss. In this work, we study the RNLS equation with the following two different types of G.

Governing equation of two-mode RNLS equation with square-root Kerr law

Equation (5) with takes the form as follows:

6

Equation (6) can be solved by employing the transformation described by

7

where are real constants. Incorporating the predefined transformations in Eq. (6), we acquire the real and imaginary parts as follows:

8

9

By applying the balance principle among the terms and in Eq. (8) provide .

Governing equation of two-mode RNLS equation with Kerr law

Eq. (5) with takes the form as follows:

10

Equation (10) can be solved by employing the transformation described by

11

where are real constants. Incorporating the transformations given in Eq. (11) into Eq. (10), we acquire the real and imaginary parts as follows:

12

13

By applying the homogeneous balancing principle between the terms and in Eq. (12) provide .

In addition, the proposed model has been discussed in literature from various aspects like, in⁴⁴ the modified Sardar sub-equation method is applied to recover soliton solutions. Where, in⁴⁵ this model is investigated applying the -expansion method. Similarly, in⁴⁶ its interaction phenomena is studied. The model under consideration in this study exemplifies the utilization of advanced integration methods to achieve numerous exact solutions.

Extraction of solutions

This section aims to calculate the solutions of two-mode RNLS equation with square root Kerr law and with Kerr law applying the proposed methods, namely mGERFM and MGERIFM.

Two-mode RNLS equation with square-root Kerr law

Application of modified generalized exponential rational function method

The general mGERFM³⁸ solution is described by

14

where

15

For , Eq. (14) is written as:

16

• Putting and in Eq. (15), gives and inserting Eq. (16) in Eq. (8) offers then we get: Soliton solution of exponential form is written as

17

The explicit hyperbolic solution

18

• Next, choosing and in Eq. (15), offers while solving Eqs. (16) and (8) provide and the following solutions written as:

The soliton solutions

19

20

• By taking and Eq. (15), offers while Eqs. (16) and (8) gives then we get:

21

Next, the hyperbolic solution

22

• Choosing and then Eq. (15), gives Eqs. (16) and (8) provide along with we get:

23

24

• Taking and then Eq. (15), gives Eqs. (16) and (8) provide along with we get:

25

26

Application of multivariate generalized exponential rational integral function approach

The general solution to MGERIFM³⁷ is written as:

27

The solution to Eq. (27) for is as follows:

28

where is defined by

29

Moreover, the solutions can be expressed as:

Case-1:   Choosing and Eq. (29) takes the following form

30

Inserting Eq. (30) into Eq. (28), implies that

31

The solutions are as follows when Eq. (31) is inserted into Eq. (8):

For we have:

32

Case-2:   Let and Eq. (29) transforms to sine hyperbolic function

33

Plugging Eq. (33) into Eq. (28), offers

34

Applying Eq. (34) to Eq. (8), we get:

For we get the solutions as follows:

35

Case-3: Taking and Eq. (29) converts to the periodic function

36

Putting Eq. (36), into Eq. (28), gives

37

Incorporating Eq. (37) in Eq. (8), we have:

For we obtain:

38

Case-4:   Choosing the parameters and Eq. (29) offers

39

Incorporating Eq. (39) into Eq. (28), we have

40

Plugging Eq. (40) into Eq. (8), give the following solution:

When we get:

41

Case-5:   Choosing the parameters and Eq. (29) offers

42

Incorporating Eq. (42) into Eq. (28), we have

43

Plugging Eq. (43) into Eq. (8), give the following solution:

When we get:

44

Next, its hyperbolic solution is given by

45

Two-mode RNLS equation with Kerr law

Application of modified generalized exponential rational function method

The general mGERFM³⁸ solution is described by

46

where

47

For , Eq. (46) is written as:

48

• Putting and in Eq. (47), gives and inserting Eq. (48) in Eq. (12) offers then we get: Soliton solution of exponential form is written as

  49

  The explicit hyperbolic solution

  50
• Next, letting and in Eq. (47), offers while solving Eqs. (48) and (12) provide and the following solutions written as: The soliton solutions

  51

  52
• By taking and Eq. (47), offers while Eqs. (48) and (12) gives and then we get:

  53

  Next, the hyperbolic solution

  54
• Choosing and then Eq. (47), gives Eqs. (48) and (12) provide along with we get:

  55

  56
• Taking and then Eq. (47), gives Eqs. (48) and (12) provide along with we get:

  57

  58

Application of multivariate generalized exponential rational integral function approach

The general solution to MGERIFM³⁷ is mentioned as:

59

For , we get

60

where is defined by

61

Moreover, the solutions can be expressed as:

Case-1:   Choosing and Eq. (61) takes the following form

62

Manipulating the Eq. (62) and Eq. (60), offer

63

By tackling the Eq. (63) and Eq. (12), provide:

For and we have:

64

65

Case-2:   Let and Eq. (61) transforms to sine hyperbolic function

66

Plugging Eq. (66) into Eq. (60), offers

67

Putting Eq. (67) to Eq. (12), give:

For and offer:

68

69

Case-3:   Taking and Eq. (61) converts to the periodic function

70

Putting Eq. (70), into Eq. (60), gives

71

On solving the Eq. (71) and Eq. (12), provide:

For and we obtain:

72

73

Case-4:   Choosing the parameters and Eq. (61) offers

74

Incorporating Eq. (74) into Eq. (60), we have

75

Plugging Eq. (75) into Eq. (12), give the following solution:

When and we get:

76

77

Case-5:   Choosing the parameters and Eq. (61) offers

78

Incorporating Eq. (78) into Eq. (60), we have

79

By solving the Eq. (79) and Eq. (12), give:

When we get:

80

Next, its hyperbolic solution is given by

81

Graphical discussion

The graphical analysis offers a more thorough comprehension of the dynamics and interactions of the soliton solutions, as evidenced by the solution profiles plots under various parameter configurations. The 3D plots offer a comprehensive understanding of the motion and development of solitons in space and time, as a result of the emergence of numerous categories of soliton solutions (periodic, dark, bright, kink, and antikink solitons) in 2D and 3D plots. These representations facilitate comprehension of the propagation, oscillation, and stability of solitons. In contrast, two-dimensional diagrams concentrate on the exact spatial distribution of solitons, which enables the analysis of the amplitude distribution, peaking, and localization degree of solitons within the system. These graphs are particularly significant for the study of soliton interaction, as they emphasize regions with high and low density, which provides insight into processes such as soliton interaction, soliton merging, energy exchange, or modulation. Figure 1 presents the behavior of the bright soliton of the solution for the parameters Bright solitons are confined wave packets formed by emphasizing nonlinear media through a balance between dispersion (or diffraction) and nonlinearity, thus maintaining their shape throughout propagation. Their stability is shown by the inverse relationship between amplitude and width and elastic interactions that maintain their properties after collision. Due to their special qualities and uses, bright solitons (which are used in fiber optics, plasma physics, and Bose-Einstein condensates) have been extensively studied. Figure 2 with parameters Fig. 3 with parameters and Fig. 4 with parameters represents the exponential solutions. The nature of such solutions is unique in nonlinear dynamics, as seen in this behavior. They emerge in settings where nonlinearity and dispersion-or diffraction-balance to produce stable, self-reinforcing structures. Figure 5 with parameters Fig. 6 with parameters Fig. 7 with parameters and Fig. 8 with parameters illustrates the behaviors of various solitary waves. Due to the balance between nonlinearity and dispersion or diffraction, solitary waves maintain their form while moving at a constant speed. They may occur in many nonlinear systems; the parameters of the medium determine their form and speed. Solitons differ from solitary waves in that solitons can disperse or change shape and exhibit elastic collisions under certain circumstances. Similarly, Fig. 9 with parameters and Fig. 10 with parameters demonstrates the behaviors of periodic waves. As periodic waveforms, they show the stability and recurrence characteristics one would expect for periodic waveforms in nonlinear dynamics and provide a potentially observed periodic soliton behavior. The diversity of soliton solutions is beneficial for future research, as it simplifies the systems employed in numerous scientific fields, including condensed matter physics, optical fibers, and fluid dynamics. Consequently, it facilitates progress in both theoretical and practical development.

Fig. 1.

Plots of solution (18).

Fig. 2.

Plots of solution (21).

Fig. 3.

Plots of solution (44).

Fig. 4.

Plots of solution (53).

Fig. 5.

Plots of solution (25).

Fig. 6.

Plots of solution (51).

Fig. 7.

Plots of solution (54).

Fig. 8.

Plots of solution (58).

Fig. 9.

Plots of solution (56).

Fig. 10.

Plots of solution (72).

Chaotic analysis with square-root Kerr law and perturbation term

In this section, we introduce the chaotic behavior with quasi-periodic by adding the perturbation term. For this consider, in Eq. (8) and with perturbation term, we get

82

and is a perturbation term that represents the outward periodic force, and depict the frequency and intensity of periodic phrase respectively. Graphical visualization of chaotic behavior is represented in 2D phase portraits, time series, and Poincare map have been sketched in the Fig. (11 AND 12). Practical uses for chaotic behavior, characterized by non-linear system interactions and sensitivity to initial conditions, are plentiful in many fields. In meteorology, chaos calls for flexible, probabilistic models that allow accurate short-term forecasts but restrict long-term projections⁴⁷. Using chaos theory, traffic flow dynamics maximize signals, ease congestion, and forecast bottlenecks, so improving urban commuting efficiency⁴⁸. Driven by intricate interplay of economic forces, investor behavior, and events, financial markets also use chaos theory to create trading algorithms, risk management techniques, and instruments for market volatility analysis. Using chaotic systems, engineers safeguard digital transactions including data movement and encryption. Policymakers and marketers study disorderly tendencies in social systems and behaviors, including the dissemination of viral trends or false information, in order to project outcomes and create winning plans. By allowing the modeling and management of city and economic development among challenging and unpredictable elements, chaos theory supports urban planning and economic policy. Eventually, art and entertainment include the creative and aesthetic aspects of chaos. Fractals and chaotic patterns are used in movies and video games to offer visually appealing designs, animations, and digital effects. Despite its natural uncertainty, chaos theory offers strong instruments for negotiating the complexity of contemporary life.

Fig. 11.

Graphs for the system (82) for and taking the initial conditions (0.009, 0.7).

Fig. 12.

Graphs for the system (82) for and taking the initial conditions (0.1, 0.2).

Sensitivity analysis

This section addresses the sensitivity analysis utilizing multiple initial conditions and the Runge-Kutta approach. By applying the Galilean transformation, the Eq. (8) is transferred into two coupled systems of equations⁴⁹.

83

The values of parameters are set as follows: and graphs have been shown in the Figs. 13, 14, 15 and 16. The figures indicate that small changes in initial conditions influence system dynamics significantly. Academics and practitioners use sensitivity analysis to understand how input parameters affect system outputs for decision-making and mathematical modeling. Multidisciplinary and versatile, this technique is crucial for many areas. Sensitivity analysis helps engineers optimize system designs by identifying performance and robustness characteristics. This tool helps environmental scientists assess climate model uncertainty to inform policy decisions. Recognizing patient-specific variables can improve drug and healthcare results. Financial modeling uses sensitivity analysis to evaluate market risks and returns, improving investment strategies and risk management. Its capacity to simplify models and improve computations helps identify critical components. Control, optimization, and resource allocation improve with sensitivity analysis, as does model prediction reliability.

Fig. 16.

Plot for the system (83).

Fig. 13.

Plot for the system (83).

Fig. 14.

Plot for the system (83).

Fig. 15.

Plot for the system (83).

Conclusions

Analytical solutions provide comprehensive understanding of the behavior of complicated nonlinear systems, thereby revealing the processes underlying events including soliton generation, wave interactions, and stability. By use of requirements for the validation of numerical techniques and approximations, these solutions ensure the precision and reliability of computational research. This research produced substantial results by offering analytical solutions to the two-mode RNLS equation with square-root Kerr law and the two-mode RNLS equation with Kerr law. This investigation underscores the effectiveness of analytical methods in clarifying the fundamental dynamics of nonlinear equations, highlighting their intrinsic behaviors, and offering valuable insights. The dynamical behaviors of the proposed model have been effectively investigated by employing integration methods, including the mGERFM and MGERIFM. This work is also unique because it looks at this problem from a qualitative point of view, including chaotic behavior and sensitivity analysis. For this purpose, we use the Galilean transformation to turn the NLODEs into two sets of equations. We look at how 2D, time series, and Poincare maps can be used as useful ways to find out what chaos is really like. Sensitivity analysis is also explored by using the varying initial conditions. Our research illustrates a wide range of solitary wave solutions and establishes the requisite conditions for their existence, thereby addressing substantial physical factors. The proposed methods results in a series of solutions that facilitate comprehension of the specific physical phenomena of the proposed model. The solutions obtained, demonstrate the efficacy of the proposed methods in extracting the solitary wave solutions of NLPDEs, as well as their potential to broaden the field of soliton theory. The 3D and 2D wave profiles have been employed to illustrate the physical attributes of the secured solutions. Additionally, our research elucidates the behavior of solitary waves in physical systems, thereby creating opportunities for further research and practical applications in related fields.

Author contributions

J.M.:Writing–original draft, Validation, Software, Resources, Methodology, Investigation, U.Y.: Writing–review & editing, Validation, Project administration, Formal analysis.M.Y.: Supervision,Conceptualization, Graphics, Visualization, Data curation.

Data availability

All data that support the findings of this study are included in the article.

Declarations

Competing interests

No conflict of interest.