Abstract
Nonlinear partial differential equations provide a fundamental mathematical framework for understanding the real world. This study presents a comprehensive analysis of perturbed nonlinear Biswas-Milovic equation with Kudryashov’s law of refractive index. By employing the traveling wave transformation, the original partial differential equation is first converted into a nonlinear ordinary differential equation. The Gaussian soliton solutions are then derived via the generalized trial equation method. To obtain its qualitative properties, a two-dimensional dynamic system is constructed accordingly. Subsequently, we establish the existence of periodic, kink and antikink soliton, and bell-shaped soliton solutions in the qualitative analysis. In particular, we obtained all traveling wave solutions to the equation in the quantitative analysis. Moreover, the exploration of chaotic dynamics is conducted by introducing specific perturbation terms, thereby revealing chaotic behaviors. To the best of our knowledge, this work is the first to provide a classification of possible traveling wave solutions to this equation and to show its chaotic properties.
1 Introduction
Nonlinear partial differential equations (NLPDEs) serve as fundamental tools for modeling complex natural phenomena that defy linear approximation. Their capacity to capture intricate, often non-intuitive dynamics makes them indispensable across a broad spectrum of scientific and engineering disciplines [1–5]. In particular, NLPDEs have been proven essential in nonlinear optics [6–13], for the reason that they govern the behaviors of light propagation in media when dispersion and nonlinearity coexist and interact.
Among the many canonical models in this field, the Biswas-Milovic equation (BME) occupies a central role, which was initially proposed in the year of 2010 [14]. The BME describes the propagation of wave packets across transcontinental and trans-oceanic distances through optical fibres [42]. As a generalization of the nonlinear Schrödinger equation, the BME incorporates power-law nonlinearities, which allows for a more flexible and realistic description of wave dynamics in non-Kerr media, and thus holds substantial importance in nonlinear optics. This significance has attracted considerable scholarly attention, with the BME being investigated in numerous studies [15–18].
In the absence of nonlinear perturbation terms, the BME in polarization preserving fibers is given by [14].where and are the spatial coordinate and temporal variable, respectively; is a complex-valued function; is the group velocity dispersion coefficient; and is the nonlinear coefficient. The -times continuous differentiability of the function is considered, thuswhere is the complex plane, and is the two-dimensional linear space.
In this work, we focus on constructing all possible traveling wave solutions to the perturbed Biswas-Milovic equation (PBME) with Kudryashov’s law of refractive index. The equation is given bywhere and are the power nonlinearity and maximum intensity, respectively; characterize distinct nonlinearity effects; is the coefficient of the self-steepening term; and are the nonlinear dispersion coefficients. In comparison with Equations 1, 3 further incorporates higher-order effects such as self-steepening and nonlinear dispersion, thus delivering a more accurate description of practical optical systems. Kudryashov’s law of refractive index mathematically describes the nonlinear relation between light intensity and the refractive index. The PBME with Kudryashov’s law of refractive index integrates the nonlinear dynamics of the BME with the intensity-dependent refractive index dictated by Kudryashov’s law. This equation enables the analysis of light propagation in nonlinear media, where the refractive index varies with light intensity [19]. Accordingly, it is essential to conduct a systematic and in-depth investigation of this equation.
Over the past several decades, the search for exact solutions to nonlinear evolution equations, particularly soliton solutions, has remained a vibrant research frontier [20–28]. A widely adopted approach to obtaining such solutions entails reducing the original NLPDE to an ordinary differential equation (ODE) via the traveling wave transformation. This reduction opens the door to a variety of analytical methods, which have been extensively employed to solve diverse NLPDEs. Some of the related studies can be found in [29–39].
Equation 3, the target equation of this work, has been studied via various approaches. In [40], Zayed, Elsayed M. E., et al. utilized the sub-ODE method to obtain a variety of soliton solutions and periodic solutions for . In [41], the -expansion method, the new Kudryashov method, and the simple equation method were employed by Lanre Akinyemi, et al., which yielded a range of exact solutions including singular soliton solutions, the bright and dark soliton solutions, rational, periodic, and exponential solutions. Subsequently, in [42], applying the first integral approach, Lanre Akinyemi, et al. retrieved several existing solutions and obtained new ones. However, existing approaches failed to acquire complete traveling wave solutions. In this paper, to address this deficiency, we employ the complete discrimination system for polynomial method (CDSPM), which was initially introduced in [43]. On the condition that the equation can be converted into the specific integral form, we can easily utilize the CDSPM to conduct the qualitative and quantitative analysis, thereby deriving all traveling wave solutions, which has been validated in [44].
The remainder of this paper is organized as follows. First, Equation 3 is transformed into an ODE. Then, via the generalized trial equation method, we derive the Gaussian soliton solutions. Thereafter, the CDSPM is applied, followed by a comprehensive qualitative analysis to elaborate on the dynamical characteristics in depth. Subsequently, we perform a quantitative analysis by classifying the traveling wave solutions of the equation, effectively verifying these qualitative conclusions. Finally, we present a further exploration of the system’s chaotic dynamics and its intrinsic complexity, thereby furnishing additional theoretical underpinnings to enhance the comprehension of nonlinear phenomena.
2 Mathematical derivation
We introduce the following transformation [41].where and are arbitrary constants.
Substituting (4) into (3) yields
Taking the real part of Equation 5, we obtainwhile the corresponding imaginary part of Equation 5 reads
Equation 7 generally admits no nontrivial solutions unless or , both of which are trivial solutions and thus excluded from the discussion in this work. To ensure Equation 7 holds for nontrivial solutions, we set the corresponding coefficients equal to zero, which also facilitates the subsequent analysis. Thus we obtain
We adopt the following transformationEquation 6 can be transformed intowhich followswhere
When , we obtainwhere
If , Equation 13 is reduced to
It is demonstrated in [45] that, for a general nonlinear equation, once it can be reduced to the form of Equation 15, the generalized trial equation method can be used to directly derive the corresponding Gaussian soliton solutions, which take the formwhere is an integral constant.
Figure 1 depicts the two-dimensional and three-dimensional representations of solution (16), while directly indicating the existence of the Gaussian bright solitons solutions to Equation 15. Different colors are adopted to illustrate distinct values of in Figure 1a. It is obvious that the position of the solitons depends on .
FIGURE 1
To conduct a comprehensive qualitative analysis of Equation 13, we first recast it in the formwhich yields
By means of the first integral method, we obtainwhereand is an integral constant.
In this study, we focus on the scenario where , namely
Here, we introduce the transformationthen Equation 21 can be rewritten aswhere .
The corresponding dynamic system readsand the Hamiltonian is given bywhich satisfies
3 Qualitative analysis
The potential energy function iswe can then derivewhere
It is demonstrated in [44] that, for a nonlinear differential equation, we can conduct a thorough qualitative analysis via CDSPM. Building on this framework, the qualitative properties of Equation 23 are presented in the subsequent discussion.
The complete discrimination system of the fourth-order polynomial takes the form [43].
From (30), there are nine cases, but two of them have no contributions to our study and are therefore omitted here.
I. ,
Supposing , , , we have , , , , which means we get two equilibrium points here (1,0) and . When (1,0) is a saddle point, while is a center. When (1,0) is a center, while is a saddle point. Figures 2a,b depict the corresponding phase portraits. The red trajectories indicate the existence of bell-shaped soliton solutions, while the blue closed orbits correspond to periodic solutions.
FIGURE 2
We further consider two additional sets of parameters. For , , , , the resulting coefficients are , , , , yielding two saddle points (0,0) and (1,0). For , , , , we get , , , , with two saddle points and (2,0). The phase portraits for these two cases are shown in Figures 2c,d. The red closed trajectories represent heteroclinic cycles, which directly comfirm the existence of kink and antikink soliton solutions.
II. ,
Supposing , , , , we have , , , which means that there exists a unique equilibrium point (1,0), it is a cuspidal point for both and . Figure 3 depicts the corresponding phase portraits.
FIGURE 3
III. ,
Supposing , , , , we have , , , , which means we get four equilibrium points here (1,0), (2,0), and . When (1,0) and are centers, while and (2,0) are saddle points. When (1,0) and are saddle points, while and (2,0) are centers. Figure 4 depicts the corresponding phase portraits. The blue trajectories indicate the existence of bell-shaped soliton solutions, while the red closed orbits correspond to periodic solutions.
FIGURE 4
IV. ,
Supposing , , , we have , , , which means we get three equilibrium points here (0,0) (1,0), and . When (0,0) is a cuspidal point (1,0) is a saddle point, and is a center. When (0,0) is a cuspidal point (1,0) is a center, and is a saddle point. Figure 5 depicts the corresponding phase portraits. The red trajectories indicate the existence of bell-shaped soliton solutions, while the blue closed obrits correspond to periodic solutions.
FIGURE 5
V. ,
Supposing , , , , we have , , which means we get two equilibrium points here (1,0), and . For both and (1,0) and are cuspidal points. Figure 6 depicts the corresponding phase portraits.
FIGURE 6
VI. ,
Supposing , , , , we have , , which means we get two equilibrium points here, , and (3,0). When , is a center, while (3,0) is a saddle point. When , is a saddle point, while (3,0) is a center. Figure 7 depicts the corresponding phase portraits. The red trajectories indicate the existence of bell-shaped soliton solutions, while the blue closed orbits correspond to periodic solutions.
FIGURE 7
VII. ,
Obviously, there exists a unique equilibrium point (0,0), which is a cuspidal point for both and . Figure 8 depicts the corresponding phase portraits.
FIGURE 8
Following the discussions above, the existence of periodic solutions and soliton solutions is established. We will validate these qualitative conclusions by deriving all exact solutions in the next section.
4 Quantitative analysis
For simplicity, we set , then Equation 23 reduces to
The integral form readswhere is an integral constant. Let
It is demonstrated in [43] that, for a general nonlinear equation, once it can be reduced to the form of Equation 39, the complete classification of its traveling wave solutions can be obtained. Therefore, we present the corresponding results below.
The complete discrimination system of the fifth-order polynomial takes the form [43].
Based on this, we analyze the exact solutions to Equation 39 in the cases below.
Case I. ,where is a real number. The corresponding solution is given by
The corresponding graphs of the singular solution (43) are shown in Figure 9.
FIGURE 9
Case II. ,where , are real numbers (, ). The corresponding solutions are given as follows.
If ,and if ,
Case III. ,where , are real numbers (, ). The corresponding solutions are given as follows.
If ,and if ,
Figure 10 demonstrates the kink and antikink soliton solutions. In particular, we compare the effects of different parameter sets on the kink and antikink soliton profiles in Figures 10a,b, respectively.
FIGURE 10
Case IV. ,where , , and are real numbers (, ). The corresponding solutions are given as follows.
If ,if ,if ,and if ,
Figures 11, 12 demonstrate the kink-antikink soliton solution and dark soliton solutions, respectively. In particular, we compare the effects of different parameter sets on the dark soliton profiles in Figure 12a.
FIGURE 11
FIGURE 12
Case V. ,where , , and are real numbers . The corresponding solution is given bywhere
Case VI. ,where , , and are real numbers . The corresponding solutions are given as follows.
If ,and if ,where
We can also derive that solution (59) is a periodic solution.
Case VII. ,where , , , and are real numbers (, , , ). The corresponding solution is given bywhere
Case VIII. ,where , , , and are real numbers . The corresponding solutions are given as follows.
If ,if ,if and ,where
Case IX. ,where , , and are real numbers (, , ). The corresponding solutions are given as follows.
If ,if ,and if ,
Case X. ,where , , , , and are real numbers. The corresponding solution can be expressed by the hyper-elliptic integral as
Case XI. ,where , , , , and are real numbers. The corresponding solution is given by
Case XII. ,where , , , , and are real numbers. The corresponding solution is given by
So far, we have derived all traveling wave solutions to the equation, and verified the conclusions from our qualitative analysis. In the next section, we aim to investigate the chaotic properties of System (24) under distinct perturbation terms.
5 Chaotic behavior
Although System (24) contains nonlinear terms, it remains a two-dimensional dynamic system. To explore its chaotic dynamics, we consider introducing a perturbation term to extend it into a three-dimensional dynamic system [46].
Here, we introduce the perturbation term , and the resulting perturbed system takes the form:where denotes the perturbation intensity, and denotes the perturbation function. In existing studies on chaotic behavior in dynamic systems, Gaussian and trigonometric functions are two widely adopted perturbation forms, typically introduced into the specific dynamic system under investigation to induce chaotic behavior. For instance, Huang et al. investigated the chaotic dynamics of their target dynamic system with a Gaussian perturbation term [46]. While, Li et al. explored the chaotic behavior presented by their target dynamic system with a trigonometric perturbation term [47]. Both perturbation forms have also been considered in [48, 49]. We therefore adopt Gaussian and trigonometric functions as the perturbation terms to investigate whether they can induce chaotic behavior in System (24). In what follows, we verify that this is indeed the case.
I. Let
If , , , , , the corresponding plots are illustrated in Figure 13. Figures 13a,b show the 2-D and 3-D phase portraits of the perturbed system. Figures 13c–f illustrate the largest Lyapunov exponent (LLE) corresponding to different parameters of the system under the preset conditions. Our results show that this perturbation term induces obvious chaotic behaviors in the equation, with the LLE exceeding 60. As a key indicator of dynamic systems, larger LLE value generally indicates higher degree of chaos in the system.
FIGURE 13
To investigate the effects of parameter variations on System (80), we present a controlled parameter comparison in Figure 14. Here, we tune each parameter individually while holding all other parameters fixed, and contrast the resulting chaotic dynamics with the baseline case shown in Figures 13a,b. This comparison indicates that the chaotic behavior of System (80) varies with parameter values, and different parameter sets correspond to different chaotic behaviors of the system.
FIGURE 14
II. Let
If , , , , , the corresponding plots are illustrated in Figure 15. Corresponding discussions are omitted here, since they exhibit the same characteristics as illustrated in Figure 13. It can also be concluded that different perturbations exert distinct effects on System (24), and the chaotic properties of this system can be further investigated by adjusting the perturbation term. Analogous to Case I, the effects of parameter variations on the system’s chaotic behavior for Case II is presented in Figure 16. This further confirms that parameter adjustments can effectively modulate the system’s chaotic properties.
FIGURE 15
FIGURE 16
6 Conclusion
This paper presents a systematic investigation of PBME with Kudryashov’s law of refractive index. The equation is first transformed into an ODE. The Gaussian soliton solutions are then derived via the generalized trial equation method. Subsequently, we construct a two-dimensional dynamic system, and establish the existence of kink and antikink soliton, bell-shaped soliton and periodic solutions. As a key result, we provide a classification of traveling wave solutions to the equation through quantitative analysis. Furthermore, the chaotic behaviors induced by specific perturbations are demonstrated via corresponding 2-D and 3-D phase portraits and the plots of the LLE. To the best of our knowledge, the Gaussian soliton solutions, the classification of traveling wave solutions, and the investigation of chaotic behavior for this equation have not been reported in previous studies, including Refs. [19, 40–42]. These findings deepen the understanding of nonlinear phenomena and hold significant potential for applications in related physical fields. It is worth noting that all results obtained in this work are derived under the condition . For other values of , one possible approach is to transform the right-hand side of Equation 19 into a polynomial of a suitable degree via an appropriate transformation, and then perform the analysis using the CDSPM corresponding to that degree. In addition, all solutions obtained in this work are single traveling wave solutions, whereas other types such as multi-soliton and non-traveling wave solutions cannot be derived using the method employed in this work. Several extensions of the present work deserve to be studied in the future. For instance, it is of great theoretical interest to identify what types of perturbation terms can induce chaotic behavior in System (24), and further to quantify how different perturbation terms modulate the properties of the resulting chaotic dynamics.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
YZ: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2026.1811376/full#supplementary-material
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Summary
Keywords
chaotic behavior, complete discrimination system for polynomial method, Gaussian soliton solution, Kudryashov’s law of refractive index, perturbed nonlinear Biswas-Milovic equation
Citation
Zhang Y (2026) Wave structures and chaotic behaviors of perturbed nonlinear Biswas-Milovic equation with Kudryashov’s law of refractive index. Front. Phys. 14:1811376. doi: 10.3389/fphy.2026.1811376
Received
14 February 2026
Revised
11 April 2026
Accepted
13 April 2026
Published
07 May 2026
Volume
14 - 2026
Edited by
Bruno Piccirillo, University of Naples Federico II, Italy
Reviewed by
Ghulam Hussain Tipu, Shanghai University, China
Ibrahim Sani Ibrahim, Northwest University Kano, Nigeria
Updates
Copyright
© 2026 Zhang.
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*Correspondence: Yuxi Zhang, zhangyuxi0723@gmail.com
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