Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning

Zhe Su; Yiying Tong; Guo-Wei Wei; Zhe Su; Yiying Tong; Guo-Wei Wei

doi:10.3934/math.20241333

AIMS Mathematics

2024, Volume 9, Issue 10: 27438-27470. doi: 10.3934/math.20241333

Previous Article Next Article

Research article Special Issues

Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning

Zhe Su ¹,
Yiying Tong ^{2
,
,},
Guo-Wei Wei ^{1,3,4
,
,}

1.
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
2.
Department of Computer Science and Engineering, Michigan State University, East Lansing, MI 48824, USA
3.
Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, USA
4.
Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA

Received: 31 July 2024 Revised: 02 September 2024 Accepted: 04 September 2024 Published: 23 September 2024
MSC : 53Z50, 55N31

Recently, topological data analysis has become a trending topic in data science and engineering. However, the key technique of topological data analysis, i.e., persistent homology, is defined on point cloud data, which does not work directly for data on manifolds. Although earlier evolutionary de Rham-Hodge theory deals with data on manifolds, it is inconvenient for machine learning applications because of the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation. In this work, we introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning. Our PHLs were constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multi-scale manifolds. To facilitate the manifold topological learning, we proposed a persistent Hodge Laplacian learning algorithm for data on manifolds or volumetric data. As a proof-of-principle application of the proposed manifold topological learning model, we considered the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlighted the power and promise of the proposed method.

Keywords:

Citation: Zhe Su, Yiying Tong, Guo-Wei Wei. Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning[J]. AIMS Mathematics, 2024, 9(10): 27438-27470. doi: 10.3934/math.20241333

Related Papers:

[1]	Yazid Alhojilan, Islam Samir . Investigating stochastic solutions for fourth order dispersive NLSE with quantic nonlinearity. AIMS Mathematics, 2023, 8(7): 15201-15213. doi: 10.3934/math.2023776
[2]	Mahmoud Soliman, Hamdy M. Ahmed, Niveen Badra, Taher A. Nofal, Islam Samir . Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method. AIMS Mathematics, 2024, 9(9): 25205-25222. doi: 10.3934/math.20241229
[3]	Ahmed M. Elsherbeny, Taher A. Nofal, Yakup Yıldırım, Ahmed H. Arnous . New solitary waveforms and their dynamics in the stochastic generalized Chen–Lee–Liu model. AIMS Mathematics, 2025, 10(3): 5197-5235. doi: 10.3934/math.2025239
[4]	Abeer S. Khalifa, Hamdy M. Ahmed, Niveen M. Badra, Wafaa B. Rabie, Farah M. Al-Askar, Wael W. Mohammed . New soliton wave structure and modulation instability analysis for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. AIMS Mathematics, 2024, 9(9): 26166-26181. doi: 10.3934/math.20241278
[5]	Wafaa B. Rabie, Hamdy M. Ahmed, Taher A. Nofal, Soliman Alkhatib . Wave solutions for the (3+1)-dimensional fractional Boussinesq-KP-type equation using the modified extended direct algebraic method. AIMS Mathematics, 2024, 9(11): 31882-31897. doi: 10.3934/math.20241532
[6]	Islam Samir, Hamdy M. Ahmed, Wafaa Rabie, W. Abbas, Ola Mostafa . Construction optical solitons of generalized nonlinear Schrödinger equation with quintuple power-law nonlinearity using Exp-function, projective Riccati, and new generalized methods. AIMS Mathematics, 2025, 10(2): 3392-3407. doi: 10.3934/math.2025157
[7]	Hajar F. Ismael, Hasan Bulut, Haci Mehmet Baskonus, Wei Gao . Dynamical behaviors to the coupled Schrödinger-Boussinesq system with the beta derivative. AIMS Mathematics, 2021, 6(7): 7909-7928. doi: 10.3934/math.2021459
[8]	Elsayed M. E. Zayed, Mona El-Shater, Khaled A. E. Alurrfi, Ahmed H. Arnous, Nehad Ali Shah, Jae Dong Chung . Dispersive optical soliton solutions with the concatenation model incorporating quintic order dispersion using three distinct schemes. AIMS Mathematics, 2024, 9(4): 8961-8980. doi: 10.3934/math.2024437
[9]	Jinfang Li, Chunjiang Wang, Li Zhang, Jian Zhang . Multi-solitons in the model of an inhomogeneous optical fiber. AIMS Mathematics, 2024, 9(12): 35645-35654. doi: 10.3934/math.20241691
[10]	Abdul Mateen, Ghulam Hussain Tipu, Loredana Ciurdariu, Fengping Yao . Analytical soliton solutions of the Kairat-Ⅱ equation using the Kumar–Malik and extended hyperbolic function methods. AIMS Mathematics, 2025, 10(4): 8721-8752. doi: 10.3934/math.2025400

Abstract

1. Introduction

Because of its potential application in high-speed communication networks, solitons in optical fiber systems have attracted a lot of attention ^[1,2,3]. Solitons are special self-sustaining wave packets that do not significantly deform when traveling great distances. They are perfect for dependable information transmission since they maintain their amplitude and shape while propagating ^[4,5,6]. Nonlinear wave phenomena have captivated the attention of physicists and mathematicians for centuries, playing a pivotal role in understanding the intricate dynamics of diverse physical systems. From the propagation of light in optical fibers to the behavior of waves in plasmas and condensed matter, nonlinear wave equations have emerged as fundamental tools for describing these complex phenomena. There are many fields that seek soliton solutions, such as plasma physics, which examines the Zakharov equation in weakly nonlinear ion-acoustic phenomena ^[7], and nonlinear optics ^[8,9,10]. Among these equations, the nonlinear Schrödinger equation (NLS) stands out as a cornerstone in the study of nonlinear wave dynamics, finding applications in diverse fields such as optics, fluid mechanics ^[11,12], and Bose–Einstein condensates.

The pure-quartic NLS has garnered significant attention in recent years due to its ability to model a variety of physical phenomena, particularly in the context of nonlinear wave dynamics ^[13,14,15], specially for Schrodinger equations ^[16,17,18], and many papers tried to provide numerical methods to solve such as ^[19,20,21]. Unlike traditional cubic NLS equations, which describe standard soliton behavior, the pure-quartic NLS introduces a higher-order nonlinearity that can lead to richer dynamics and more complex wave structures. This nonlinearity allows for the modeling of phenomena such as wave collapse and the formation of localized structures in various media, including optical fibers and Bose-Einstein condensates. Incorporating nonlocal effects into the pure-quartic NLS is essential for accurately describing systems where interactions extend beyond immediate spatial or temporal neighborhoods. Nonlocality is particularly relevant in optical systems, where the propagation of light can be influenced by the nonlocal response of the medium, leading to phenomena that are not captured by local models.

Nonlinear dynamical systems exhibit a wide range of behaviors, including bifurcations, chaotic dynamics, and soliton formation, which are essential in fields such as optics, fluid dynamics, and economic models. Bifurcation theory plays a crucial role in understanding stability transitions and complex system behavior. For instance, previous studies such as ^[22] have analyzed the bifurcation dynamics of the Kopel triopoly model, highlighting the effects of parameter variations on stability. Similarly, the discrete-time Lotka–Volterra model ^[23] has been studied using nonstandard finite difference discretization methods to explore its dynamical properties and bifurcations, providing valuable insights into the role of discretization in nonlinear systems.

Additionally, ensemble classifier design methods, such as those based on the perturbation binary salp swarm algorithm ^[24], have been developed to handle complex classification problems by leveraging perturbation-based optimization techniques. Inspired by these advancements, our study investigates the impact of fractional dispersion and weak nonlocality on soliton formation, exploring how nonlinear interactions influence soliton stability and bifurcation structures within a fractional NLS framework. Understanding these effects contributes to the broader discussion of nonlinear wave phenomena and their stability under varying system parameters. The original model as mentioned in ^[25,26] is:

$\begin{eqnarray} i q_t + \frac{1}{24}C_1 q_{xxxx}+ C_2|q|^2q + C_3q (|q|^2)_{xx} = 0. \end{eqnarray}$

(1.1)

The transition from the classical pure-quartic NLS to its fractional counterpart is motivated by the need to accurately model nonlocal and memory effects in optical wave propagation. In many optical systems, such as highly dispersive fiber optics and photonic crystals, wave dynamics exhibit anomalous dispersion and long-range interactions that cannot be fully captured using integer-order derivatives. The incorporation of a fractional derivative into the pure-quartic NLS serves to generalize the standard model by introducing a history-dependent influence on soliton behavior, so we modify the equation to be:

$\begin{eqnarray} i D_t^{^{\beta}} q + \frac{1}{24}C_1 D_{xxxx}^{\beta} q + C_2|q|^2q + C_3q D_{xx}^{\beta}|q|^2 = 0. \end{eqnarray}$

(1.2)

The model in the context of the $\beta$ -fractional derivative, a generalization of the classical derivative, is discussed in this article. The model equation describes the wave profile of the electric field in an optical fiber, which is represented by the complex-valued function

$q = q(x,t).$

The model incorporates several key elements, including the $\beta$ -fractional derivative $D^\beta_t$ , which governs memory effects and nonlocal dispersion in the system. The coefficients $C_1$ – $C_3$ represent the strength of fourth-order dispersion, Kerr nonlinearity, and nonlocal interactions, respectively. The model also accounts for spatially extended (weak nonlocal) interactions through the term $D^\beta_{xx} q$ . The present study focuses on the pure-quartic NLS with weak nonlocality and fractional dispersion, which effectively captures the influence of fourth-order dispersion on soliton dynamics. However, it does not explicitly account for higher-order dispersion terms such as fifth-order dispersion or additional nonlinear interactions like self-steepening or Raman scattering effects. The authors suggest that future work could extend the current model by incorporating these higher-order effects, leading to a more general fractional NLS. This could be achieved through numerical simulations and analytical techniques, which could provide insights into the stability, collision dynamics, and spectral properties of soliton solutions in highly nonlinear optical systems ^[27,28,29].

The study explores the concept of fractional derivatives, which offer a powerful framework for capturing nonlocal effects in real-world systems. Fractional calculus has numerous applications, including modeling wave propagation through incompressible fluids and shallow water ^[30], the behavior of molecules and materials ^[31,32], and applications in circuit theory and biology ^[33].

The generalized $\beta$ fractional derivative nonlinear Schrödinger (GBFNS) equation presents a generalized framework where soliton properties can be continuously tuned via the fractional derivative parameter $\beta$ . This feature allows for a deeper exploration of the transition between classical solitons and rogue-like structures under varying degrees of nonlocality and dispersion. Furthermore, symbolic computation methods, similar to those used in the derivation of lump solutions, such as the (3+1)-dimensional Hirota–Satsuma–Ito equation ^[34], and lump solutions in spatially symmetric generalized Kadomtsev–Petviashvili models ^[35], could be extended to fractional wave equations to identify novel solution families. Moreover, Darboux transformations have been widely employed to generate $N$ -soliton solutions in integrable systems, including modified Korteweg–de–Vries-type equations ^[36,37]. Incorporating Darboux-based approaches into the fractional-order framework may enable the construction of multi-soliton and breather solutions within the GBFNS equation. Future work could explore how these methods can be adapted to study the interaction dynamics of fractional solitons, including the potential emergence of rogue wave-type structures in nonlocal optical systems. The key contributions of this work are now clearly stated, emphasizing the combination of fractional derivatives, weak nonlocality, and the improved modified extended tanh-function method (IMETM) to obtain soliton solutions. Unlike previous studies, our approach considers higher-order dispersion effects in fractional systems, leading to new insights into soliton dynamics. Additionally, we discuss how fractional-order dispersion modifies soliton behavior, providing a broader understanding of nonlinear wave propagation in optical and complex media.

The paper is structured as follows: Section 2 provides preliminary definitions and properties of the fractional beta derivative. Section 3 presents the IMETM as a powerful technique for solving nonlinear fractional differential equations. Section 4 demonstrates the application of the IMETM to the GBFNS equation, leading to the derivation of novel solitary wave solutions. Section 5 provides a graphical illustration of some obtained solutions showing the characteristics of solutions under different fractional orders. Section 6 discusses the obtained results with previous literature. Finally, Section 7 concludes the paper, summarizing the main findings.

2. Preliminaries

Definition 1. ^[38] Let $f$ be a function such that

$f : [a, \infty) \to \mathbb{R} .$

Then, the beta derivative of a function $f$ is defined as

$D_t^\beta f(t) = \lim\limits_{\epsilon \to 0} \frac{f\left( t + \epsilon \left( t + \frac{1}{\Gamma(\beta)} \right)^{1 - \beta} \right) - f(t)}{\epsilon},$

for all $t \geq a$ , $\beta \in (0, 1]$ . Then, if the limit of the above exists, $f$ is said to be $\beta$ -differentiable.

The fractional derivative $D^{\beta}$ used in this study satisfies several important properties, which we summarize below:

$\begin{align} D^{\beta} \left( af(t) + bg(t) \right) & = a D^{\beta} f(t) + b D^{\beta} g(t), \; \; \; \forall a,b \in \mathbb{R}, \quad \text{(linearity)}, \\[8pt] D^{\beta} \left( f(t) g(t) \right) & = f(t) D^{\beta} g(t) + g(t) D^{\beta} f(t), \quad \text{(product rule)}, \\[8pt] D^{\beta} \left( \frac{f(t)}{g(t)} \right)& = \frac{g(t) D^{\beta} f(t) - f(t) D^{\beta} g(t)}{(g(t))^2}, \quad \text{(quotient rule),} \\[8pt] D^{\beta} f(t)& = (t+ \frac{1}{\Gamma(\beta)})^{1-\beta} \frac{df(t)}{dt}, \quad \text{(fundamental definition)}, \\[8pt] D^{\beta} \left( f(g(t)) \right) & = g'(t)^\beta D^{\beta} f(g(t)), \quad \text{(fractional chain rule)}. \end{align}$

3. The referenced methodology: IMETM

In the works ^[39,40,41], the IMETM methodology is thoroughly articulated. Consider the non-linear partial differential equation (NLPDE) shared below:

$\begin{equation} \mathcal{Y}(U, D_t^{^{\beta}} U , D_{x}^{\beta} U, D_{xx}^{\beta} U, D_{tt}^{\beta} U, ...) = 0. \end{equation}$

(3.1)

Implementing the approach advanced in this study necessitates traversing through enumerated procedures:

Step 1: To begin, use the wave transformation explained below to convert the NLPDE in Eq (3.1) to its corresponding ordinary differential equation (ODE):

$\begin{eqnarray*} \label{eq:wave_transform} U(x, t) = u(z)e^{i\phi},\quad z = \frac{h \left(\frac{1}{\Gamma (\beta )}+x\right)^{\beta }}{\beta },\quad \phi = \theta +\frac{\omega \left(\frac{1}{\Gamma (\beta )}+t\right)^{\beta }}{\beta }. \end{eqnarray*}$

The wave propagation direction is represented by $h$ , the order of fractional derivative is denoted as $\beta$ , wave number is indicated by $\omega$ , and the phase constant can be referred to as $\theta$ . Subsequently, Eq (3.1) transforms into:

$\begin{equation} \mathcal{R}(u, u', u^{''}, u^{(3)}, u^{(4)}, ...) = 0. \end{equation}$

(3.2)

Step 2: Represent the ODE solution with the formula:

$\begin{equation} u(z) = s_0 + \sum\limits_{j = 1}^{N} s_{2j} \frac{1}{\lambda^j(z)} + \sum\limits_{j = 1}^{N} s_{2j-1} \lambda^{j}(z), \end{equation}$

(3.3)

where $s_{2j}$ and $s_{2j-1} \neq 0$ , and at same time, $\lambda(z)$ fulfills the differential equation:

$\begin{equation} \lambda'(z) = \sqrt{p_0 + p_1 \lambda(z) + p_2 \lambda^2(z) + p_3 \lambda^3(z) + p_4 \lambda^4(z) }. \end{equation}$

(3.4)

Step 3: The choice of $N$ in Eq (3.3) hinges on the balance method, derived from analysis on Eq (3.2).

3.1. Balance principle for determining $N$

To ensure a consistent solution structure, we apply the balance principle, which determines the highest power $N$ in the assumed series solution. This principle equates the dominant terms in the highest-order derivative and the nonlinear terms. The balance principle requires comparing:

(1) The highest-order derivative term in the governing equation.

(2) The highest-order nonlinear term in the equation.

Let $m$ be the highest order of differentiation appearing in the equation; $p$ be the highest exponent of $q$ in the nonlinear terms; $k$ be an integer shift factor that depends on the structure of the nonlinear term (e.g., if a nonlinear term involves a derivative of $q$ , $k$ accounts for that differentiation). From the governing equation, the highest-order derivative term contributes a leading order of $N - m$ , while the highest nonlinear term contributes an order of $pN + k$ . Setting these powers equal gives the balance condition:

$\begin{equation} N - m = pN + k. \nonumber \end{equation}$

Solving for $N$ , we obtain

$\begin{equation} N = \frac{m - k}{1 - p}. \nonumber \end{equation}$

For a valid solution, $N$ must be a non-negative integer, which imposes constraints on the system parameters. Applying the balance condition ensures the validity of the solution ansatz and helps determine the appropriate power series expansion.

Step 4: Create a set of nonlinear algebraic equations by inserting the proposed solutions into Eq (3.3) and utilizing Eq (3.4) to substitute in Eq (3.2). Afterwards, ensure that each coefficient of $\lambda^h(z)$ for $h = 0, 1, 2, \ldots$ equals zero through equating.

Step 5: Use software like Mathematica to solve the algebraic system, determining the coefficients $s_{2j}, s_{2j-1}$ , $\omega$ , and $k$ .

Step 6: Distinct numerical values for $p_0$ – $p_4$ yield different types of solutions:

Case 1. With $p_0 = p_1 = p_3 = 0$ ,

$\begin{align*} \lambda(z)& = \sqrt{-\frac{p_2}{p_4}} \text{sech}(\sqrt{p_2} z), \quad p_2 > 0, \quad p_4 < 0,\\ \lambda(z)& = \sqrt{-\frac{p_2}{p_4}} \text{sec}(\sqrt{-p_2} z),\quad p_2 < 0, \quad p_4 > 0. \end{align*}$

Case 2. For $p_1 = p_3 = 0$ , $p_0 = \frac{p_2^2}{4p_4}$ ,

$\begin{align*} \lambda(z)& = \sqrt{\frac{-p_2}{2p_4}} \text{tanh}(\sqrt{\frac{-p_2}{2}} z), \quad p_2 < 0, \quad p_4 > 0,\\ \lambda(z)& = \sqrt{\frac{p_2}{2p_4}} \text{tan}(\sqrt{\frac{p_2}{2}} z), \quad p_2 > 0, \quad p_4 > 0. \end{align*}$

Case 3. For $p_4 = p_3 = 0$ ,

$\begin{align*} \lambda(z)& = \frac{-p_1}{2p_2} \text{sinh}(2\sqrt{p_2} z) - \frac{p_1}{2 p_2}, \quad p_2 > 0, \quad p_0 = 0,\\ \lambda(z) & = e^{\sqrt{p_2} z} - \frac{p_1}{2 p_2}, \quad p_2 > 0, \quad p_0 = \frac{p_1^2}{4p_2}. \end{align*}$

Case 4. When $p_{0} = p_{1} = 0$ ,

$\begin{equation*} \lambda(z) = \frac{-p_{2}}{p_{3}} (\tanh(\frac{\sqrt{p_2}}{2} z) + 1), \quad p_2 > 0, \quad p_4 = \frac{p^2_3}{4p_2}. \end{equation*}$

Case 5. For the condition $p_{1} = p_{3} = 0$ ,

$\begin{eqnarray*} &&\lambda(z) = \sqrt{\frac{-p_2 m^{2}}{p_4(2m^{2}-1)}} \text{cn}(\sqrt{\frac{p_2}{(2m^{2}-1)}} z),\quad p_2 > 0,\; \; p_4 < 0 , \; \; p_0 = \frac{p_2^{2} m^{2}(1-m^{2})}{p_4 (2m^{2}-1)^{2}}, \\ &&\lambda(z) = \sqrt{-\frac{m^{2}}{p_4(2-m^{2})}} \text{dn}(\sqrt{\frac{p_2}{(2-m^{2})}} z),\quad p_2 > 0,\; \; p_4 < 0 ,\; \; p_0 = \frac{p_2^{2} (1-m^{2})}{p_4 (2-m^{2})^{2}}, \\ &&\lambda(z) = \sqrt{-\frac{p_{2}m^{2}}{p_4(1+m^{2})}} \text{sn}(\sqrt{\frac{-p_2}{(1+m^{2})}} z),\quad p_2 < 0,\; \; p_4 > 0 ,\; \; p_0 = \frac{p_2^{2} m^{2}}{p_4 (1+m^{2})^{2}}. \end{eqnarray*}$

The provides clarity on how the Jacobi elliptic functions behave as the parameter $m$ approaches 0 or 1. As seen some of them oscillate between trigonometric functions and hyperbolic ones.

Table 1. Behavior of Jacobi elliptic functions as

$m$ approaches 0 and 1.

Function	Limit as $m \to 0$	Limit as $m \to 1$
$\text{sn}(u, m)$	$\text{sn}(u, 0) = \sin(u)$	$\text{sn}(u, 1) = \tanh(u)$
$\text{cn}(u, m)$	$\text{cn}(u, 0) = \cos(u)$	$\text{cn}(u, 1) = 0$
$\text{dn}(u, m)$	$\text{dn}(u, 0) = 1$	$\text{dn}(u, 1) = \text{sech}(u)$
$\text{sc}(u, m)$	$\text{sc}(u, 0) = 1$	$\text{sc}(u, 1)$ = $\infty$
$\text{cs}(u, m)$	$\text{cs}(u, 0) = 1$	$\text{cs}(u, 1) = 0$
$\text{ns}(u, m)$	$\text{ns}(u, 0) = 1$	$\text{ns}(u, 1) = 1$
$\text{nc}(u, m)$	$\text{nc}(u, 0) = 1$	$\text{nc}(u, 1) = 0$

| Show Table

DownLoad: CSV

These transformations provide a clearer interpretation of the solutions in the context of both periodic and hyperbolic behaviors, enhancing the understanding of the dynamics captured by the $\beta$ -fractional derivative NLS.

Step 7. Integrating the multiple coefficients $s_j$ and $s_{j+1}$ with the solutions obtained, numerous solutions to Eq (3.1) are extracted.

The IMETM has proven to be a valuable tool for deriving soliton solutions for partial differential equations. One of the key strengths of the IMETM is its ability to find a wide range of soliton solutions, including bright, dark, and singular solitons, as well as periodic solutions and trigonometric solutions. This is compared to many other methods such as Kudryashov method, sine-Gordon expansion, or variational approaches that may struggle to obtain such a diverse set of solutions. However, it is also important to acknowledge the limitations of the IMETM. When the balance parameter $N$ is large, the method can become increasingly complex and difficult to apply. This is because the number of terms in the solution expansion grows rapidly with $N$ , making it challenging to obtain a tractable solution. In such cases, other methods may be more suitable, and a combination of approaches may be necessary to fully understand the behavior of the system.

4. Applications of the studied model

We seek to obtain exact soliton solutions for Eq (1.2) through the following approach:

$\begin{eqnarray} q(x,t) = Q(z) e^{i\phi},\; \; \; z = \frac{h \left(\frac{1}{\Gamma (\beta )}+x\right)^{\beta }}{\beta },\; \; \; \text{and}\; \; \; \phi = \theta +\frac{\omega \left(\frac{1}{\Gamma (\beta )}+t\right)^{\beta }}{\beta }. \end{eqnarray}$

(4.1)

The phase term $\phi$ primarily accounts for wave oscillations and soliton velocity. However, in fractional systems, this phase function plays a more complex role. The fractional exponent in $\phi$ modifies the dispersion relation, making the soliton velocity explicitly dependent on the fractional order $\beta$ . This modification introduces anomalous dispersion effects, enabling solitons to propagate at non-integer power scaling rates rather than the linear propagation observed in classical systems. Unlike standard solitons, which obey a fixed dispersion-speed relationship, fractional solitons exhibit memory-dependent velocity adjustments, leading to variable soliton width and amplitude. As a result, the phase term ensures that the soliton profile dynamically adapts to fractional-order dispersion, allowing for enhanced stability under varying nonlocal and dispersive conditions. The parameter $h$ is a scaling factor that influences the soliton amplitude and width, playing a crucial role in shaping the soliton's spatial structure. The frequency parameter $\omega$ determines the propagation speed and oscillation rate of the soliton, directly affecting its phase evolution. The constant $\theta$ represents the initial phase shift, which influences the soliton's initial position and interaction with the surrounding medium. These parameters collectively govern the soliton's behavior in a fractional system, ensuring adaptability under different dispersion and nonlocality conditions.

By inserting Eq (4.1) into Eq (1.2), we are able to transform the fractional NLPDE into a ODE, resulting in the following outcome:

$\begin{align} C_1 h^4 Q^{(4)}(z)+48 C_3 h^2 Q(z)^2 Q''(z)-24 Q(z) \left(\omega -2 C_3 h^2 Q'(z)^2\right)+24 C_2 Q(z)^3 = 0. \end{align}$

(4.2)

To carry out the suggested technique, it is essential to compute an integer value for $N$ . The calculation of this value involves balancing $Q^{(4)}$ with $Q(z)^2 Q''(z)$ in Eq (4.2) and determining that

$N = 1.$

Afterwards, we can express the solution to the resulting ordinary differential equation as follows:

$\begin{equation} Q(z) = s_0 + s_1 \lambda(z) + \frac{s_2}{\lambda(z)}. \end{equation}$

(4.3)

Equations (4.3) and (3.4) are substituted into Eq (4.2) to solve Eq (1.2). The system of nonlinear algebraic equations that emerges from equating the coefficients of $\lambda(z)$ to zero is handled by Mathematica software packages. To obtain exact soliton solutions, we employed Mathematica to solve the resulting algebraic systems derived from the IMETM. The computational process involved symbolic manipulation, equation simplification, and the use of polynomial system solvers to determine the unknown parameters. The complexity of solving these algebraic systems depends on the number of unknown parameters and the degree of the nonlinear terms. In our case, the symbolic computation required solving polynomial equations of a degree of 6, with multiple coupled nonlinear constraints. The worst-case computational complexity for polynomial equation solving is exponential in the number of variables, but Mathematica optimizes this process using Gröbner bases and elimination techniques. On a standard machine (Intel Core i5-7200u, 8GB RAM), the execution time for solving the algebraic system varied between 1 and 5 minutes, depending on the specific parameter choices and system constraints. Larger solution spaces and higher-order polynomial terms required more computation time due to increased symbolic simplifications.

This yields the following outcomes for Eq (1.2):

Case 1. $p_0 = p_1 = p_3 = 0$ ,

$\begin{eqnarray*} s_0 = 0,\; \; \; s_2 = 0, \; \; \; \omega = -\frac{C_3 h^2 p_2^2 s_1^2}{4 p_4},\; \; \; p_4 = -\frac{6 C_3 s_1^2}{C_1 h^2},\; \; \; p_2 = \frac{C_2}{C_3 h^2}. \nonumber \end{eqnarray*}$

For Eq (1.2), we are able to provide both a singular solution and a bright soliton solution:

$\begin{align} q_1(x,t) & = \sqrt{\frac{C_1 C_2}{6 C_3^2}} \, \text{sech}\left(\frac{1}{\beta} \sqrt{\frac{C_2}{C_3}} \left( \frac{1}{\Gamma(\beta)} + x \right)^{\beta} \right) e^{\mathrm{i} \phi} , \; \; \; \text{for}\; C_1 C_2 > 0, \, C_2 C_3 > 0, \end{align}$

(4.4)

$\begin{align} q_2(x,t)& = \sqrt{\frac{C_1 C_2}{6 C_3^2}} \, \text{sec}\left(\frac{1}{\beta} \sqrt{\frac{-C_2}{C_3}} \left( \frac{1}{\Gamma(\beta)} + x \right)^{\beta} \right) e^{\mathrm{i} \phi} ,\; \; \; \text{for}\; C_1 C_2 > 0, \; C_2 C_3 < 0. \end{align}$

(4.5)

Case 2. $p_1 = p_3 = 0, p_0 = \frac{p_2^2 }{4p_4}$ ,

$\begin{eqnarray*} s_0 = 0, \; \; \; s_1 = \frac{2 p_4 s_2}{p_2},\; \; \; \omega = -4 C_3 h^2 p_4 s_2^2, \; \; \; p_2 = -\frac{C_2}{2 C_3 h^2}, \; \; \; p_4 = -\frac{C_1 h^2 p_2^2}{24 C_3 s_2^2}. \nonumber \end{eqnarray*}$

Following this, singular soliton and singular periodic solutions can be offered for Eq (1.2):

$\begin{eqnarray} q_3(x,t) = \sqrt{\frac{- C_1 C_2}{6 C_3^2}} \, \text{csch}\left(\frac{1}{\beta} \sqrt{\frac{C_2}{C_3}} \left( \frac{1}{\Gamma(\beta)} + x \right)^{\beta} \right) e^{\mathrm{i} \phi}, \end{eqnarray}$

(4.6)

for $C_2 C_3 > 0, \; C_1 C_2 < 0,$

$\begin{eqnarray} q_4(x,t) = \sqrt{\frac{C_1 C_2}{6 C_3^2}} \, \text{csc}\left(\frac{1}{\beta} \sqrt{\frac{C_2}{C_3}} \left( \frac{1}{\Gamma(\beta)} + x \right)^{\beta} \right) e^{\mathrm{i} \phi} , \end{eqnarray}$

(4.7)

for $C_2 C_3 < 0, \; C_1 C_2 > 0.$

Case 3. $p_3 = p_4 = 0$ . For $p_0 = 0$ , we get this set of solutions:

$\begin{eqnarray*} s_0 = \frac{p_1 s_1}{2 p_2}, \; \; \; s_2 = 0, \; \; \; \omega = \frac{C_1 h^4 p_2^3-12 C_3 h^2 p_1^2 s_1^2}{24 p_2}, \; \; \; C_2 = -4 C_3 h^2 p_2, \end{eqnarray*}$

and for

$p_0 = \dfrac{p_1^2}{4p_2} ,$

we get this set of solutions:

$\begin{eqnarray*} s_0 = \frac{p_2 s_2}{p_1}, \; \; \; s_1 = 0,\; \; \; \omega = -\frac{C_3 h^2 p_2^3 s_2^2}{2 p_1^2}, \; \; \; C_1 = -\frac{24 C_3 p_2 s_2^2}{h^2 p_1^2}, \; \; \; C_2 = -\frac{1}{2} C_3 h^2 p_2. \end{eqnarray*}$

Subsequently, hyperbolic and exponential solutions can be presented for Eq (1.2):

$\begin{align} q_5(x,t) = \frac{p_1 s_1}{2 p_2} \sinh \left(\frac{2 h \sqrt{p_2} \left(\frac{1}{\Gamma (\beta )}+x\right)^{\beta }}{\beta }\right) e^{\mathrm{i} \phi} , \end{align}$

(4.8)

for $p_0 = 0, p_2 > 0,$

$\begin{align} q_6(x,t) = s_2 \left(\frac{1}{e^{\frac{h \sqrt{p_2} \left(\frac{1}{\Gamma (\beta )}+x\right)^{\beta }}{\beta }}-\frac{p_1}{2 p_2}}+\frac{p_2}{p_1}\right) e^{\mathrm{i} \phi} , \end{align}$

(4.9)

for

$p_2 > 0,\; \; \; p_0 = \dfrac{p_1^2}{4p_2}.$

Case 4. $p_0 = p_1 = 0$ ,

$\begin{eqnarray*} s_0 = \frac{\sqrt{p_2 p_4} s_1}{2 p_4},\; \; \; s_2 = 0,\; \; \; \omega = -\frac{C_3 h^2 p_2^2 s_1^2}{8 p_4},\; \; \; p_4 = -\frac{6 C_3 s_1^2}{C_1 h^2},\; \; \; p_2 = -\frac{2 C_2}{C_3 h^2}. \end{eqnarray*}$

Following this, a dark soliton solution is demonstrated in Eq (1.2),

$\begin{eqnarray} q_7(x,t) & = \frac{\sqrt{C_1 C_2}}{2 C_3 \sqrt{3}} \, \tanh \left( \frac{1}{\beta} \sqrt{-\frac{C_2}{2 C_3}} \left( \frac{1}{\Gamma(\beta)} + x \right)^{\beta} \right) e^{\mathrm{i} \phi} , \end{eqnarray}$

(4.10)

for

$p_3^2 = 4p_2 p_4,\; \; \; C_2 C_3 < 0 ,\; \; \; C_1 C_2 > 0.$

Case 5. $p_3 = p_1 = 0$ ,

$\begin{eqnarray*} s_0 = 0, \; \; \; s_2 = 0, \; \; \; \omega = -\frac{C_3 h^2 \left(p_2^2+4 p_0 p_4\right) s_1^2}{4 p_4}, \; \; \; C_1 = -\frac{6 C_3 s_1^2}{h^2 p_4}, \; \; \; C_2 = C_3 h^2 p_2. \end{eqnarray*}$

After that, Eq (1.2) can have Jacobi elliptic solutions as shown:

$\begin{align} q_8(x ,t) = \sqrt{\frac{C_1 C_2 m^2}{6 C_3^2 \left(2 m^2-1\right)}} \text{cn}\left(\frac{\left(x+\frac{1}{\Gamma (\beta )}\right)^{\beta } \sqrt{\frac{C_2}{C_3}}}{\beta }, m\right) e^{\mathrm{i} \phi} , \end{align}$

(4.11)

for

$\frac{C_1 C_2 }{2 m^2-1} > 0,\; \; \; \frac{C_2}{C_3} > 0,\; \; \; 0 < m < 1;$

$\begin{align} q_9(x ,t) = \sqrt{\frac{C_1 m^2}{6 C_3 \left(2-m^2\right)}} \text{dn}\left(\frac{\left(x+\frac{1}{\Gamma (\beta )}\right)^{\beta } \sqrt{\frac{C_2}{\left(2-m^2\right) C_3}}}{\beta }, m\right) e^{\mathrm{i} \phi} , \end{align}$

(4.12)

for

$\frac{C_1 }{ C_3 } > 0, \; \; \; \frac{C_2}{ C_3} > 0, \; \; \; 0 < m < 1;$

$\begin{align} q_{10}(x ,t) = \sqrt{\frac{C_1 C_2 m^2}{6 C_3^2 \left(m^2+1\right)}} \text{sn}\left(\frac{\left(x+\frac{1}{\Gamma (\beta )}\right)^{\beta } \sqrt{-\frac{C_2}{\left(m^2+1\right) C_3}}}{\beta }, m\right) e^{\mathrm{i} \phi}, \end{align}$

(4.13)

for

$C_1 C_2 > 0 , \; \; \; \frac{C_2}{ C_3} < 0 ,\; \; \; 0 < m < 1.$

5. Graphical visualization of solutions

Graphical representations are utilized to showcase the features of the obtained results for selected solutions. In , was simulate the bright optical soliton solution described by Eq (4.4) with different values of $\beta$ , using

$C_1 = 1.65,\; \; C_2 = 1,\; \; C_3 = 2.71.$

represents contour plots for different values of $\beta$ to compare between the widths of solitons.

Figure 1. Graphical visualization of hyperbolic solution Eq (4.4).

DownLoad: Full-Size Img PowerPoint

Figure 2. Graphical contour plot of hyperbolic solution Eq (4.4).

DownLoad: Full-Size Img PowerPoint

Similarly, Figure 3 illustrates the dark optical soliton solution of Eq (4.10) with

$C_1 = 2.48,\; \; C_2 = 0.75,\; \; C_3 = -1.09,$

for various $\beta$ values. represents contour plots for different values of $\beta$ to compare between the widths of solitons for the dark soliton solution.

Figure 3. Simulation of another hyperbolic solution Eq (4.10).

DownLoad: Full-Size Img PowerPoint

Figure 4. Simulation of contour plot solution Eq (4.10).

DownLoad: Full-Size Img PowerPoint

Additionally, Figure 5 presents a graph of the periodic solution of Eq (4.11) with

$C_1 = 1.35,\; \; C_2 = 1,\; \; C_3 = 2.61,\; \; \; \text{and}\; \; \; m = 0.84,$

demonstrating the effect of different $\beta$ values. The 2D graphs are drawn at $t = 4$ . presents a graph of the singular soliton solution of Eq (4.6) with $C_1 = -1.15, \; \; C_2 = 1.14, \; \; C_3 = 1.14,$ demonstrating the effect of different $\beta$ values. The 2D graphs are drawn at $t = 4$ .

Figure 5. Visualization of periodic solution Eq (4.11).

DownLoad: Full-Size Img PowerPoint

Figure 6. Visualization of singular soliton solution Eq (4.6).

DownLoad: Full-Size Img PowerPoint

Figure 7 presents a graph of the trigonometric solution of Eq (4.7) with

$C_1 = 1.38,\; \; C_2 = 1.14,\; \; C_3 = 1.5,$

demonstrating the effect of different $\beta$ values. The 2D graphs are drawn at $t = 4$ .

Figure 7. Visualization of trigonometric solution Eq (4.7).

DownLoad: Full-Size Img PowerPoint

The capability to manipulate and fine-tune soliton properties and make use of the fractional derivative parameter $\beta$ creates new opportunities for optical communication system design and optimization for photonic devices. The graphical representations show how this parameter affects soliton solutions, highlighting the significance of including fractional calculus in modeling dispersive and nonlocal systems. One of the key findings of this study is that small variations in the fractional order parameter $\beta$ lead to significant changes in soliton characteristics, including amplitude, width, and propagation speed. This sensitivity poses potential challenges for designing and optimizing optical systems, as minor fluctuations in $\beta$ could impact soliton stability and interaction dynamics. However, this property also provides a distinct advantage: the ability to fine-tune soliton properties by adjusting $\beta$ , which is not feasible in classical integer-order models. In optical fiber communication systems, such control could allow for adaptive dispersion management and tunable soliton-based signal transmission. Additionally, fractional-order models enable a broader range of soliton behaviors, making them highly versatile for photonic device applications. To address the potential challenges of sensitivity to $\beta$ , future research could explore stabilization techniques such as external feedback control, engineered dispersion profiles, or hybrid integer-fractional models that balance flexibility with robustness. Experimental validation of fractional soliton dynamics in structured optical media would also provide insights into practical implementation strategies.

6. Discussion

The importance of using fractional calculus to model dispersive and nonlocal systems lies in their complex dynamics, characterized by non-integer orders. Dispersive systems exhibit frequency-dependent propagation, while nonlocal systems show interactions that extend beyond immediate spatial or temporal neighborhoods.

Previous studies, such as ^[25,26], explored soliton dynamics in traditional models without fractional derivatives and found that soliton stability could be maintained under certain conditions. However, these models did not capture the nuanced behavior observed in fractional calculus. Our results show that the parameter $\beta$ significantly influences soliton characteristics, allowing better control over their amplitude and shape.

Graphical representations of soliton solutions obtained through fractional calculus provide insights into these systems, aiding the development of advanced photonic technologies. For instance, at ( $\beta = 1$ ) Eq (4.4) yields a complete bright soliton, while varying $\beta$ results in distinct wave shapes. Similarly, Eq (4.10) produces a complete dark soliton at ( $\beta = 1$ ) with varying wave forms at other $\beta$ values. Moreover, Eq (4.11) indicates that while $\beta$ affects wave propagation, it does not alter wave shape, contrasting with findings in ^[25,26], which maintained shape consistency regardless of parameter variations.

The soliton solutions obtained in this study exhibit distinct differences in amplitude, shape, and stability compared to previously known solutions in ^[25,26]. One of the key distinctions arises from the inclusion of the $\beta$ -fractional derivative, which introduces additional flexibility in controlling soliton characteristics. In classical pure-quartic soliton solutions, such as those in ^[25], soliton profiles remain nearly symmetric with a fixed amplitude and shape determined by the Kerr nonlinearity and fourth-order dispersion. In contrast, our results show that fractional-order dispersion, controlled by $\beta$ , allows for tunability in both soliton amplitude and width. Specifically, as $\beta$ decreases from 1, solitons exhibit an increase in peak intensity while becoming more localized, which is a unique feature not observed in integer-order models. Furthermore, compared to ^[26], which considers weak nonlocal effects in an integer-order framework, our solutions reveal a broader variety of soliton types, including bright, dark, and singular solitons, as well as Jacobi elliptic function solutions. The presence of fractional nonlocality results in more gradual soliton decay and enhanced localization effects. Graphical comparisons indicate that the solitons obtained here exhibit greater robustness against perturbations, suggesting improved stability under weak nonlocal interactions. Additionally, the phase structure of solitons in our model is more dynamically adjustable, allowing for better control over their propagation behavior. Unlike the prior works where soliton characteristics were primarily dictated by the Kerr coefficient and fourth-order dispersion, our results demonstrate that fractional derivatives enable a continuous transformation between different soliton regimes, making the solutions more adaptable to practical optical applications. These findings highlight the potential advantages of fractional-order models in designing and optimizing optical communication systems.

7. Conclusions

The analytical solutions derived in this study provide valuable insights into the dynamics of the GBFNS, particularly in optical fiber systems with weak nonlocal effects. The use of the IMETM has yielded a variety of soliton solutions, including bright, dark, and singular solitons, as well as hyperbolic, trigonometric, and Jacobi elliptic solutions. Notably, the parameter $\beta$ significantly influences the shape, amplitude, stability, and interaction behaviors of the solitons.

The fractional-order parameter, $\beta$ , controls the impact on the solitons' characteristics. As $\beta$ approaches unity, the fractional derivative converges to the classical integer-order derivative, and the solutions resemble those obtained from traditional methods. However, for fractional orders far from unity, the solutions exhibit distinct features, highlighting the nonlocal effects in wave dynamics.

The derived solutions include various soliton types, each with unique characteristics. Bright solitons show a localized peak, dark solitons have a dip in the background intensity, and singular solitons possess an infinite peak at a specific point. The diversity of these solutions underscores the complexity of the GBFNS equation in modeling nonlinear wave phenomena.

Comparing our solutions to those from traditional integer-order derivatives reveals significant differences ^[25,26]. Fractional-order derivative solutions demonstrate higher localization, faster propagation, and more complex interactions. This comparison emphasizes the importance of fractional derivatives in modeling real-world systems with substantial nonlocal effects.

The derived solitons in this study have significant real-world implications, particularly in nonlinear optics and fiber optic communications. The bright and dark solitons can be utilized in dispersion-managed fiber systems to maintain signal integrity over long distances, reducing transmission losses and distortion. Additionally, the presence of fractional dispersion makes these solitons highly relevant for mode-locked lasers, where tunable pulse durations are essential for ultrafast optical signal processing. Beyond optics, the findings have applications in plasma physics and Bose–Einstein condensates, where similar nonlinear Schrödinger-type equations govern wave propagation. The inclusion of weak nonlocality and fractional derivatives also introduces unique dispersion properties that could be exploited in metamaterials and engineered optical systems for advanced wave manipulation.

Furthermore, the soliton stability analysis ensures their robustness, making them suitable for optical computing and photonic crystal applications. By addressing these real-world connections, our study bridges the gap between mathematical soliton theory and practical implementation in modern photonics and wave dynamics.

While this method produces diverse solutions, including bright, dark, and singular solitons, large values of $N$ , determined by the balance rule, lead to more complex systems that are challenging to solve.

Author contributions

Mahmoud Soliman: formal analysis, software, writing—original draft; Hamdy M. Ahmed: methodology, validation, writing—review & editing; Niveen Badra: supervision, visualization; M. Elsaid Ramadan: investigation, writing—original draft; Islam Samir: investigation, writing—review & editing; Soliman Alkhatib: conceptualization, writing—review & editing. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, et al., Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), 1–35.
[2]	H, Adams, A. Tausz, M. Vejdemo-Johansson, JavaPlex: A research software package for persistent (co) homology, Mathematical Software–ICMS 2014, Seoul, South Korea, 2014,129–136. https://doi.org/10.1007/978-3-662-44199-2_23
[3]	D. N. Arnold, R. S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), 1–155. https://doi.org/10.1017/S0962492906210018 doi: 10.1017/S0962492906210018
[4]	U. Bauer, Ripser: efficient computation of vietoris-rips persistence barcodes, J. Appl. Comput. Topology, 5 (2021), 391–423. https://doi.org/10.1007/s41468-021-00071-5 doi: 10.1007/s41468-021-00071-5
[5]	P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77–102.
[6]	H. Cai, C. Shen, T. Y. Jian, X. J. Zhang, T. Chen, X. Q. Han, et al., Carsidock: A deep learning paradigm for accurate protein-ligand docking and screening based on large-scale pre-training, Chem. Sci., 15 (2024), 1449–1471. https://doi.org/10.1039/D3SC05552C doi: 10.1039/D3SC05552C
[7]	Z. X. Cang, L. Mu, G.-W. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLoS Comput. Biol., 14 (2018), e1005929. https://doi.org/10.1371/journal.pcbi.1005929 doi: 10.1371/journal.pcbi.1005929
[8]	Z. X. Cang, L. Mu, K. D. Wu, K. Opron, K. Xia, G.-W. Wei, A topological approach for protein classification, Computational and Mathematical Biophysics, 3 (2015), 140–162. https://doi.org/10.1515/mlbmb-2015-0009 doi: 10.1515/mlbmb-2015-0009
[9]	Z. X. Cang, G.-W. Wei, Topologynet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLoS Comput. Biol., 13 (2017), e1005690. https://doi.org/10.1371/journal.pcbi.1005690 doi: 10.1371/journal.pcbi.1005690
[10]	Z. X. Cang, G.-W. Wei, Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction, Int. J. Numer. Meth. Bio., 34 (2018), e2914. https://doi.org/10.1002/cnm.2914 doi: 10.1002/cnm.2914
[11]	G. Carlsson, Topology and data, B. Am. Math. Soc., 46 (2009), 255–308.
[12]	D. Chen, J. Liu, G.-W. Wei, multi-scale topology-enabled structure-to-sequence transformer for protein-ligand interaction predictions, Nat. Mach. Intell., 6 (2024), 799–810. https://doi.org/10.1038/s42256-024-00855-1 doi: 10.1038/s42256-024-00855-1
[13]	D. Chen, J. X. Zheng, G.-W. Wei, F. Pan, Extracting predictive representations from hundreds of millions of molecules, J. Phys. Chem. Lett., 12 (2021), 10793–10801. https://doi.org/10.1021/acs.jpclett.1c03058 doi: 10.1021/acs.jpclett.1c03058
[14]	H. Chen, Y. Zhang, W. H. Zhang, P. X. Liao, K. Li, J. L. Zhou, et al., Low-dose CT via convolutional neural network, Biomed. Opt. Express, 8 (2017), 679–694. https://doi.org/10.1364/BOE.8.000679 doi: 10.1364/BOE.8.000679
[15]	J. H. Chen, Y. C. Qiu, R. Wang, G.-W. Wei, Persistent laplacian projected omicron ba. 4 and ba. 5 to become new dominating variants, Comput. Biol. Med., 151 (2022), 106262. https://doi.org/10.1016/j.compbiomed.2022.106262 doi: 10.1016/j.compbiomed.2022.106262
[16]	J. H. Chen, R. Wang, M. L. Wang, G.-W. Wei, Mutations strengthened SARS-CoV-2 infectivity, J. Mol. Biol., 432 (2020), 5212–5226. https://doi.org/10.1016/j.jmb.2020.07.009 doi: 10.1016/j.jmb.2020.07.009
[17]	J. H. Chen, G.-W. Wei, Omicron BA. 2 (B. 1.1. 529.2): high potential for becoming the next dominant variant, J. Phys. Chem. Lett., 13 (2022), 3840–3849. https://doi.org/10.1021/acs.jpclett.2c00469 doi: 10.1021/acs.jpclett.2c00469
[18]	J. H. Chen, R. D. Zhao, Y. Y. Tong, G.-W. Wei, Evolutionary de rham-hodge method, Discrete Cont. Dyn-B, 26 (2021), 3785–3821. https://doi.org/10.3934/dcdsb.2020257 doi: 10.3934/dcdsb.2020257
[19]	M. Desbrun, E. Kanso, Y. Y. Tong, Discrete differential forms for computational modeling, In: ACM SIGGRAPH 2006 Courses, New York: Association for Computing Machinery, 2006, 39–54. https://doi.org/10.1145/1185657.1185665
[20]	T. K. Dey, F. T. Fan, Y. S. Wang, Computing topological persistence for simplicial maps, In: Proceedings of the thirtieth annual symposium on Computational geometry, New York: Association for Computing Machinery, 2014, 345–354. https://doi.org/10.1145/2582112.2582165
[21]	J. Dodziuk, Finite-difference approach to the hodge theory of harmonic forms, Am. J. Math., 98 (1976), 79–104. https://doi.org/10.2307/2373615 doi: 10.2307/2373615
[22]	R. Dong, A faster algorithm of up persistent laplacian over non-branching simplicial complexes, 2024, arXiv: 2408.16741. https://doi.org/10.48550/arXiv.2408.16741
[23]	H. Edelsbrunner, J. Harer, Persistent homology-a survey, In: Surveys on discrete and computational geometry: twenty years later, Singapore: Contemporary Mathematics, 2008, 257–282. https://doi.org/10.1090/conm/453/08802
[24]	P. G. Francoeur, T. Masuda, J. Sunseri, A. Jia, R. B. Iovanisci, I. Snyder, et al., Three-dimensional convolutional neural networks and a cross-docked data set for structure-based drug design, J. Chem. Inf. Model., 60 (2020), 4200–4215. https://doi.org/10.1021/acs.jcim.0c00411 doi: 10.1021/acs.jcim.0c00411
[25]	K. O. Friedrichs, Differential forms on riemannian manifolds, Commun. Pur. Appl. Math., 8 (1955), 551–590. https://doi.org/10.1002/cpa.3160080408 doi: 10.1002/cpa.3160080408
[26]	A. Gaulton, A. Hersey, M. Nowotka, A.P. Bento, J. Chambers, D. Mendez, P. Mutowo, F. Atkinson, L.J. Bellis, E. Cibrián-Uhalte, M. Davies, The ChEMBL database in 2017, Nucleic acids research, 45 (2017), D945–D954. https://doi.org/10.1093/nar/gkw1074 doi: 10.1093/nar/gkw1074
[27]	R. Ghrist, Barcodes: The persistent topology of data, B. Am. Math. Soc., 45 (2008), 61–75. https://doi.org/10.1090/S0273-0979-07-01191-3 doi: 10.1090/S0273-0979-07-01191-3
[28]	A. B. Gülen, F. Mémoli, Z. C. Wan, Y. S. Wang, A generalization of the persistent laplacian to simplicial maps, The 39th International Symposium on Computational Geometry (SoCG 2023), Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2023, 37: 1–37: 17. https://doi.org/10.4230/LIPIcs.SoCG.2023.37
[29]	J. Irwin and B. Shoichet, ZINC - a free database of commercially available compounds for virtual screening, Journal of chemical information and modeling, 45 (2005), 177–182. https://doi.org/10.1021/ci049714+ doi: 10.1021/ci049714+
[30]	M. Khovanov, A categorification of the jones polynomial, Duke Math. J., 101 (2000), 359–426. https://doi.org/10.1215/S0012-7094-00-10131-7 doi: 10.1215/S0012-7094-00-10131-7
[31]	S. Kim, P.A. Thiessen, E.E. Bolton, J. Chen, G. Fu, A. Gindulyte, L. Han, J. He, S. He, B.A. Shoemaker, J. Wang, PubChem substance and compound databases, Nucleic acids research, 44 (2016), D1202–D1213. https://doi.org/10.1093/nar/gkv951 doi: 10.1093/nar/gkv951
[32]	Z. M. Lin, H. Akin, R. Rao, B. Hie, Z. K. Zhu, W. T. Lu, et al., Language models of protein sequences at the scale of evolution enable accurate structure prediction, BioRxiv, 2022 (2022), 500902. https://doi.org/10.1101/2022.07.20.500902 doi: 10.1101/2022.07.20.500902
[33]	J. Liu, J. Y. Li, J. Wu, The algebraic stability for persistent laplacians, 2023, arXiv: 2302.03902. https://doi.org/10.48550/arXiv.2302.03902
[34]	J.-B. Liu, X. Wang, J. D. Cao, The coherence and properties analysis of balanced $2^{p}$ -ary tree networks, IEEE T. Netw. Sci. Eng., 11 (2024), 4719–4728. https://doi.org/10.1109/TNSE.2024.3395710 doi: 10.1109/TNSE.2024.3395710
[35]	J.-B. Liu, X. Zhang, J. D. Cao, L. P. Chen, Mean first-passage time and robustness of complex cellular mobile communication network, IEEE T. Netw. Sci. Eng., 11 (2024), 3066–3076. https://doi.org/10.1109/TNSE.2024.3358369 doi: 10.1109/TNSE.2024.3358369
[36]	R. Liu, X. Liu, J. Wu, Persistent path-spectral (PPS) based machine learning for protein-ligand binding affinity prediction, J. Chem. Inf. Model., 63 (2023), 1066–1075. https://doi.org/10.1021/acs.jcim.2c01251 doi: 10.1021/acs.jcim.2c01251
[37]	X. Liu, H. T. Feng, J. Wu, K. L. Xia, Persistent spectral hypergraph based machine learning (PSH-ML) for protein-ligand binding affinity prediction, Brief. Bioinform., 22 (2021), bbab127. https://doi.org/10.1093/bib/bbab127 doi: 10.1093/bib/bbab127
[38]	Z. H. Liu, M. Y. Su, L. Han, J. Liu, Q. F. Yang, Y. Li, et al., Forging the basis for developing protein-ligand interaction scoring functions, Acc. Chem. Res., 50 (2017), 302–309. https://doi.org/10.1021/acs.accounts.6b00491 doi: 10.1021/acs.accounts.6b00491
[39]	R. MacPherson, B. Schweinhart, Measuring shape with topology, J. Math. Phys., 53 (2012), 073516. https://doi.org/10.1063/1.4737391 doi: 10.1063/1.4737391
[40]	F. Mémoli, Z. C. Wan, Y. S. Wang, Persistent laplacians: properties, algorithms and implications, SIAM J. Math. Data Sci., 4 (2022), 858–884. https://doi.org/10.1137/21M1435471 doi: 10.1137/21M1435471
[41]	Z. Y. Meng, K. L. Xia, Persistent spectral-based machine learning (perspect ml) for protein-ligand binding affinity prediction, Sci. Adv., 7 (2021), eabc5329. https://doi.org/10.1126/sciadv.abc5329 doi: 10.1126/sciadv.abc5329
[42]	K. Mischaikow, V. Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom., 50 (2013), 330–353. https://doi.org/10.1007/s00454-013-9529-6 doi: 10.1007/s00454-013-9529-6
[43]	C. B. Morrey, A variational method in the theory of harmonic integrals, ii, Am. J. Math., 78 (1956), 137–170. https://doi.org/10.2307/2372488 doi: 10.2307/2372488
[44]	D. D. Nguyen, Z. X. Cang, G.-W. Wei, A review of mathematical representations of biomolecular data, Phys. Chem. Chem. Phys., 22 (2020), 4343–4367. https://doi.org/10.1039/C9CP06554G doi: 10.1039/C9CP06554G
[45]	D. D. Nguyen, Z. X. Cang, K. D. Wu, M. L. Wang, Y. Cao, G.-W. Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R grand challenges, J. Comput. Aided Mol. Des., 33 (2019), 71–82. https://doi.org/10.1007/s10822-018-0146-6 doi: 10.1007/s10822-018-0146-6
[46]	D. D. Nguyen, K. F. Gao, M. L. Wang, G.-W. Wei, MathDL: mathematical deep learning for D3R grand challenge 4, J. Comput. Aided Mol. Des., 34 (2020), 131–147. https://doi.org/10.1007/s10822-019-00237-5 doi: 10.1007/s10822-019-00237-5
[47]	D. D. Nguyen, G.-W. Wei, DG-GL: Differential geometry-based geometric learning of molecular datasets, Int. J. Numer. Meth. Bio., 35 (2019), e3179. https://doi.org/10.1002/cnm.3179 doi: 10.1002/cnm.3179
[48]	E. Panagiotou, K. C. Millett, P. J. Atzberger, Topological methods for polymeric materials: characterizing the relationship between polymer entanglement and viscoelasticity, Polymers, 11 (2019), 437. https://doi.org/10.3390/polym11030437 doi: 10.3390/polym11030437
[49]	T. Papamarkou, T. Birdal, M. M. Bronstein. G. E. Carlsson, J. Curry, Y. Gao, et al., Position: Topological Deep Learning is the New Frontier for Relational Learning, The 41st International Conference on Machine Learning, Vienna, Austria, 2024, 39529–39555.
[50]	C. S. Pun, K. Xia, S. X. Lee, Persistent-homology-based machine learning and its applications–a survey, 2018 arXiv: 1811.00252. https://doi.org/10.48550/arXiv.1811.00252
[51]	M. M. Rana, D. D. Nguyen, Geometric graph learning with extended atom-types features for protein-ligand binding affinity prediction, Comput. Biol. Med., 164 (2023), 107250. https://doi.org/10.1016/j.compbiomed.2023.107250 doi: 10.1016/j.compbiomed.2023.107250
[52]	E. Ribando-Gros, R. Wang, J. H. Chen, Y. Y. Tong, G.-W. Wei, Combinatorial and hodge laplacians: Similarity and difference, SIAM Rev., 66 (2024), 575–601. https://doi.org/10.1137/22M1482299 doi: 10.1137/22M1482299
[53]	G. Schwarz, Hodge decomposition–A method for solving boundary value problems, Berlin: Springer, 1995. https://doi.org/10.1007/BFb0095978
[54]	L. Shen, H. S. Feng, F. L. Li, F. C. Lei, J. Wu, G.-W. Wei, Knot data analysis using multi-scale gauss link integral, P. Nati. A. Sci., In press, 2024.
[55]	L. Shen, J. Liu, G.-W. Wei, Evolutionary khovanov homology, AIMS Mathematics, 9 (2024), 26139–26165. https://doi.org/10.3934/math.20241277 doi: 10.3934/math.20241277
[56]	C. Shonkwiler, Poincaré duality angles on Riemannian manifolds with boundary dissertation, University of Pennsylvania, PhD Thesis, University of Pennsylvania, 2009.
[57]	M. Y. Su, Q. F. Yang, Y. Du, G. Q. Feng, Z. H. Liu, Y. Li, et al., Comparative assessment of scoring functions: the CASF-2016 update, J. Chem. Inf. Model., 59 (2019), 895–913. https://doi.org/10.1021/acs.jcim.8b00545 doi: 10.1021/acs.jcim.8b00545
[58]	Z. Su, Y. Y. Tong, G.-W. Wei, Hodge decomposition of single-cell RNA velocity, J. Chem. Inf. Model., 64 (2024), 3558–3568. https://doi.org/10.1021/acs.jcim.4c00132 doi: 10.1021/acs.jcim.4c00132
[59]	J. Townsend, C. P. Micucci, J. H. Hymel, V. Maroulas, K. D. Vogiatzis, Representation of molecular structures with persistent homology for machine learning applications in chemistry, Nat. Commun., 11 (2020), 3230. https://doi.org/10.1038/s41467-020-17035-5 doi: 10.1038/s41467-020-17035-5
[60]	R. Wang, J. H. Chen, G.-W. Wei, Mechanisms of SARS-CoV-2 evolution revealing vaccine-resistant mutations in Europe and America, J. Phys. Chem. Lett., 12 (2021), 11850–11857. https://doi.org/10.1021/acs.jpclett.1c03380 doi: 10.1021/acs.jpclett.1c03380
[61]	R. Wang, D. D. Nguyen, G.-W. Wei, Persistent spectral graph, Int. J. Numer. Meth. Bio., 36 (2020), e3376. https://doi.org/10.1002/cnm.3376 doi: 10.1002/cnm.3376
[62]	R. Wang, R. D. Zhao, E. Ribando-Gros, J. H. Chen, Y. Y. Tong, G.-W. Wei, Hermes: Persistent spectral graph software, Found. Data Sci., 3 (2021), 67–97. https://doi.org/10.3934/fods.2021006 doi: 10.3934/fods.2021006
[63]	L. Wasserman, Topological data analysis, Annu. Rev. Stat. Appl., 5 (2018), 501–532. https://doi.org/10.1146/annurev-statistics-031017-100045 doi: 10.1146/annurev-statistics-031017-100045
[64]	X. Q. Wei, G.-W. Wei, Persistent topological Laplacians–a survey, 2023, arXiv: 2312.07563. https://doi.org/10.48550/arXiv.2312.07563
[65]	X. Q. Wei, G.-W. Wei, Persistent sheaf Laplacian, Found. Data Sci., 2024 (2024), 033. https://doi.org/10.3934/fods.2024033 doi: 10.3934/fods.2024033
[66]	M. Wójcikowski, P. J. Ballester, P. Siedlecki, Performance of machine-learning scoring functions in structure-based virtual screening, Sci. Rep., 7 (2017), 46710. https://doi.org/10.1038/srep46710 doi: 10.1038/srep46710
[67]	K. L. Xia, G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Int. J. Numer. Meth. Bio., 30 (2014), 814–844. https://doi.org/10.1002/cnm.2655 doi: 10.1002/cnm.2655
[68]	W. T. Yang, R. G. Parr, R. Pucci, Electron density, Kohn–Sham frontier orbitals, and Fukui functions, J. Chem. Phys., 81 (1984), 2862–2863. https://doi.org/10.1063/1.447964 doi: 10.1063/1.447964
[69]	R. D. Zhao, M. Desbrun, G.-W. Wei, Y. Y. Tong, 3D hodge decompositions of edge-and face-based vector fields, ACM T. Graphic., 38 (2019), 181. https://doi.org/10.1145/3355089.3356546 doi: 10.1145/3355089.3356546
[70]	A. Zomorodian, G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249–274. https://doi.org/10.1007/s00454-004-1146-y doi: 10.1007/s00454-004-1146-y

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)