A new two-parameter hyperbolic secant distribution through solitary wave solution for the Landau–Ginzburg–Higgs equation

Abstract

This paper examines the Landau–Ginzburg–Higgs (LGH) equation that arises in mathematical physics. We introduce vital solitary wave solution for the LGH model using the unified solver strategy. We depict the dynamic evolution of the discovered soliton wave. To statistically model the situation, a new two-parameter hyperbolic secant (TPHS) distribution is also derived based on the solution of the nonlinear LGH equation. The unit, generalized, and wrapped two-parameter hyperbolic secants are presented using the TPHS distribution. The probability density function (PDF) plots for each of the models under consideration are shown, demonstrating how adaptable their shapes are to represent various scenarios. Moreover, the hazard rate function (HRF) plots of several considered models are shown.

Introduction

The development of nonlinear partial differential equations (NPDEs) is important to current mathematical physics and applied science, providing a transformational framework to model complicated real-world events that linear theories cannot grasp.^1–3 Unlike their linear counterparts, NPDEs take into consideration the intricate interconnections, self-organization, and feedback mechanisms seen in nonlinear systems, resulting in the production of diverse and often surprising behaviors such as solitons, chaos, and pattern formation. These equations have transformed our understanding of a wide range of topics, including fluid dynamics, plasma physics, nonlinear optics, biology, finance, and quantum field theory, many areas of physics, by giving powerful tools for describing evolution, diffusion, and wave propagation in nonlinear systems.^4,5 The ongoing development of analytical, numerical, and stochastic methodologies for solving NPDEs demonstrates deep innovation, allowing researchers to integrate theoretical insights with experimental realities and construct new technologies based on the dynamics of nonlinearity.^6–8

Mathematical models are critical to understanding how solitary waves form, evolve, and interact within nonlinear physical systems.^9,10 Researchers can capture the delicate balance between nonlinearity and dispersion that leads to coherent solitary formations by developing governing equations, such as nonlinear partial differential equations derived from conservation laws, dispersion relations, or energy principles. These models give a rigorous framework for forecasting wave stability, amplitude-velocity relationships, and long-term dynamics, which are frequently unavailable through experimentation alone.^11,12 Furthermore, mathematical modeling makes it possible to investigate parameter regimes, perturbations, and stochastic influences, providing profound insights into the behavior of solitary waves in practical contexts as ocean engineering, optical fibers, plasma physics, and Bose-Einstein condensates.^13–15

Condensed matter physics and high-energy field theory ideas are profoundly combined in the Landau–Ginzburg–Higgs (LGH) model, which offers a cohesive framework for comprehending spontaneous symmetry breaking and its physical effects.^16,17 Fundamentally, the “Mexican hat” shape is the result of a complex scalar field interacting with a gauge field under the influence of a potential that favors non-zero vacuum expectation values. The LGH model is a theoretical framework that brings together concepts from condensed matter physics, classical field theory, and particle physics. Its physical meaning is to understand how fields go through phase transitions, how order emerges, and how localized structures like solitons, vortices, and domain walls form in nonlinear systems. Because of its structure, the field can develop a non-zero ground state, which in particle physics generates mass for the gauge bosons and causes topological defects or vortices to form in condensed matter systems like superconductors. The LGH model, which captures both macroscopic representations of order in materials and the underlying dynamics of fields, is a cornerstone of modern theoretical physics, bridging the gap between microscopic interactions and observable macroscopic processes. The LGH model is given as follows^16,17:

${\varphi }_{tt}-{\varphi }_{xx}+{\gamma }^2{\varphi }^3-{\lambda }^2\varphi =0,$

(1.1)

ϕ(x, t) denotes the ion-cyclotron wave electrostatic potential, whereas λ and γ are real parameters controlling dispersion (mass parameter) and nonlinearity, respectively. In this study, we use unified solver techniques ¹⁸ to generate solitary wave solutions for a new important stochastic solution for the LGH model. The recommended solver has various advantages over previous sophisticated algorithms, including the elimination of time-consuming and difficult calculations while producing precise results via the use of physical parameters. They will also act as box solvers, or pre-built functions, for many equations and systems encountered in applied research. This solver is simple, dependable, and durable. This approach is highly useful for mathematicians, physicists, and engineers in displaying unique intriguing occurrences in real-world circumstances.

Since our aim is to explore a new statistical distribution that belongs to the family of hyperbolic distributions using solutions of the Landau–Ginzburg–Higgs equation, a brief description of this kind of distributions is given. The probability distribution, which is hyperbolic, is continuous. It is characterized by a hyperbola in the logarithm of the probability density function. As a result, the distribution falls exponentially more slowly than the normal distribution. Thus, the hyperbolic distribution is significant because it can more accurately predict and manage risk by modeling phenomena like financial market returns and turbulent wind speeds that have a higher probability of extreme values than the normal distribution. Its semi-heavy tails, which capture these more frequent large values, are one of its defining characteristics. This property is essential for comprehending how intricate financial and natural systems behave. In the literature, the study in Ivanov ¹⁹ uses the semi-hyperbolic distribution, which is a subclass of the generalized hyperbolic distribution class. In particular, they derive analytical expressions for the first and second lower partial moments of the cumulative distribution function. In Daghistani et al., ²⁰ the authors use the nonlinear evolution equation to introduce a hyperbolic secant-squared distribution. Specifically, the hyperbolic secant-squared (HSS) distribution’s probability density function has been established for this equation. A novel framework based on statistics and nonlinear partial differential equations is presented in Alharbi et al. ²¹ The probability density function of the hyperbolic secant (HS) distribution for the nonlinear Phi-4 equation has been determined. However, the study in Fajardo and Farias ²² discusses how Brazilian asset returns can be fitted using the Generalized Hyperbolic Distributions. Also, for the daily log-returns of seven of the most liquid mining stocks listed on the Johannesburg Stock Exchange, the authors in Konlack and Wilcox ²³ describe the calibration of the univariate and multivariate generalized hyperbolic distributions and their hyperbolic, variance gamma, normal inverse Gaussian, and skew Student’s t-distribution subclasses. In addition, in order to analyze the impact on investment, the study in Núñez and Sánchez-Ruenes ²⁴ suggests building diversified portfolios by incorporating each variable into the portfolio and modeling the returns of the BRIC countries’ oil mixes and indexes under a Normal Inverse Gaussian (NIG) distribution, a prominent member of the generalized hyperbolic family.

In this paper, significantly, our work establishes a novel structure between nonlinear differential equations and statistical modeling by deriving a new two-parameter hyperbolic secant (TPHS) distribution directly from the Landau-Ginzburg-Higgs (LGH) solution. Building on this, we further introduce three new distributions for extending the TPHS distribution, namely, the unit TPHS (UTPHS), wrapped TPHS (WTPHS), and generalized TPHS (GTPHS) distributions, which extend the utility of the solution framework to bounded, circular, and generalized statistical contexts. This combined analytical and statistical contribution strengthens the current work’s intersecting relevance.

The remainder of this paper is organized as follows. First, the unified solver technique is described. Next, the solitary wave solutions for the LGH model is provided. The waves behavior for the presented solutions is then presented. The new two-parameter hyperbolic secant (TPHS) distribution is described. Subsequently, the moments for the TPHS distribution are provided. We introduce the unit TPHS model, which is derived by transforming the support of the TPHS distribution to the unit interval. Also, we introduce a further extension, the generalized TPHS model. The wrapped TPHS distribution for modeling circular data is addressed. Finally, we wrap up our material with a few final thoughts.

Unified solver

The Duffing equation is a cornerstone in the study of nonlinear dynamics, best known for simulating oscillators with a cubic nonlinearity, sometimes known as a “hardening” or ”softening” spring. Its major discovery is that it demonstrates how deterministic systems can display extremely rich, complicated behavior, including as harmonic oscillations, chaotic motion, and period-doubling bifurcations, which are all driven by the interaction of damping and driving forces. This simple yet powerful equation provides a fundamental paradigm for understanding a wide range of physical phenomena, including the response of mechanical structures and electrical circuits, as well as the dynamics of orbital and microscopic magnetic systems. Here, we introduce the unified solver technique ¹⁸ for the following Duffing equation:

$A{\varphi }^{″}(\xi )+B{\varphi }^3(\xi )+C\varphi (\xi )=0.$

(2.1)

The solutions of this equation are

$\varphi (\xi )=\pm \sqrt{\frac{-2C}{B}}sech\left(\pm \sqrt{\frac{-C}{A}}\xi \right).$

(2.2)

Analytical solutions

Using a transformation:

$\varphi (x,t)=\varphi (\xi ),\xi =x-wt,$

(3.1)

where w is the wave speed. Substituting equation (3.1) into equation (3.1) yields

$(w^2-1){\varphi }^{″}+{\gamma }^2{\varphi }^3-{\lambda }^2\varphi =0.$

(3.2)

Therefore, the solutions of (3.2) are

${\varphi }_{\mathrm{1,2}}(\xi )=\pm \frac{\lambda }{\gamma }\sqrt{2}sech\left(\pm \frac{\lambda }{\sqrt{w^2-1}}\xi \right),-1>w\text{or}1<w.$

(3.3)

Consequently, the solutions of (1.1) are

${\varphi }_{\mathrm{1,2}}(x,t)=\pm \frac{\lambda }{\gamma }\sqrt{2}sech\left(\pm \frac{\lambda }{\sqrt{w^2-1}}(x-wt)\right),-1>w\text{or}1<w.$

(3.4)

Waves behavior

The waves emerging from the Landau–Ginzburg–Higgs (LGH) equation are fundamental nonlinear excitations that connect condensed matter physics and field theory. These waves explain the dynamic evolution of the complex order parameter involved in spontaneous symmetry breaking and phase transitions. In the LGH framework, the combination of the nonlinear self-interaction term with the spatial-temporal fluctuations gives rise to localized and coherent wave patterns such as solitons. These waveforms depict the fundamental physics of superconductivity, superfluidity, and even cosmological events such as topological flaws in the early universe. The balance of dispersion, nonlinearity, and potential energy produces steady wave propagation, which distinguishes the transition between ordered and disordered phases. Thus, studying waves in the Landau–Ginzburg–Higgs equation yields important insights into the mechanisms of energy localization, pattern development, and the emergence of coherence in complex nonlinear systems.

The MATLAB release is used to construct 2D charts of selected solutions that show the propagation of the solitary waves for the LGH. Figure 1 illustrates the bell-shaped soliton wave of the solution ϕ₁(x, t) with different values of t with phase shift. The strength of the nonlinear interaction between wave components is indicated by the nonlinearity coefficient γ. This coefficient is motivated by the need to take into consideration the self-modulation effect, which happens when a wave’s amplitude influences its own phase or propagation speed. As γ increases, the amplitude of the wave decreases, and the profile becomes broader and flatter as shown in Figure 2. In field theory, λ is related to the mass of the scalar field and determines how quickly perturbations decay or oscillate in time and space. As λ increases, the amplitude of the wave increases, and the profile becomes narrower and sharper as shown in Figure 3.

Figure 1.

2D bell-shaped soliton wave of solution (3.4) with different values of time t for λ = 1, γ = 0.7, and w = 1.5.

Figure 2.

2D bell-shaped soliton wave of solution (3.4) with different values of parameter γ for λ = 3 and w = 2.

Figure 3.

2D bell-shaped soliton wave of solution (3.4) with different values of parameter λ for γ = 2 and w = 1.5.

New two- parameter hyperbolic secant

This section presents a new two-parameter hyperbolic secant (TPHS) distribution, which is obtained as a solution to the nonlinear LGH equation (1.1), to statistically model the situation. The solution of the LGH equation in equation (3.4),ϕ_1,2(x, t = 0), allows us to calculate the corresponding probability density function (PDF) for the TPHS model. In light of this, we could have

${\int }_{-\infty }^{\infty }\frac{\lambda }{\gamma }\sqrt{2}\text{sech}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)dx=\frac{\sqrt{2}\pi \sqrt{{\omega }^2-1}}{\gamma }.$

As a direct result, with $\gamma /\sqrt{2}\pi \sqrt{{\omega }^2-1}$ as the normalizing constant, we get our distribution’s PDF as

$f(x)=\frac{\lambda }{\pi \sqrt{{\omega }^2-1}}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right),-\infty <x<\infty ,\lambda >0,\omega >1.$

(5.1)

The corresponding cumulative distribution function (CDF), hazard rate function (HRF), and quantile function, respectively, are given by

$F(x)=\frac{2}{\pi }\arctan \left(\exp \left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)\right),$

(5.2)

$h(x)=\frac{\frac{\lambda }{\pi \sqrt{{\omega }^2-1}}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)}{1-2\pi \arctan \left(\exp \left(\lambda x\sqrt{{\omega }^2-1}\right)\right)},$

(5.3)

$x=Q(u)=\frac{\sqrt{{\omega }^2-1}}{\lambda }\ln \left[\tan \left(\frac{\pi }{2}u\right)\right],0<u<1.$

(5.4)

The PDF and HRT plots of the TPHS distribution are shown in Figures 4 and 5. The PDF and HRF of the TPHS distribution take several shapes. As a result, every parameter affects the HRF and PDF shapes to cover a large range of models using the TPHS distribution.

Figure 4.

The plots of PDF for the TPHS model.

Figure 5.

The plots of HRT for the TPHS model.

Moments of TPHS model

The TPHS model’s moments are covered in this section. In the following proposition, we now offer raw moments of the TPHS model.

Proposition 6.1

The raw moments around the origin, E[X ^r ], for a TPHS model are given by, for r = 1, 2, … ,

$E[X^r]=\left\{\begin{array}{cc}0,\hfill & \text{if\hspace{0.17em}}r\text{\hspace{0.17em}is\hspace{0.17em}odd},\hfill \\ \frac{4\mathrm{\Gamma }(r+1)\beta (r+1)}{\pi }{\left(\frac{\sqrt{{\omega }^2-1}}{\lambda }\right)}^r,\hfill & \text{if\hspace{0.17em}}r\text{\hspace{0.17em}is\hspace{0.17em}even}\hfill \end{array}\right.$

(6.1)

where $\beta (s)={\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^s$ is the Dirichlet beta function.

Proof. Using the definition of the rth moment of X, we obtain

$E[X^r]=\frac{\lambda }{\pi \sqrt{{\omega }^2-1}}{\int }_{-\infty }^{\infty }{x}^r\text{sech}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)dx.$

If r is odd, then $x^r\text{sech}\left(\lambda x/\sqrt{{\omega }^2-1}\right)$ is an odd function, and the integral over (−∞, ∞) equals zero. If r is even, then

$E[X^r]=\frac{2\lambda }{\pi \sqrt{{\omega }^2-1}}{\int }_0^{\infty }{x}^r\text{sech}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)dx.$

Now, substitute $u=\lambda x/\sqrt{{\omega }^2-1}$ , so that $dx=\sqrt{{\omega }^2-1}/\lambda du$ . Then

$E[X^r]=\frac{2}{\pi }{\left(\frac{\sqrt{{\omega }^2-1}}{\lambda }\right)}^r{\int }_0^{\infty }{u}^r\text{sech}(u)du.$

Using the identity $\text{sech}(u)=2{\sum }_{k=0}^{\infty }{(-1)}^k{e}^{-(2k+1)u}$ , we have

$E[X^r]=\frac{4}{\pi }{\left(\frac{\sqrt{{\omega }^2-1}}{\lambda }\right)}^r\sum _{k=0}^{\infty }{(-1)}^k{\int }_0^{\infty }{u}^r{e}^{-(2k+1)u}du.$

Evaluating the integral using the Gamma function,

${\int }_0^{\infty }{u}^r{e}^{-(2k+1)u}du=\frac{\mathrm{\Gamma }(r+1)}{{(2k+1)}^{r+1}},$

so

$E[X^r]=\frac{4}{\pi }{\left(\frac{\sqrt{{\omega }^2-1}}{\lambda }\right)}^r\mathrm{\Gamma }(r+1)\sum _{k=0}^{\infty }\frac{{(-1)}^k}{{(2k+1)}^{r+1}}.$

The series ${\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^s$ is the Dirichlet beta function β(s). Combining these results, we obtain

$E[X^r]=\left\{\begin{array}{cc}0,\hfill & \text{if\hspace{0.17em}}r\text{\hspace{0.17em}is\hspace{0.17em}odd},\hfill \\ \frac{4}{\pi }\mathrm{\Gamma }(r+1)\beta (r+1){\left(\frac{\sqrt{{\omega }^2-1}}{\lambda }\right)}^r,\hfill & \text{if\hspace{0.17em}}r\text{\hspace{0.17em}is\hspace{0.17em}even}.\hfill \end{array}\right.$

This completes the proof. □

The unit TPHS model

In this section, we provide a new bounded model that is obtained by Logistic transformation from the TPHS distribution defined by equation (5.1). We refer to the unit two-parameter hyperbolic secant (UTPHS) distribution as our proposed model.

Now, we apply the Logistic transformation as: Y = 1/1 + e^−X. The inverse transformation is $X=\ln \left(Y/1-Y\right)$ , with derivative dX/dY = 1/Y(1 − Y), and using the identity sech (u) = 2/e ^u + e^−u, then the transformed PDF is given as

$f(y)=\left(\frac{2\lambda }{\pi \sqrt{{\omega }^2-1}}\right)\frac{y^{\frac{\lambda }{\sqrt{{\omega }^2-1}}-1}{(1-y)}^{\frac{\lambda }{\sqrt{{\omega }^2-1}}-1}}{y^{\frac{2\lambda }{\sqrt{{\omega }^2-1}}}+{(1-y)}^{\frac{2\lambda }{\sqrt{{\omega }^2-1}}}},0<y<1,\lambda >0,\omega >1.$

(7.1)

The CDF corresponding to equation (7.1) is

$F(y)=\frac{2}{\pi }\arctan \left({\left(\frac{y}{1-y}\right)}^{\frac{\lambda }{\sqrt{{\omega }^2-1}}}\right).$

(7.2)

The hazard rate function (HRF) is given as

$h(y)=\frac{\xi y^{\xi -1}{(1-y)}^{\xi -1}}{\left(y^{2\xi }+{(1-y)}^{2\xi }\right)\arctan \left({\left(\frac{1-y}{y}\right)}^{\xi }\right)},$

(7.3)

where $\xi =\lambda /\sqrt{{\omega }^2-1}$ .

In Figures 6 and 7, we demonstrate the plots of PDF and HRT of the unit TPHS model. It is obvious that the two parameters in the model have an effect on the shape of the PDF and HRF. The effect of the parameter λ  on the shape of PDF and HRF is more than of the parameter ω. The PDF of the UTPHS model takes several shapes, which give more flexibility in the modeling.

Figure 6.

The plots of PDF for the UTPHS model.

Figure 7.

The plots of HRF for the UTPHS model.

The generalized TPHS model

In this part, we introduce the generalized two-parameter hyperbolic secant (GTPHS) distribution.

A random variable X is considered to have a GTPHS distribution with three-parameter Δ = (λ, ω, θ), if its CDF takes the following form:

$G(x)={[F(x)]}^{\theta }={\left[\frac{2}{\pi }\arctan \left(\exp \left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)\right)\right]}^{\theta },-\infty <x<\infty ,\lambda >0,\omega >1,\theta >0,$

(8.1)

where θ is an additional shape parameter, and F(x) is the CDF of the TPHS distribution defined by equation (5.2).

Next, the relevant PDF is provided by

$g(x)=\frac{\theta \lambda }{\pi \sqrt{{\omega }^2-1}}{\left(\frac{2}{\pi }\right)}^{\theta -1}{\left[\arctan \left(\exp \left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)\right)\right]}^{\theta -1}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right).$

(8.2)

The HRF of the GTPHS model has the following form:

$h_g(x)=\frac{\frac{\theta \lambda }{\pi \sqrt{{\omega }^2-1}}{\left(\frac{2}{\pi }\right)}^{\theta -1}{\left[\arctan \left(\exp \left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)\right)\right]}^{\theta -1}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)}{1-{\left[\frac{2}{\pi }\arctan \left(\exp \left(\frac{\lambda x}{\sqrt{{\omega }^2-1}}\right)\right)\right]}^{\theta }}.$

(8.3)

In Figures 8–11, we present the PDF and HRT plots of the generalized TPHS model. Each parameter in the model has an effect on the shape of the PDF and HRF. The effect of the parameters λ  and ω  on the flat of PDF is more than that of the parameter θ.

Figure 8.

The plots of PDF for the GTPHS model.

Figure 9.

The plots of PDF for the GTPHS model.

Figure 10.

The plots of HRF for the GTPHS model.

Figure 11.

The plots of HRF for the GTPHS model.

The wrapped TPHS model

In this part, the wrapped two-parameter hyperbolic secant (WTPHS) distribution is defined by its probability density function (PDF) and cumulative distribution function (CDF). These functions are derived from specific summations. Below, we present the precise mathematical formulas for each function.

The following is a description of the wrapping technique. Assume that f(x), as described by equation (5.1), is a valid PDF. Additionally, if f(x) is specified on the entire real line, then the PDF of the wrapped distribution is defined as

$f_{\text{wrapped}}(\theta )=\sum _{k=-\infty }^{\infty }f\left(\theta +2k\pi ;\lambda ,\omega \right),0\le \theta <2\pi ,$

(9.1)

The PDF of the WTPHS model is then derived by substituting equation (5.1) into equation (9.1), resulting in the following expression:

$f_{\text{wrapped}}(\theta )=\frac{1}{2\pi }+\frac{1}{\pi }\sum _{n=1}^{\infty }\text{sech}\left(\frac{n\pi \sqrt{{\omega }^2-1}}{2\lambda }\right)\cos (n\theta ),0\le \theta <2\pi ,\lambda >0,\omega >1.$

(9.2)

The CDF for the WTPHS model can be obtained as follows:

$\begin{array}{cc}\hfill F_{\text{wrapped}}(\theta )& =\sum _{k=-\infty }^{\infty }F\left(\theta +2k\pi ;\lambda ,\omega \right)-\sum _{k=-\infty }^{\infty }F\left(2k\pi ;\lambda ,\omega \right)\hfill \\ \hfill & =\frac{\theta }{2\pi }+\frac{1}{\pi }\sum _{n=1}^{\infty }\frac{1}{n}\text{sech}\left(\frac{n\pi \sqrt{{\omega }^2-1}}{2\lambda }\right)\sin (n\theta ).\hfill \end{array}$

(9.3)

Finally, the HRF of the WTPHS model is defined as

$h_{\text{wrapped}}(\theta )=\frac{\frac{1}{2\pi }+\frac{1}{\pi }{\sum }_{n=1}^{\infty }\text{sech}\left(\frac{n\pi \sqrt{{\omega }^2-1}}{2\lambda }\right)\cos (n\theta )}{1-\left[\frac{\theta }{2\pi }+\frac{1}{\pi }{\sum }_{n=1}^{\infty }\frac{1}{n}\text{sech}\left(\frac{n\pi \sqrt{{\omega }^2-1}}{2\lambda }\right)\sin (n\theta )\right]}.$

(9.4)

Figure 12 shows some examples of the plots for the PDF for the WTPHS model. The PDF of the WTPHS model generates a symmetric circular distribution, controlled by λ and ω, making it suitable for modeling directional data.

Figure 12.

The plots of PDF for the WTPHS model.

Conclusion

In this work, the Landau–Ginzburg–Higgs (LGH) equation was solved using the new unified silver technique. We presented the solitary wave solutions that form the basis of this study’s analysis. We came to the conclusion that examining waves in the Landau–Ginzburg–Higgs equation provides crucial information about how energy is localized, patterns form, and coherence emerges in intricate nonlinear systems. The behavior of the solutions when the proper free parameter values were used was displayed in the 2D plots. The effectiveness of the suggested solver method demonstrates that it can be used for a wide range of nonlinear models that appear in numerous nonlinear study domains. Additionally, to statistically model the situation, a new two-parameter hyperbolic secant (TPHS) distribution is obtained from the solution of the nonlinear LGH equation. The general form of its moments is then found. We also calculated the unit two-parameter hyperbolic secant (UTPHS) distribution using a logistic transformation from the TPHS distribution. Furthermore, we introduced the generalized two-parameter hyperbolic secant (GTPHS) distribution and the wrapped two-parameter hyperbolic secant (WTPHS) distribution. Several plots of the four models under consideration are displayed to show their overall behaviors.

In future work, the mathematical and inferential characteristics of the suggested TPHS, UTPHS, WTPHS, and GTPHS models, such as moment generating function, skewness, kurtosis, entropy, parameter estimation techniques, and creating regression models based on the UTPHS for bounded data, would be worth investigating. Furthermore, the practical utility of the WTPHS model for circular data.