This study explores the nonlinear dynamics of a nerve impulse neuron model governed by a partial differential equation (PDE) with beta-fractional derivatives, allowing the inclusion of nonlocal temporal dependence in the model formulation. The model exhibits rich dynamical behavior, including chaos, multistability, and chaos control, alongside the emergence of solitary and periodic wave solutions within the cell membrane. Through a suitable transformation, the PDE is transformed into an ordinary differential equation (ODE), which is further reduced to a planar dynamical system via the Galilean transformation. The stability of the system’s equilibrium points is assessed through eigenvalue analysis of the associated Jacobian matrix. Numerical solutions of the reduced integer-order ODE system are performed using a Runge-Kutta method, and chaotic dynamics are induced via a time-dependent periodic forcing term. Chaos is consistently identified using standard diagnostics, including phase portraits, time-series analysis, Lyapunov exponents, Poincaré maps, bifurcation diagrams, power spectra, return maps, and recurrence plots. The Pyragas time-delayed feedback control method is then applied to stabilize the system, resulting in stabilized periodic states. To explore wave solutions, variational and Hamiltonian approaches are employed to derive analytical expressions for bright, bright-dark, kinky-bright, and periodic wave solutions within the fractional-order framework. The influence of the fractional order on temporal dynamics and spatial wave propagation is illustrated through two- and three-dimensional graphical representations. These wave structures may be interpreted as mathematical analogs of excitation, inhibition, and rhythmic firing mechanisms in neuronal dynamics. The results provide new insights into the nonlinear features of the fractional neuron model, linking memory effects, nonlinear excitability to mathematically meaningful interpretations of neuronal firing and signal transmission.

The new Hamiltonian amplitude equation effectively expresses the modulated wave instability and addresses the ill-posedness of the unstable nonlinear Schrödinger equation. This equation simulates nonlinear optical pulse propagation, fiber optic communication engineering, self-phase modulation, and modulated wave train instability. The unified tanh approach is used in this article to establish broad-spectral soliton solutions to the stated model in terms of hyperbolic and trigonometric functions. The solutions enfolded several free parameters associated with the model and the procedure, and specific values of these parameters result in some novel and typical soliton solutions that are examined in the texts. Additionally, the effect of the stretch coordinate ε is examined. The effects of stretching coordinates are determined by sketching three- and two-dimensional plots for different values of ε. The stability analysis of the gained solutions is examined, and the Hamiltonian function is discussed. Furthermore, we proceed to the bifurcation analysis of the model that is explored. To analyze the dynamic behavior of the solitons in nonlinear optics and other fields, the stability of the equilibrium points is evaluated, and a graphical representation of the system's phase diagram is provided.

This study employs the unified method to analyze fractional-order DNA systems and constructs various soliton solutions. These include kink, anti-kink, singular, singular-periodic, periodic, and hybrid solitons in different parameter regimes. Graphical depictions of these solutions demonstrate the system’s spatiotemporal dynamics, revealing dual wave behaviors and the influence of physical parameters on soliton formation. The results indicate that the dual behavior of kink-antikink solitons may offer insights into the formation of an open-state configuration within the DNA double helix. Specifically, the amplitude of the anti-kink wave profile increases as the distance between the DNA strands grows, and the soliton profile shifts with varying stiffness and cross-sectional area. Additionally, the oscillatory wave remains unaffected by stiffness and area in terms of amplitude, though its profile undergoes shifts under varying conditions. This study provides a mathematical framework that bridges applied mathematics and molecular biology, enabling the exploration of DNA dynamics.

The study aims to explore the intricate dynamics of various types of pulses within a nonlinear, lossy electrical transmission line, describing the propagation of electrical solitons in nonlinear dispersive media. In pursuit of these aims, an analytical method known as the unified method is employed to the nonlinear, lossy transmission lines periodically loaded with symmetric voltage-dependent capacitances. By applying Kirchhoff's law in the continuum limit, a nonlinear partial differential equation for the voltage on the transmission line is derived. Subsequently, with the aid of the unified method, a variety of electrical soliton pulses, including dark, bright, singular, periodic singular, periodic W-shaped, and periodic waves, are obtained from the voltage equation. Additionally, stability analysis of the model is assessed using linear stability analysis technique, confirming the stability of dispersion. The coefficients of the nonlinear transmission lines model are found to play a substantial role in changing the shapes of the solitons. By adjusting the parameters for the obtained analytic solutions for nonlinear transmission lines to suitable values, the model can effectively modify the characteristics of the waves and produce desired wave profiles. The received solitons could find their applications in telecommunication systems to carry information and increase the bit rate of data. The ability of electrical soliton pulses to propagate with minimal dispersion makes them an efficient approach for transmitting data modulated as short pulses over long distances.

Qualitative analysis in mathematical modeling has become an important research area within the broad domain of nonlinear sciences. In the realm of qualitative analysis, the bifurcation method is one of the significant approaches for studying the structure of orbits in nonlinear dynamical systems. To apply the bifurcation method to the (2 + 1)-dimensional double-chain Deoxyribonucleic Acid system with beta derivative, the bifurcations of phase portraits and chaotic behaviors, combined with sensitivity and multi-stability analysis of this system, are examined. Initially, the bifurcations of phase portraits are visually identified at the obtained equilibrium points of a planar dynamical system via both Hamiltonian and Jacobian algorithms. The obtained results indicate Jacobian algorithm is more efficient in identifying the stability of bifurcations than the Hamiltonian algorithm for this system. Subsequently, by introducing an external perturbation term into the planar dynamical system, the chaotic behavior is effectively identified by using a variety of tools, such as two- and three-dimensional phase portraits, time series, Lyapunov exponents, and Poincaré maps. The findings suggest that the perturbed dynamical system deviates from regular patterns and exhibits behavior ranging from periodic to quasi-periodic and from quasi-periodic to chaotic. Finally, the sensitivity and multi-stability of the system are examined using the Runge-Kutta method to assess the model's response to minor variations in initial conditions through numerical solutions, revealing that the model is sensitive and multi-stable. The outcomes of this study will enhance a relationship between applied mathematicians and experimental biologists, helping to explore hidden features of Deoxyribonucleic Acid through the studied model.

This study investigates wave solutions to the higher order Ramani equation with beta derivative via the generalized Kudryashov and the extended sinh-Gordon equation methods. The higher order Ramani equation with beta derivative is reduced into an ordinary differential equation (ODE) with the use of a fractional transformation involving the definition of beta derivative. Hereafter, via the generalized Kudryashov method and the extended sinh-Gordon equation method, some novel solutions are constructed of the reduced ODE in terms of trigonometric functions, hyperbolic functions, and their combinations. Then, the considered wave transformation is set back to the solutions of reduced ODE. As a consequence, all explored wave solutions of the fractional Ramani equation are found to be novel in terms of beta derivative and applied methods sense. To demonstrate the fractional effects of the explored wave solutions, the three-dimensional (3D) and their two-dimensional (2D) cross-sectional line plots are presented under the particular selection of any fractional values within beta is an element of(0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (\mathrm{0,1}$$\end{document}). The 3D and 2D cross-sectional line plots of some of the achieved novel solutions confirm the underlying mechanisms of the model. With an increase in fractional parameters, the kink or anti-kink profile takes on complete form, and smoothness rises. Conversely, the singular-periodic wave solutions show an increase in smoothness and periodicity. Furthermore, a thorough comparison of all the investigated solutions to the equation under consideration that integrates beta derivative (BD), conformable derivative (CD), and M-truncated derivative (MTD) is included in this paper. The findings for the model's BD, CD, and MTD investigate how the fractional parameter affects the wave profile's amplitude, using graphs to illustrate this effect by designating exact fractional parameter values. The obtained findings demonstrate the ability of the implemented techniques to identify wave solutions with fractional derivatives for the nonlinear sixth-order Ramani equation, which are practically useful for utilizing optical fiber.

The study aims to explore the intricate dynamics of various types of pulses within a nonlinear, lossy electrical transmission line, describing the propagation of electrical solitons in nonlinear dispersive media. In pursuit of these aims, an analytical method known as the unified method is employed to the nonlinear, lossy transmission lines periodically loaded with symmetric voltage-dependent capacitances. By applying Kirchhoff’s law in the continuum limit, a nonlinear partial differential equation for the voltage on the transmission line is derived. Subsequently, with the aid of the unified method, a variety of electrical soliton pulses, including dark, bright, singular, periodic singular, periodic W-shaped, and periodic waves, are obtained from the voltage equation. Additionally, stability analysis of the model is assessed using linear stability analysis technique, confirming the stability of dispersion. The coefficients of the nonlinear transmission lines model are found to play a substantial role in changing the shapes of the solitons. By adjusting the parameters for the obtained analytic solutions for nonlinear transmission lines to suitable values, the model can effectively modify the characteristics of the waves and produce desired wave profiles. The received solitons could find their applications in telecommunication systems to carry information and increase the bit rate of data. The ability of electrical soliton pulses to propagate with minimal dispersion makes them an efficient approach for transmitting data modulated as short pulses over long distances.

This paper revisits the distribution of thermodynamic variables within initial protoplanets formed via gravitational instability (GI) across a broad mass spectrum ranging from 0.3MJ to 10MJ (where 1MJ denotes 1 Jupiter mass, equal to 1.8986×1030 g), using the Homotopy Analysis Method (HAM), a novel approach in this context. Concerning heat transfer within the protoplanets, consideration is given to the convective mode. Our findings reveal a noteworthy alignment between the results obtained via the HAM, utilizing only the first four terms (third approximation), and numerical outcomes. The HAM is found to demonstrate rapid convergence towards the exact solution, showcasing its effectiveness in obtaining such solutions to nonlinear problems. Comparative graphical illustrations of approximate series solutions obtained via HAM and the Adomian decomposition method highlight the superior accuracy and analytical depth of HAM within this framework. This establishes HAM as a powerful and insightful tool for studying astrophysical phenomena. The method offers significant advantages in accuracy and analytical depth over traditional methods, demonstrating its potential for broader applications in astrophysical research and providing robust solutions where conventional approaches may fall short.

The study seeks to obtain new analytical solutions for the (2+1)-dimensional cubic Klein–Gordon (cKG) equation using the beta derivative. By applying the unified method to the equation, various types of solitons have been generated, including periodic solitons, periodic solitons with equal and unequal wavelengths, bright solitons, and periodic singular solitons with unequal wavelengths. To demonstrate the fundamental dynamics of the soliton family, three-dimensional and two-dimensional graphs showcasing various novel solutions that satisfy the relevant equations are provided. In relation to fractionality, the bright waveform retains its overall shape, but its smoothness improves as the fractional parameters increase. Conversely, periodic wave solutions show enhanced periodicity as the fractional parameters rise. Additionally, the study provides a comprehensive comparison of solutions derived from models utilizing conformable, M-truncated, and beta derivatives. The investigation explores the effect of the fractional parameter on soliton amplitude, using graphs to illustrate this impact by assigning specific values to the fractional parameter. The properties of the waves can be modified through changes to the model's parameters to produce the appropriate wave profiles. Consequently, the solutions we obtained could be particularly valuable for analyzing physical problems associated with nonlinear complex dynamical systems.

This study aims to explore the intricate behavior of soliton-like pulses within a nonlinear and lossy electrical transmission line. The transmission line is described by a mathematical model called the “beta derivative”, which is specifically designed to characterize the propagation of electrical solitons in media with both nonlinearity and dispersion properties. To achieve this goal, the (G′/G2)-expansion method is applied to a periodically loaded nonlinear beta derivative lossy transmission line with symmetric voltage-dependent capacitances. Soliton-like pulses, including periodic singular, dark, bright, and singular waves, are obtained using the (G′/G2)-expansion method based on the voltage equation. Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying relevant equations are provided to understand the underlying mechanisms of the soliton family. In terms of fractionality, the profiles of dark or bright waveforms maintain their complete forms, but their smoothness increases as the fractional parameters increase. On the other hand, periodic wave solutions exhibit an increase in periodicity. By adjusting the parameters of the model, the characteristics of the waves can be modified to generate desired wave profiles. Additionally, the study involves a comprehensive comparison among solutions derived from models incorporating the beta, conformable, and M−truncated derivatives. It examines how the amplitude of the solitons is influenced by the fractional parameter, utilizing graphs to visualize this impact by assigning specific fractional parameter values. These pulse-like solitons offer promising potential for integration within telecommunication systems, where they can function as information carriers, facilitating heightened data bit rates. By leveraging the findings of this study, advancements in telecommunication technology can be pursued, leading to enhanced efficiency and improved performance.

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional Kudryashov-Sinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional KudryashovSinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

This study uses the Hirota bilinear method and Maple, a symbolic computation program, to derive lump solutions for a new integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations. Furthermore, lump solutions with free parameters have been constructed using the dimensionally reduced new form of the (3 + 1)-dimensional Boussinesq equation. The derived lump solutions show it has two trough positions and one crest position. The amplitudes and shapes of lump waves don't vary during propagation but they change their positions. By making three-dimensional, two-dimensional, and density plots for specific values of the relevant free parameters, the propagations of the obtained lump wave solutions are displayed. They also demonstrate how the trough and crest positions of a lump wave change over time with constant velocity. The phase shifts, propagation directions, and energy distributions can be seen from the graphical outputs that the free parameters of the model play a significant role in changing the shapes and amplitudes of the waves. The resulting solutions and their physical characteristics may help to understand how the waves propagate in shallow water in oceanography.

This study investigates some analytic solutions and phase portraits to the diffusive predator-prey system in studying the spatiotemporal dynamics of a predator-prey community in ecology through an analytical approach and a qualitative theory of planar dynamical systems, respectively. To accomplish such aims, a simple wave transformation is applied to the diffusive predator-prey system for converting it into a system of ordinary differential equations with a planar dynamical system for analyzing the behavior of bifurcation properties and analytic solutions. Then, the analytical unified method is employed to the attained system. The applied wave transformation is put back to the obtained solutions of the system of ordinary differential equations. Finally, the analytic solutions namely, kink, anti-kink, singular, periodic-singular, and sinusoidal wave solutions are attained to the considered system. All the constructed wave solutions are found to be new from the viewpoint of the application of the unified method. In order to verify the biological wave phenomena of predator and prey populations, some graphs are presented for illustrating the analytic solutions, which have interesting implications in ecology. The effects of free parameters and wave celerity on the attained solutions are demonstrated graphically along with their physical descriptions. The graphical outputs reveal that the predator and prey population densities are changed with the change in the free parameters. Wave solutions to the fractional diffusive predator-prey system are also reported with Atangana conformable derivative sense. As the value of the fractional parameter increases, the smoothness of the anti-kink wave profile is found to increase gradually, but the steepness decreases. For the sinusoidal wave profile, the periodicity and smoothness increase as the value of the fractional parameter increases. Thus, the present study may enrich the interpretation of the spatiotemporal dynamics of predator-prey interaction in a real environment. It is also found that the applied method and the relevant transformation are effective and easy to use for acquiring new analytic solutions to the diffusive predator-prey system over the other analytic methods. The method of interest is novel and efficient because it overcomes the weaknesses and deficiencies of the other analytic methods. Therefore, the method can be applied to further studies to explain various physical phenomena arising in ecology.

The study aims to explore obliquely propagating optical wave solutions to the (2 + 1)-dimensional chiral nonlinear Schrodinger (NLS) equation in both the absence and presence of the Atangana derivative. In order to convert the classical-order chiral nonlinear Schrodinger equation to an ordinary differential equation, a transformation associated with wave obliqueness is applied. Hereafter, the unified method is applied to the reduced equation. As outcomes, the dark, periodic singular with unequal wave length, periodic with equal wavelength, periodic, unsmooth periodic, singular periodic soliton solutions are received to the ordinary differential equation. Later, the acquired solutions are then put to the applied transformation associated with obliqueness. Moreover, the fractional-order chiral NLS equation is solved by using the spatiotemporal Atangana derivative with oblique wave transformation and the unified method. In terms of wave obliqueness, fractionality, and applied technique sense, all generated wave solutions are revealed to be novel. Along with their physical explanations, the impacts of obliqueness and fractionality on the solutions are graphically illustrated. It is exposed that as obliqueness and fractionality increase, the optical wave phenomena change. Additionally, it is discovered that the employed method can be used to obtain novel optical soliton features to the chiral nonlinear Schrodinger equation with or without fractional and obliqueness constraints. It can be assured that the utilized method is more powerful than the other methods. As a result, the method can be used in future research to explain the many physical phenomena that arise in optical fiber communication networks.

The study aims to explore obliquely propagating optical wave solutions to the (2 + 1)-dimensional chiral nonlinear Schrödinger (NLS) equation in both the absence and presence of the Atangana derivative. In order to convert the classical-order chiral nonlinear Schrödinger equation to an ordinary differential equation, a transformation associated with wave obliqueness is applied. Hereafter, the unified method is applied to the reduced equation. As outcomes, the dark, periodic singular with unequal wave length, periodic with equal wavelength, periodic, unsmooth periodic, singular periodic soliton solutions are received to the ordinary differential equation. Later, the acquired solutions are then put to the applied transformation associated with obliqueness. Moreover, the fractional-order chiral NLS equation is solved by using the spatiotemporal Atangana derivative with oblique wave transformation and the unified method. In terms of wave obliqueness, fractionality, and applied technique sense, all generated wave solutions are revealed to be novel. Along with their physical explanations, the impacts of obliqueness and fractionality on the solutions are graphically illustrated. It is exposed that as obliqueness and fractionality increase, the optical wave phenomena change. Additionally, it is discovered that the employed method can be used to obtain novel optical soliton features to the chiral nonlinear Schrödinger equation with or without fractional and obliqueness constraints. It can be assured that the utilized method is more powerful than the other methods. As a result, the method can be used in future research to explain the many physical phenomena that arise in optical fiber communication networks.

In this study, lump and two classes of interaction, multi-stripe, and breather wave solutions for the (3+1)-dimensional generalized shallow water equation are presented via the Hirota bilinear method. Interaction solutions are found between one-lump and one-stripe, and one-lump and two-stripes solutions by combining a quadratic function and an exponential function, and a quadratic function and a hyperbolic cosine or double exponential functions, respectively. Dynamical behaviours of some obtained valid solutions are presented through some graphs. The physical interpretation of fission-fusion dynamics is also explained graphically through lump-kink interaction solutions. During the fission-fusion interaction process, it is seen that stripe solitons split into a stripe and a lump soliton, and then the lump and stripe solitons fuse together. During this process, a rogue wave is found between one lump and twin stripes soliton at t=0. Furthermore, multi-stripe and breather wave solutions are investigated by choosing the appropriate functions and the values for the free parameters. The multi-stripe waves are found to be nonsingular and rectangular hyperbolic shaped. On the other hand, breather waves are found to be periodic, which can evolve periodically along a straight line in the xy-plane. The produced wave solutions might be helpful to understand the propagation behaviour of waves in shallow water.

In this paper, an improvement has been made to the approximation technique of a complex domain through the stairstep approach to have a considerable accuracy, minimize computational cost, and avoid the hardship of manual work.A novel stair-step representation algorithm is used in this regard, where the entire procedure is carried out through our developed MATLAB routine. Arakawa C-grid is used in our approximation with（1/120）° grid resolution. As a test case, the method is applied to approximate the domain covering the area between 15°–23°N latitudes and 85°–95°E longitudes in the Bay of Bengal. Along with the approximation of the land-sea interface, coastal stations are also identified. Approximated land-sea interfaces and coastal stations are found to be in good agreement with the actual ones based on the similarity index, overlap fraction, and extra fraction criteria. The method can be used for approximating an irregular geometric domain to employ the finite difference method in solving problems related to long waves. As a test case, shallow water equations in Cartesian coordinates are solved on the domain of interest for simulating water levels due to the nonlinear tide-surge interaction associated with the storms April 1991 and AILA, 2009 along the coast of Bangladesh. The same input except for the discretized domain and bathymetry as that of Paul et al.（2016） is used in our simulation. The results are found to be in reasonable agreement with the observed data procured from Bangladesh Inland Water Transport Authority.

The propagation of optical solitons via nonlinear metamaterials with cubic-quintic nonlinearity, detuning intermodal dispersion, self steepening effect, and nonlinear third and fourth-order dispersions is the focus of this study. To find the optical solitons and other solutions, the extended sinh-Gordon equation expansion method is applied to the aforementioned model. As a result, dark, bright, combined dark–bright, singular, combined singular soliton, and singular periodic wave solutions are obtained. To our best knowledge, the application of the method to the model, and the acquired combined soliton solutions are novel. To understand the nonlinear propagation theory of solitons in metamaterials, the reported outcomes can be enriched by the soliton theory.

In this study, a fourth-order nonlinear wave equation with variable coefficients was investigated. Through appropriate choice of the free parameters and using the simplified linear superposition principle (LSP) and velocity resonance (VR), the examined equation can be considered as Hirota–Satsuma–Ito, Calogero–Bogoyavlenskii–Schiff and Jimbo–Miwa equations. The main objective of this study was to obtain novel resonant multi-soliton solutions and investigate inelastic interactions of traveling waves for the above-mentioned equation. Novel resonant multi-soliton solutions along with their essential conditions were obtained by using simplified LSP, and the conditions guaranteed the existence of resonant solitons. Furthermore, the obtained solutions were used to investigate the dynamic and fission behavior of Y-type multi-soliton waves. For an accurate investigation of physical phenomena, appropriate free parameters were chosen to ascertain the impact on the speed of traveling waves and the initiation time of fission. Three-dimensional and contour plots of the obtained solutions are presented in Figures 1–6. Additionally, two nonlinear equations were formulated and investigated using VR, and the related soliton molecules were simultaneously extracted. The reported resonant Y-type multi-soliton waves and equations are new and have not been previously investigated. They can be used to explain modeled physical phenomena and can provide information about dynamic behavior of shallow water waves.