Narrow Results

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

A microcrystalline material refers to a crystallized substance or rock comprised of tiny crystals that can only be observed under a microscope. The strain wave equation is a fourth-order nonlinear partial differential equation encountered in the examination of non-dissipative strain wave propagation within microstructured solids. In this paper, the transmission of waves in microcrystalline materials is dictated by the non-dissipative case of strain wave equation’s structure, accounting for multiple dimensions within microcrystalline structures. The simplest equation method is employed to extract multi-soliton solutions, while the modified Sardar subequation method is applied to identify additional soliton solutions, including bright, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Furthermore, the dynamical system bifurcation theory approach is utilized to investigate the phase diagrams of the governing equation. Further elaboration on the physical dynamical representation of the presented solutions is provided through profile illustrations. A comparison with the existing literature is also provided, highlighting the efficacy of our work. The significance of the acquired outcomes lies in their capacity to portray a wide array of intricate and diverse phenomena observed in both mathematical and physical systems.

The Master Stability Function (MSF) is a crucial tool for understanding the synchronization behavior of coupled nonlinear oscillators. In recent years, the field of fractional calculus and its applications in nonlinear dynamics gained significant attention, leading to an expanded focus on the network dynamics of such systems. This study aims to explore this emerging area by deriving the MSF for coupled fractional-order nonlinear oscillators and investigating their relationship with coupling strength and fractional order. To provide a comprehensive comparison, we utilize the well-known nonlinear oscillators to study the differences and similarities between integer-order and fractional-order MSFs. Our analysis reveals that, similar to integer-order systems, fractional-order coupled nonlinear oscillators exhibit MSFs that can be characterized by the presence of negative values within a finite interval of the normalized coupling parameter. This negative region is crucial as it indicates stable synchronization. Furthermore, we categorize the fractional-order MSFs using the same classifications applied to integer-order MSFs. This classification helps in systematically understanding and comparing the synchronization properties of both types of systems. Our findings are supported by extensive numerical simulations, which demonstrate that the majority of fractional-order coupled oscillators exhibit higher classes of MSF. This higher classification suggests that fractional-order systems have a superior ability to achieve synchronization compared to their integer-order counterparts. In summary, this study underscores the significance of fractional calculus in enhancing our understanding of synchronization in complex systems. The derived MSF for fractional-order nonlinear oscillators provides valuable insights into their dynamic behavior and opens new avenues for research in various scientific and engineering disciplines.

Megastable oscillations are a subject of significant research interest due to their broad range of potential applications. Typically, megastable systems are driven into oscillation by a forcing term. In this paper, we propose a novel megastable oscillator that utilizes a combination of Signum and trigonometric functions. To the best of our knowledge, no 3D megastable system has been found to exhibit hyperchaotic behavior without any forcing term. We demonstrate the megastability of our oscillator using phase portraits and basins of attraction and confirm the oscillations using the Hamiltonian energy method. We also conduct a stability analysis to explore the system’s nature and investigate the impact of parameters using a bifurcation diagram. Furthermore, we present a Lyapunov spectrum to identify regions of chaos, hyperchaos, and periodic oscillations. The results we obtain demonstrate the complexity of the system and its sensitivity to initial conditions, making it well-suited for applications such as random number generation and secure communication.

The analysis of nonlinear models and their complex geometrical shapes with globalized piecewise fractional operators for past heredity characteristics serve as the novelty of and motivation for this research. We explore the dynamics of a financial model in the aforementioned framework, using the well known and extensively applied numerical technique of the fractional piecewise Adams–Bashforth method. The effect of the employed parameters on price index, interest rate, investment demand, variation, and expenditure per investment from the saving amount along with the elasticity of the commercial markets demand are investigated. Various fractional orders are simulated in the sense of the Caputo and Atangana–Baleanu Caputo derivatives in the first and second subintervals, respectively. Moreover, the piecewise existence and uniqueness of the solution for the proposed model are established. The concept of Ulam–Hyers stability is applied to the obtained solution. This study will be helpful for beginners to complex geometrical and crossover dynamical problems in various physical and biological sciences.

The primary objective of this study is to analyze the Hyers–Ulam stability of fractional derivatives for the Black–Scholes model, involving two underlying asset systems and utilizing Caputo fractional derivatives. We employ a fixed-point approach to examine the existence and uniqueness of solutions and to investigate the Hyers–Ulam stability of the given problem. Additionally, we analyze the graphical behavior of the obtained results, demonstrating that the analytical method is highly efficient and delivers precise results for determining approximate numerical solutions. The findings highlight the significant role of fractional approaches in studying nonlinear systems of scientific and physical importance. Furthermore, the graphical analysis, considering various fractional orders and parameter values, unveils new insights and intriguing phenomena associated with the Black–Scholes model.

In this study, we focus on examining Milne’s error bounds through the use of Katugampola fractional integral operators. To attain this, we commence by presenting a novel integral identity, which will serve as the basis for proving different new Milne-type inequalities for differentiable $s$-convex functions, thereby generalizing certain known results in the literature. The study concludes with a few applications to special means.

Vector-borne diseases pose substantial risks to public health and have a high impact on different sectors of life around the world. Hence, it is valuable to construct a conceptual framework for comprehending the dynamics of these infections to introduce efficacious public health control strategies. In this paper, we structure a model for a vector-borne infection to conceptualize the complex dynamics of dengue, including drug resistance and insecticide. The recommended model of dengue fever is presented through a non-integer derivative to capture the role of memory, drug resistance and insecticide. We examine the positivity and boundedness of the solution using analytic skills and the threshold parameter is determined via the next-generation matrix approach. The well-known theorem of fixed-point is utilized to evaluate the solution’s existence and uniqueness. We have provided sufficient conditions for the Ulam–Hyers stability. To delve further into the intricacies, we employ a numerical technique to elucidate how various input factors influence the infection dynamics. The impact of fractional-order parameters, vaccination rates, the rate of immunity waning, biting rates, resistance to drugs and insecticides is visualized. Our analysis has underscored the pivotal significance of certain parameters of the system, which can exert substantial influence over the intensity of dengue fever. In contrast, our findings suggest that parameters such as vaccination rates, the memory index, and fractional order parameters, along with treatment and insecticide policies, hold promise as effective control measures in mitigating the spread and impact of dengue infection.

Finding the solution to constrained monotone nonlinear equations has become more challenging as the number of dimensions rises. In this paper, a projection-based spectral conjugate gradient algorithm for solving monotone nonlinear equations with convex constraints is proposed. This algorithm features two bounded spectral parameters designed by using the quasi-Newton condition and the double-truncating technique. An improved conjugate parameter ensures that the search direction satisfies the sufficient descent and trust-region properties. The global convergence of the algorithm is established under reasonable assumptions. Numerical experiments demonstrate that it is promising and robust for large-scale nonlinear equations with convex constraints in both fixedly and randomly initial point strategies, as well as for the sparse signal reconstruction problem.

This paper characterizes the Lorentzian manifolds endowed with a semi-symmetric non-metric $\rho$-connection (briefly, ssnmρc). First, the existence of semi-symmetric non-metric connection (ssnmc) on Lorentzian manifold is established, and it is shown that an $n$-dimensional Lorentzian manifold equipped with an ssnmρc is a generalized Robertson–Walker spacetime. We also establish the condition for a Lorentzian manifold together with an ssnmρc to be a Robertson–Walker spacetime. In this series, the properties of Ricci semi-symmetric Lorentzian manifold endowed with an ssnmρc, almost Ricci solitons and gradient Ricci solitons are explored. Finally, a non-trivial example of Lorentzian manifold admits an ssnmρc, which is constructed to verify some of our results.

The present work constitutes a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems. It gives a new interpretation of the concept of constant expansion in the homotopy perturbation method.

Most engineering problems, especially heat transfer equations are mostly nonlinear. Variational iteration method (VIM) and homotopy perturbation method (HPM) are useful numerical and analytical methods for solving nonlinear heat transfer equations. Perturbation methods depend on a small parameter, which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two newly powerful analytical methods are introduced to solve nonlinear heat transfer problems in our work; one is the VIM and the other one is the HPM. The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM makes a difficult problem much easier to be evaluated. In this research, these methods are used to solve nonlinear ordinary differential system for melting of a finite slab.

The factorization method of the Hamiltonian in quantum mechanics is used to solve a particular type of Riccati equation. In this paper, He's homotopy perturbation method is applied to a general Riccati equation. The obtained solutions are in recursive sequence forms and, in many cases, these solutions lead to an exact solution. This property generally means that one can solve completely the eigenvalue problem for the Hamiltonian operator. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.

A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein–Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

For the purpose of constructing the exact periodic solutions of nonlinear wave equations, it has been proposed to use a method known as the Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. It has been demonstrated that the periodic solutions obtained using this method contain both solitary wave solutions and shock wave solutions in some instances.

To find periodic solutions of nonlinear equations, the Jacobi elliptic function expansion method is used. This strategy is more extensive than expanding the hyperbolic tangent series. This approach can produce periodic shock wave solutions such as solitary wave solutions.

In the last decade, many potent analytical methods have been utilized to find the approximate solution of nonlinear differential equations. Some of these methods are energy balance method (EBM), homotopy perturbation method (HPM), variational iteration method (VIM), amplitude frequency formulation (AFF), and max–min approach (MMA). Besides the methods mentioned above, the Akbari–Ganji method (AGM) is a highly efficient analytical method to solve a wide range of nonlinear equations, including heat transfer, mass transfer, and vibration problems. In this study, it was constructed the approximate analytic solution for movement of two mechanical oscillators by employing the AGM. In the derived analytical method, both oscillator motion equations and the sensitivity analysis of the frequency were included. The AGM was validated through comparison against Runge–Kutta fourth-order numerical method and an excellent agreement was achieved. Based on the results, the highest sensitivity of the oscillation frequency is related to the mass. As $\alpha$ and $\beta$ increase, the slope of the system velocity and acceleration will increase.

Significance: The thermal energy transfer in nanofluid flow over an exponentially stretching surface has crucial practical configurations in various industrial processes, and it has potential applications in heat exchangers, chemical engineering, energy harvesting, and material processing. Purpose: This study is devoted to exploring the features of free convection in the thermally stratified, unsteady flow of Casson fluid over an inclined, exponentially stretching surface. Moreover, the implications of nonlinear thermal radiation, activation energy, and thermal/salute stratification effects are examined over the thermal energy transport distributions. Diffusion-thermo and thermo diffusion impressions are also taken into consideration. Methodology: By introducing reasonable transformations, partial differential equations are altered into ordinary differential equations. A nonlinear system of differential equations is solved numerically by employing the Midrich numerical technique. Findings: The impacts of diverse fluid parameters like the Soret/Dufour number, temperature difference parameter, radiation parameter, thermal/salute stratification parameter, magnetic parameter, and Prandtal number are assessed and depicted in plots by explaining the physical justifications of each parameter. Also, numerical values of sink friction and local Nusselt and Sherwood numbers are computed and examined for different values of pertinent variables involved in the problems. It is found that the rate of thermal energy transport is significantly enhanced by the larger estimation of the radiation parameter. Furthermore, it is perceived that the escalation in the temperature ratio constant leads to increased thermal convection in the fluid, while the larger thermal stratification constant decays the rate of heat transport in the fluid. Additionally, the rate of thermal transport is de-escalated due to the escalation in the intensity of thermal stratification parameter.

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse or instability point) is one of the key problems in power system operations and planning such an index indicates the severity of a power system with regard to voltage collapse or instability. While much work has been done on the former problem, the latter problem poses a more severe challenge both to the engineering and mathematical communities.

This paper first presents a review of the main methods for detecting Hopf bifurcation in the general field of bifurcation theory and then attempts to generalize the previously studied fold detection and index methods to the Hopf case. The proposed Hopf test functions should be of wide interest while the proposed test function based index method gives an indication of the distance of the current operating point from the Hopf instability. Promising numerical results have been obtained using some standard dynamical test systems.

We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), which is obtained by extending the space from ℝ³ to ℂ³, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of two strong attractors unrelated to the Lorenz attractor. We calculate its Lyapunov exponents and show that two of them are 0, and the other four are double and negative. Hence the attractors are nonchaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (𝕀LM), which is defined in the same space ℂ³ by multiplying with the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show evidence that the basin of attraction of each attractor is a big open set of ℂ³. The chaotic limit set L_ℂ ⊂ ℂ³ obtained from the classical Lorenz attractor L₀ of (LM) by moving it with the (𝕀LM) has two positive Lyapunov exponents, but only captures a set of 6D-volume 0 in its basin of attraction. Hence this attractor may be hyperchaotic in ℝ⁵.