Search Results (49)

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the $\beta$-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the $\beta$-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering. Full article

The paper consists of various types of wave solutions for the truncated M-fractional Bateman–Burgers equation, a significant mathematical physics equation. This model describes the nonlinear waves and solitons in different physical fields such as optical fibers, plasma physics, fluid dynamics, traffic flow, etc. Through the application of the ${exp}_a$ function method and the modified simplest equation method, we are able to obtain exact series of soliton solutions. The results differ from the current solutions of the Bateman–Burgers model because of the fractional derivative. The achieved results could be helpful in various engineering and scientific domains. The Mathematica software is used to assist in obtaining and verifying the exact solutions and to obtain contour plots of the solutions in two and three dimensions. To ensure that the model in question is stable, a stability analysis is also carried out using the modulation instability method. Future research on the system in question and related systems will benefit from the findings. The methods used are simple and effective. Full article

Varshavskii’s ‘Diffusion Theory’, less investigated due to its limited international visibility, can offer one of the simplest and, on the other hand, high-accuracy methods for evaluating the ignition delay of fossil fuel and biofuel droplets, including their blend. In this study, experimental pre-tests were conducted to determine pre-existing subject knowledge on stationary droplet combustion at ambient pressure and temperatures varying from 935 to 1010 K followed by simulation of droplet ignition times. The test fuels were mineral diesel (DF), RME and a 20% RME blend with DF. Simulations were performed for isobaric conditions. Using the detailed transport model and detailed chemical kinetics, the necessary rearrangements were made for the governing equations to meet the criteria for modern fuels (biodiesel, diesel, and blend). The influence of different physical parameters, such as droplet radius, or initial conditions, on the ignition delay time was investigated. The high sensitivity of the proposed methodology to experimental results was substantiated. Full article

This paper investigates the bifurcation dynamics and optical soliton solutions of the integrable quintic Kundu–Eckhaus (QKE) equation with an M-fractional derivative. By adding quintic nonlinearity and higher-order dispersion, this model expands on the nonlinear Schrödinger equation, which makes it especially applicable in explaining the propagation of high-power optical waves in fiber optics. To comprehend the behavior of the connected dynamical system, we categorize its equilibrium points, determine and analyze its Hamiltonian structure, and look at phase diagrams. Moreover, integrating along periodic trajectories yields soliton solutions. We achieve this by using the simplest equation approach and the modified extended Tanh method, which allow for a thorough investigation of soliton structures in the fractional QKE model. The model provides useful implications for reducing internet traffic congestion by including fractional temporal dynamics, which enables directed flow control to avoid bottlenecks. Periodic breather waves, bright and dark kinky periodic waves, periodic lump solitons, brilliant-dark double periodic waves, and multi-kink-shaped waves are among the several soliton solutions that are revealed by the analysis. The establishment of crucial parameter restrictions for soliton existence further demonstrates the usefulness of these solutions in optimizing optical communication systems. The theoretical results are confirmed by numerical simulations, highlighting their importance for practical uses. Full article

New types of truncated M-fractional wave solitons to the simplified Modified Camassa–Holm model, a mathematical physics model, are obtained. This model is used to explain the unidirectional propagation of shallow water waves. The required solutions are obtained by utilizing the simplest equation, the Sardar subequation, and the generalized Kudryashov schemes. The obtained results consist of the dark, singular, periodic, dark-bright, and many other analytical solitons. Dynamical behaviors of some obtained solutions are represented by two-dimensional (2D), three-dimensional (3D), and Contour graphs. An effect of fractional derivative is shown graphically. The results are newer than the existing results of the governing equation. Obtained solutions have much importance in the various areas of applied science as well as engineering. We concluded that the utilized methods are helpful and applicable for other partial fractional equations in applied science and engineering. Full article

In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim is to construct closed-form solutions of the equation using a powerful technique, namely the Lie group analysis method. Firstly, we derive the Lie point symmetries of the equation. Thereafter, the equation is reduced to non-linear ordinary differential equations using symmetry reductions. Furthermore, the solutions of the equation are derived using the extended Jacobi elliptic function technique, the simplest equation method, and the power series method. In conclusion, we construct conservation laws for the equation using Noether’s theorem and the multiplier approach, which plays a crucial role in understanding the behavior of non-linear equations, especially in physics and engineering, and these laws are derived from fundamental principles such as the conservation of mass, energy, momentum, and angular momentum. Full article

Kinematics is a hot topic in robotic research, serving as a foundational step in the synthesis and analysis of robots. Forward kinematics and inverse kinematics are the prerequisite and foundation for motion control, trajectory planning, dynamic simulation, and precision guarantee of robotic manipulators. Both of them depend on the displacement models. Compared with the previous work, finite screw is proven to be the simplest and nonredundant mathematical tool for displacement description. Thus, it is used for displacement modelling of serial robots in this paper. Firstly, a finite-screw-based method for formulating displacement model is proposed, which is applicable for any serial robot. Secondly, the procedures for forward and inverse kinematics by solving the formulated displacement equation are discussed. Then, two typical serial robots with three translations and two rotations are taken as examples to illustrate the proposed method. Finally, through Matlab simulation, the obtained analytical expressions of kinematics are verified. The main contribution of the proposed method is that finite-screw-based displacement model is highly related with instantaneous-screw-based kinematic and dynamic models, providing an integrated modelling and analysis methodology for robotic mechanisms. Full article

Chitosan is a biopolymer that can be subjected to a variety of chemical modifications to generate new materials. The properties of modified chitosan are affected by its degree of deacetylation (DDA), which corresponds to the percentage of D-glucosamine monomers in its polymeric structure. Potentiometric titration is amongst the simplest, most readily available, and most cost-effective methods of determining the DDA. However, this method often suffers from a lack of precision, especially for modified chitosan resins. This is in large part because the equation used to calculate the DDA does not consider the molecular weight of the chemically modified monomeric units. In this paper, we introduce a new equation that is especially suited for modified chitosan bearing three different types of monomers. To test this equation, we prepared naphthalene–chitosan resins and subjected them to potentiometric titration. Our results show that our new equation, which is truer to the real structure of the polymeric chains, gives higher DDA values than those of the routinely used equations. These results show that the traditional equations underestimate the DDA of modified chitosan resins. Full article

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave variables is investigated using the Painlevé test. Periodic and solitary wave solutions of the studied equation are presented. The investigated equation belongs to the class of generalized nonlinear Schrödinger equations and may be used for the description of optical solitons in a nonlinear medium. Full article

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width. Full article

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as $u=b{e}^{-a{x}^2}$, $a<0$, $a,b$ being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form $u=b{e}^{-a\frac{x^2}{2}}$, respectively, for the third-order PDE, $u=A{e}^{i\Phi }$, where the amplitude b and the phase function a are complex-valued functions, $A>0$, and $\Phi$ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation. Full article

The advent of the Internet of Things brought a new age of interconnected device functionality, ranging from personal devices and smart houses to industrial control systems. However, increased security risks have emerged in its wake, in particular self-replicating malware that exploits weak device security. Studies modeling malware epidemics aim to predict malware behavior in essential ways, usually assuming a number of simplifications, but they invariably simplify the single most important subdynamics of malware: random propagation. In our previous work, we derived and presented the first exact mathematical model of random propagation, defined as the subdynamics of propagation of a malware model. The propagation dynamics were derived for the SIS model in discrete form. In this work, we generalize the methodology of derivation and extend it to any Markov chain model of malware based on random propagation. We also propose a second method of derivation based on modifying the simplest form of the model and adjusting it for more complex models. We validated the two methodologies on three malware models, using simulations to confirm the exactness of the propagation dynamics. Stochastic errors of less than 0.2% were found in all simulations. In comparison, the standard nonlinear model of propagation (present in ∼95% of studies) has an average error of 5% and a maximum of 9.88% against simulations. Moreover, our model has a low mathematical trade-off of only two additional operations, being a proper substitute to the standard literature model whenever the dynamical equations are solved numerically. Full article

This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case. Full article

We prove that the simplest gravitational symmetry-reduced models describing cosmology and black hole mechanics are invariant under the Schrödinger group. We consider the flat FRW cosmology filled with a massless scalar field and the Schwarzschild black hole mechanics and construct their conserved charges using the Eisenhart–Duval (ED) lift method in order to show that they form a Schrödinger algebra. Our method illustrates how the ED lift and the more standard approach analyzing the geometry of the field space are complementary in revealing different sets of symmetries of these systems. We further identify an infinite-dimensional symmetry for those two models, generated by conserved charges organized in two copies of a $\mathfrak{Witt}$ algebra. These extended charge algebras provide a new algebraic characterization of these homogeneous gravitational sectors. They guide the path to their quantization and open the road to non-linear extensions of quantum cosmology and quantum black hole models in terms of hydrodynamic equations in field space. Full article

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions. Full article