Auxiliary Equation Research Articles

Nonlocal symmetry and group invariant solutions of dissipative (2+1)-dimensional AKNS equation

In this paper, the nonlocal symmetry of dissipative (2+1)-dimensional AKNS equation is constructed by auxiliary equation method. In order to better study new group invariant solutions of AKNS equation with the help of nonlocal symmetry, we introduce an new auxiliary dependent variable, the (2+1) dimensional AKNS equation is extended to a new closed prolonged system. Therefore, the nonlocal symmetry of the original equation is the Lie point symmetry of the new prolonged system. Then, by solving the initial-value problems, the finite symmetry transformation for the prolonged system is derived. Furthermore, by using symmetry reduction method to the prolonged system, the interaction solution between cnoidal waves and solitary waves is given.

Abundant soliton solutions of the modified KdV-KP equation

The precise solutions of the (2+1)-dimensional modified KdV-KP equation are examined in this study. Three reliable methods, namely, the modified auxiliary equation scheme, extended rational sine–cosine methodology, and extended rational sinh–cosh technique, are utilized for the first time to extract multiple soliton solutions for the governing model. These types of solutions have applications in plasma, solid state, neuronal, biological production, and diffusion processes. Graphical simulations of the obtained solutions are included in the form of 3 dimensional surface plots to reveal their physical structure and dynamical features.

Iterated Perturbation Theory for Mott Insulators in a Static Electric Field with Optical Phonons

This article aims to compare the so‐called iterated perturbation theory (IPT) and auxiliary master equation approach (AMEA) impurity solvers for a Mott insulating system driven out of equilibrium by a static electric field. Electronic heat bath and optical phonons are employed as dissipation mechanism of the current‐induced Joule heat that the excited electrons of the lattice experience as the result of the field's driving. Despite its simplicity, the IPT approach yields results that qualitatively are in good agreement with those obtained within the AMEA impurity solver, although fails to reproduce some correlation effects.

Dynamical study of groundwater systems using the new auxiliary equation method

In this research, the exact solitary wave solutions to the non-linear problem of underground water levels are found. This study examines the transport of solutes in groundwater systems with variable density flow. The mathematical equation that is used to explain how groundwater moves through an aquifer is known as the groundwater flow equation and is used in hydrogeology. The auxiliary equation method is used to gain the analytical solutions to the underlying model equation. These solutions are gained in the form of hyperbolic, trigonometric, exponential, and rational function solutions. Mathematica generates two-dimensional and three-dimensional graphs with suitable parameter values. The resulting solutions are also useful for researching wave interactions in several novel structures.

The auxiliary equation approach for solving reaction-diffusion equations

The paper concerns the auxiliary equation method and proposes an approach for finding the most general nonlinear term that can generalize a nonlinear differential equation, so that it keeps solutions expressed in terms of the same auxiliary equation. More precisely we will consider the second order reaction-diffusion equations and we will find the most general nolinear term of this type of equations, for which the solutions can be expressed in terms of the Riccati equation. The procedure is exemplified on Fitzhugh-Nagumo, Dodd-Bullough-Mikhailov, and Klein-Gordon models, seen as reaction-diffusion equations.

Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.

On the Van der Waals model on granular matters with truncated M-fractional derivative

In this work, exact solutions of the Van der Waals model (vdWm) are investigated with a new algebraic analytical method. The closed-form analysis of the vdW equation arising in the context of the fluidized granular matter is implemented under the effect of time-fractional M-derivative. The vdWm is a challenging problem in the modelling of molecules and materials. Noncovalent Van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science and physics. The auxiliary equation which is known as a direct analytical method is constructed for the nonlinear fractional equation. The process includes a transformation based on Weierstrass and Jacobi elliptic functions. Wave solutions of the model are analytically verified for the various cases. Then, graphical patterns are presented to show the physical explanation of the model interactions. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter and other areas of vdWm applications.

Distinct wave profiles relating to a coupled of Schrödinger equations depicting the modes in optics

Diverse aspects of real-world problems are portrayed through nonlinear Schrodinger equations. This exploration considers a pair of fractional order Schrodinger equations describing important nonlinear instability phenomena which are related to heat pulse, quantum condensates, nonlinear acoustics, hydrodynamics, and nonlinear optics. The improved auxiliary equation and improved tanh schemes are imposed on the governing model which generate a bulk of innovative accurate wave solutions. Plenty of wave solutions achieved in this study could be utilized to characterize the interior mechanisms of intricate phenomena relating to the considered nonlinear models. Some of the solutions are selected to figure out in the sense of contours, 3D and 2D outlines after assigning suitable values to the involved free parameters. The directed techniques are employed to explore appropriate wave solutions of the above-stated nonlinear model for the first time ensuring the unique solutions rather than the results in earlier studies. This present work highlights the competency of used methods celebrating huge appropriate wave solutions of suggested complex nonlinear models and gives a guideline for related future research.

Exact traveling wave solutions of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation using extended trial equation method and modified auxiliary equation method

Exact traveling wave solutions of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation using extended trial equation method and modified auxiliary equation method

The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.

On the numerical resolution of the second-order viscoacoustic and viscoelastic anisotropic wave equations using the recursive convolution method

Accurate and efficient seismic wave modeling is fundamental for the high-resolution seismic full-waveform inversion and correct interpretation of seismic data. Based on the viscoelastic mechanism of the Generalized Standard Linear Solid and the constitutive relationship between stress and strain, we demonstrate a generalized recursive formula using the Taylor series expansion to directly compute the temporal convolution in the constitutive equation, rather than solving auxiliary differential equations of the memory variables in the conventional method. New sets of the second order viscoacoustic and viscoelastic anisotropic wave equations are established with the new recursive temporal convolutions, which completely replaces the traditional auxiliary differential equations of the memory variables. The error analysis shows the proposed method can offer better accuracy than the existing recursive methods, and is very competitive with the widely used memory variable method. The numerical results comparing the different methods show the accuracy and feasibility of the generalized recursive convolution method to simulate the viscoacoustic and viscoelastic anisotropic waves.

Wave Solutions for a (2 + 1)-Dimensional Burgers–KdV Equation with Variable Coefficients via the Functional Expansion Method

A (2 + 1)-dimensional fourth order Burgers–KdV equation with variable coefficients (vcBKdV) is studied here and interesting wave-type solutions with variable amplitudes and velocities are reported. The model has been not previously studied in the chosen form and it presents a twofold interest: as a model describing a rich variety of phenomena and as a higher-order equation solving difficulties generated by the presence of the variable coefficients. The novelty of our approach is related to the use of the functional expansion, a solving method based on an auxiliary equation that generalizes other approaches, such as, for example, the G′G proved here. We use a similarity reduction with a nonlinear wave variable that leads to a determining system that it is not usually algebraic, but an over-determined system of partial differential equations. It depends on 14 constant or functional parameters and can generate much richer classes of solutions. Three such classes of solutions, corresponding to the case when a specific form of the generalized reaction–diffusion equation is used as auxiliary equation, are considered. The influence on the dynamical behavior of two important factors, the choices of the auxiliary equation and the form of solution, are studied by providing graphical representations of specific solutions for various values of the parameters.

A unified higher-order unsplit CFS-PML technique for solving second-order seismic equations using discontinuous Galerkin method

The absorbing boundary conditions play a key role in numerical simulations. The complex frequency-shifted perfectly matched layer (CFS-PML), one of the absorbing boundary conditions, has attracted widespread attention because it can efficiently attenuate near-grazing incident waves and evanescent waves. In practical application, the distinct boundary reflections still exist using the CFS-PML in a thin PML region. To further enhance the absorbing performance, we have developed a novel unified high-order unsplit CFS-PML technique. The high-order CFS-PML combined with the discontinuous Galerkin (DG) method is applied to solve the second-order seismic wave equation on an unstructured mesh. In this work, we establish a unified framework for high-order unsplit CFS-PML formulations. The classical CFS-PML is the first-order form of our proposed unsplit high-order approach. By using auxiliary variables and auxiliary ordinary differential equations (AODEs), a higher-order CFS-PML formulation can be reduced to a first-order CFS-PML formula. For an Nth-order AODE CFS-PML, it will lead to (2N-1) additional variables and (2N-1) auxiliary differential equations. Taking second- and third-order as examples, we have derived the high-order unsplit AODE CFS-PML expressions in detail. The DG method can effectively deal with complex geological structures, such as caves, cracks, faults, etc., which yields more accurate numerical solutions. We incorporate the high-order unsplit AODE CFS-PML technique into the DG method to approximate seismic wave equations. Their strengths are combined to calculate the propagating wavefield in more complex finite models and obtain better results. In addition, the stability conditions of the second- and third-order CFS-PMLs are provided. We perform the high-order unsplit CFS-PML implementation for seismic wave modeling in several models. The isotropic and anisotropic examples prove that the high-order unsplit CFS-PML enjoys better absorbing performance than the classical CFS-PML. The numerical result for the graben model certifies the feasibility and flexibility of our proposed high-order CFS-PML combined with the DG method.

Formation of solitons for the modified nonlinear Schrödinger equation

The study of modified nonlinear Schrödinger equation plays a significant role in the description of wave propagation through optical waveguides and rogue waves in ocean. The main aim of this work is to obtain solitons as well as other types of exact wave solutions to the modified nonlinear Schrödinger equation. The accurate traveling wave solutions are obtained using the [Formula: see text]-exponential expansion method and the modified auxiliary equation method. Consequently, soliton solutions and periodic wave solutions have been retrieved. The graphical illustrations of the results are provided for suitable values of the parameters involved in the solutions to explain the dynamical nature of the considered equation.

Analysis of bifurcation, chaotic structures, lump and $ M-W $-shape soliton solutions to $ (2+1) $ complex modified Korteweg-de-Vries system

<abstract><p>This research focuses on the fascinating exploration of the $ (2+1) $-dimensional complex modified Korteweg-de Vries (CmKdV) system, exhibiting its complex dynamics and solitary wave solutions. This system is a versatile mathematical model that finds applications in various branches of physics, including fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two newly developed methodologies, namely the auxiliary equation (AE) method and the Hirota bilinear (HB) method, are implemented for the construction of novel solitons in various formats. Numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, singular, periodic, combo, $ W $-shape, mixed trigonometric, exponential, hyperbolic, and rational, based on the proposed methods. Furthermore, we also find some lump solutions, including the periodic cross rational wave, the homoclinic breather (HB) wave solution, the periodic wave solution, the $ M $-shaped rational wave solution, the $ M $-shaped interaction with one kink wave, and the multiwave solution, which are not documented in the literature. In addition, we employ the Galilean transformation to derive the dynamic framework for the presented equation. Our inquiry includes a wide range of topics, including bifurcations, chaotic flows, and other intriguing dynamic properties. Also, for the physical demonstration of the acquired solutions, 3D, 2D, and contour plots are provided. The resulting structure of the acquired results can enrich the nonlinear dynamical behaviors of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.</p></abstract>

Novel wave solutions for the sixth-order Boussinesq equation arising in nonlinear lattice dynamics

<p>This study examines a class of Boussinesq equations with sixth-order using two promising analytical methods. The equation in question is among the frontier evolution equations with significant relevance in nonlinear lattice dynamics. To study this model, the Kudryashov method and the modified auxiliary equation method are employed due to their analytical precision in constructing several exact wave solutions for the model under examination. As expected, the methods yield many valid solution sets that satisfy all the underlying assumptions of the model. Finally, some of the obtained wave solutions are graphically illustrated, taking into account the parameter values of the model.</p>

Threshold analysis of an algae-zooplankton model incorporating general interaction rates and nonlinear independent stochastic components

<abstract><p>The stochastic nature of ecological systems is fundamental to their modeling and understanding. In this paper, we introduce a comprehensive algae-zooplankton model that incorporates general interaction rate and second-order independent stochastic components. Our model's perturbation component encompasses both white noise and jump processes, enabling us to account for various sources of variability and capture a wide range of potential fluctuations in the system. By utilizing an auxiliary equation, we establish a global threshold for the stochastic system, distinguishing between scenarios of extinction and ergodicity. This threshold serves as a critical determinant of the system's long-term behavior and sheds light on the delicate balance between population persistence and decline in ecological communities. To elucidate the impact of noise on the dynamics of algae and zooplankton, we present a series of numerical illustrations. Through these simulations, we highlight how noise influences not only the extinction time but also the shape of the stationary distribution. Our findings underscore the significant role of stochasticity in shaping ecological dynamics and emphasize the importance of considering noise effects in ecological modeling and management practices.</p></abstract>

Relay model of a fading neuron

This study is a continuation of M. M. Preobrazhenskaya’s work “Relay System of Differential Equations with Delay as a Perceptron Model”, which aimed to combine approaches related to artificial neural networks and modeling biological neurons using differential equations with delay. The model of a single neuron was proposed, which allows for the existence of special modes called “aging” and “dying” behavior of the neuron. The study found a certain range of parameters where the “dying” mode of the neuron exists and numerically demonstrated the existence of the “aging” mode. Purpose. We will unify the concepts of “aging” and “dying” neurons into the term “freezing” neuron. For this neuron, we will analytically construct a solution and find the range of parameters for its existence and stability, which will extend the results of the reference article. Methods. To study this model, an auxiliary equation obtained by exponential substitution in the original equation is considered. Then, the method of step integration of a differential equation with delay and the introduction of additional functions are used. Results. A solution of the “freezing” neuron type for the original model is constructed, and the range of parameters for the existence and stability of this solution is described. Conclusion. The study obtained an extension of results for solutions of a special type in the model proposed by M. M. Preobrazhenskaya.

Exponential Response Matrix Spectral Nodal Method for the Discrete Ordinates Neutral Particle Transport Model

In this work, the response matrix‐exponential nodal method is presented to solve fixed source neutral particle transport problems with isotropic scattering and discrete ordinates formulation in two‐dimensional Cartesian geometry. The method is based on the response matrix scheme by using the general solution of the transverse integrated SN nodal equations with exponential approximations for the leakage terms. This approach eliminates the need for additional auxiliary equations found in the spectral Green’s function (SGF) schemes, streamlining internodal computation to a matrix‐vector multiplication. The formulation includes a lambda parameter possible determined by the medium characteristics, e.g., the scattering ratio coefficient and/or empirical methods. Two model problems with low absorption rates were investigated to analyze the performance of the proposed method. The numerical solutions exhibited superior accuracy for the coarsest spatial grid configurations in both homogeneous and heterogeneous study cases. However, refining the spatial grid revealed a relative loss in accuracy compared to the RM‐CN method, potentially due to increased truncation error propagation. Nevertheless, all numerical solutions converge to the reference solution when refining the spatial discretization grid. Comparison of both response matrix schemes demonstrated similar convergence rates and computational efficiency in terms of iterations and CPU consumption.

Application of financial management in regional economic development based on VAR modeling

Abstract This paper constructs a vector autoregressive (VAR) model between the level of financial management and regional GDP. Through the Granger causality test, the panel data at the financial management level and regional economic phenomena are studied, and the empirical distributions in the stationary situation are given by combining the stochastic simulation method. The auxiliary regression equations are established separately for constrained and unconstrained conditions, and the test statistic is constructed to determine whether there is Granger causality between the variables. Finally, from the actual data, a one standard deviation shock is applied to LATS and LGDP, and a response model with a lag of 100 periods is chosen to obtain the impulse response of the level of financial management and regional economic growth. The results show that the impact of the regional economy on the level of financial management is zero in the first period and then increases continuously, reaching a maximum response of 0.0233 in the 49th period, which indicates that there is a bidirectional promotional effect between the level of financial management and the regional economy and that the effect is not very strong but lasts for a long time.