Dimensional Equation Research Articles

New (3+1)-dimensional integrable generalized KdV equation: Painlevé property, multiple soliton/shock solutions, and a class of lump solutions

The present work aims to examine a newly proposed (3+1)-dimensional integrable generalized Korteweg-de Vries (gKdV) equation. By employing the Weiss- Tabor-Carnevale technique in conjunction with Kruskal ansatz, we establish the com- plete integrability of the suggested model by demonstrating its ability to satisfy the Painlev´e property. The bilinear form of the (3+1)-dimensional gKdV equation is em- ployed to construct multiple soliton solutions. By manipulating the various values of the corresponding parameters, we generate a category of lump solutions that exhibit localization in all dimensions and algebraic decay.

Travelling Wave Solutions for Some Time-Fractional Nonlinear Differential Equations

This study employs the powerful generalized Kudryashov method to address the challenges posed by fractional differential equations in mathematical physics. The main objective is to obtain new exact solutions for three important equations: the (3+1)-dimensional time fractional Jimbo-Miwa equation, the (3+1)-dimensional time fractional modified KdV-Zakharov-Kuznetsov equation, and the (2+1)-dimensional time fractional Drinfeld-Sokolov-Satsuma-Hirota equation. The generalized Kudryashov method is highly versatile and effective in addressing nonlinear problems, making it a pivotal component in our research. Its adaptability makes it useful in diverse scientific disciplines. The method simplifies complex equations, improving our analytical capabilities and deepening our understanding of system dynamics. Additionally, we define fractional derivatives using the conformable fractional derivative framework, providing a strong foundation for our mathematical investigations. This paper examines the effectiveness of the generalized Kudryashov method in solving complex challenges presented by fractional differential equations and aims to provide guidance for future studies.

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

In this paper, we investigate and analyze various nonlinear phenomena of a new (2+1)-dimensional KdV equation with variable coefficients, and successfully obtain breather/rogue wave solutions and interaction solutions of the KdV equation by using the bilinear neural network method and symmetry transformation. Subsequently, we analyze the dynamical characteristics and evolution process of these obtained solutions through the 3-D animations, and find a series of interesting nonlinear phenomena concerning breather/rogue waves, such as fission, regeneration, annihilation, collision, and controllable interaction phenomena on nonzero backgrounds. This paper provides a more intuitive understanding for the nonlinear phenomena of these obtained solutions, and these nonlinear phenomena have potential application value in fluid dynamics, elastic mechanics and other fields of nonlinear science.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

Exact solutions of steady Euler equations in two dimension by functional separation

Exact solutions of two dimensional Euler equations for incompressible fluid flows are found by the functional separation of variables. More specifically, two independent variables are considered in the Cartesian coordinates, polar coordinates and also general orthogonal curvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several new solutions are obtained and other classical solutions are rediscovered.

Novel interaction solutions to the (3+1)-dimensional Hirota bilinear equation by bilinear neural network method

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the ([Formula: see text])-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields.

Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT-symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.

Solitary wave solutions of Camassa–Holm nonlinear Schrödinger and $$(3+1)$$-dimensional Boussinesq equations

Solitary wave solutions of Camassa–Holm nonlinear Schrödinger and $$(3+1)$$-dimensional Boussinesq equations

Bipolar electron waveguides in two-dimensional materials with tilted Dirac cones

We show that the (2+1)-dimensional massless Dirac equation, which includes a tilt term, can be reduced to the biconfluent Heun equation for a broad range of scalar confining potentials, including the well-known Morse potential. Applying these solutions, we investigate a bipolar electron waveguide in 8–Pmmn borophene, formed by a well and barrier, both described by the Morse potential. We demonstrate that the ability of two-dimensional materials with tilted Dirac cones to localize electrons in both a barrier and a well can be harnessed to create pseudogaps in their electronic spectrum. These pseudogaps can be tuned through varying the applied top-gate voltage. Potential opto-valleytronic and terahertz applications are discussed.

Non‐singular complexiton, singular complexiton and complex multiple soliton solutions to the (3 + 1)‐dimensional nonlinear evolution equation

This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs).

Flow Reversal of Unsteady Oscillatory Flow in Combined Convection Fully Developed Vertical Channel

Numerical simulations of flow and heat transfer in an unsteady oscillatory fully developed combined convection are conducted. The temperature on the right wall varies with sinusoidal rhythm over time within a locked mean temperature, whereas the temperature on the left wall remains constant. Both walls are also considered to be stationary. A numerical solution for the momentum and energy equations obtained using the Runga- Kutta method is used to analyze the effect of periodic oscillation on the velocity and temperature configurations, as well as the progression of heat generations. The combined convection process of heat transfer and its flow pattern in a vertical channel is important in many applications, including environmental, industrial, and engineering. This study is prompted by a problem with the heat transfer process, which is difficult, expensive, and time-consuming. The purpose of this study is to develop a mathematical model, find a numerical solution, and conduct parametric studies on the double-diffusive fully developed vertical channel with unsteady oscillatory flow, as well as to investigate the effect of various parameters on the velocity and temperature profiles. The effect of different dimensional parameters is tested and the flow reversal phenomenon is discussed. The dimensional equations are transformed into non-dimensional equations using the similarity technique. Then, the built-in program dsolve in Maple is used to numerically solve the boundary value problem (BVP). A validation study is conducted on a previously reported problem to confirm the validity of the present calculation. The flow and temperature numerical findings are represented visually. It is found that a large development of flow is caused by the plate temperature oscillation with high amplitude, therefore, the nature of the flow differs from the non-oscillating case.

Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

On the Painlevé integrability and nonlinear structures to a (3 + 1)-dimensional Boussinesq-type equation in fluid mediums: Lumps and multiple soliton/shock solutions

This work examines the Painlevé integrability of a (3 + 1)-dimensional Boussinesq-type equation. Using the Mathematica program, we rigorously establish Painlevé's integrability for the suggested problem. By utilizing Hirota's bilinear technique, we obtain the dispersion relations and phase shifts, which enable us to derive multiple soliton solutions. In addition, we systematically derive a wide range of lump solutions using the Maple symbolic computation. The investigation extends to encompass a variety of exact solutions with distinct structural features, including kink, periodic, singular, and rational solutions. This comprehensive analysis illustrates the profound richness of the model's dynamics and its potential to elucidate diverse nonlinear wave phenomena across various physical contexts. Therefore, the results that we will obtain play a vital role in understanding the mechanism of generation and propagation of many mysterious phenomena that arise in various scientific fields, including plasma physics, fluid mechanics, and the propagation of waves on the surfaces of seas and oceans to optical fibers.

Breather and soliton solutions of a generalized (3 + 1)-dimensional Yu–Toda–Sasa–Fukuyama equation

Fluid mechanics is a branch of physics that focuses on the study of the behavior and laws of motion of fluids, including gases, liquids, and plasmas. The Yu–Toda–Sasa–Fukuyama equation, a class of Kadomtsev–Petviashvili type equations, is a significant integrable model with applications in fluids and other fields. In this paper, we study breather and soliton solutions of a generalized (3 + 1)-dimensional YTSF equation. By utilizing the Hirota bilinear method and Painlevé analysis, we construct solutions in the form of trigonometric and hyperbolic functions and analyze the interaction between waves graphically. We consider the characteristics of wave distribution along characteristic lines to obtain the distance between each wave and the angle generated, which is beneficial for understanding the ocean wave superposition effect. Additionally, we examine the dynamic characteristics of the wave, such as amplitude, velocity, period, shape, position, width, and phase. Furthermore, we investigate the effects of the system parameters on solitons and breathers.

Analyzing Soliton Solutions of the Extended (3 + 1)-Dimensional Sakovich Equation

This work focuses on the utilization of the generalized exponential rational function method (GERFM) to analyze wave propagation of the extended (3 + 1)-dimensional Sakovich equation. The demonstrated effectiveness and robustness of the employed method underscore its relevance to a wider spectrum of nonlinear partial differential equations (NPDEs) in physical phenomena. An examination of the physical characteristics of the generated solutions has been conducted through two- and three-dimensional graphical representations.

Groundwater–surface water exchange from temperature time series: A comparative study of heat tracer methods

The importance of the interaction between groundwater and surface water is increasingly being recognized for both understanding and managing water systems. Many efforts have been made to characterize and quantify groundwater–surface water exchange. In particular, temperature–based methods have quickly established themselves given their monetary and practical advantages. In the last 15 years, several methods for interpreting passive temperature time series measured in the streambed have been developed. Still, the benchmarking of these methods has only been carried out in specific and distinct hydrological conditions.This article aims to fill this research gap by benchmarking the performance of six commonly used methods for deriving seepage fluxes using a two-year-long temperature time series covering various meteorological and thermal conditions. This work compares three analytical methods that calculate seepage flux using the amplitude and/or phase of the temperature signals, and three numerical methods that use different schemes to inversely solve the one–dimensional heat transport equation in the streambed. The temperature measurements were made in the context of an international dispute between Chile and Bolivia over the status and use of the waters of the Silala River, Northern Chile. Flux estimations are tested against Darcy’s flux derived from measured hydraulic gradient and hydraulic conductivity.Flux estimations from the benchmarked methods ranged from −0.5 to 3.5 m/d (with positive fluxes directed downwards), whereas fluxes estimated using Darcy’s law ranged from 0.5 to 6 m/d. Results show that the amplitude method is the best–performing method. This method is best suited for estimating the direction of the fluxes, while the method using both the thermal amplitude and phase is best suited for monthly flux trends, and the combination of a Local Polynomial (LP) method and a Maximum Likelihood Estimator (MLE) method (LPMLEn) is appropriate for estimating flux in transient conditions.The use of heat as a tracer proved to be an effective tool for monitoring groundwater–surface water exchange in a river reach for two years, and yielded exchange flux estimates with lower point-scale variability than Darcy’s law.

Distinctive approach for MHD natural bioconvection flow of Reiner–Rivlin nano-liquid due to an isothermal sphere

The analysis of heat excess through non-Newtonian fluid is noticed proper and promising arena for biomedical technological applications than Newtonian liquid. Herein, a type of non-Newtonian (Reiner–Rivlin) nanomaterial has been scrutinized with gyrotactic microorganisms around an isothermal sphere with surface of zero mass flux. Being mindful of that the model of Buongiorno’s nanofluid is utilized. Non-dimensional formulas for the basic dimensional partial differential equation system are obtained using appropriate non-similarity transformations. MATLAB function bvp4c code is adopted to get the non-similar solutions. The computations disclosed that the velocity of Reiner–Rivlin nanofluid and its gradient accelerate away from the surface of sphere to infinity and reduces in proximity to the sphere surface for higher values of Reiner–Rivlin factor K. The rate of surface density transport, rate of surface heat transport, and gradient of velocity diminish as the stream wise coordinate boosts. As the problem is subjected to special circumstances, there is a remarkable agreement between the acquired and prior numerical calculations resulting from the comparison.

Evolution of an interacting fermion–antifermion pair in the near-horizon of the BTZ black hole

In this study, we investigate the evolution of a composite system comprising a fermion–antifermion pair engaged in Cornell-type non-minimal interaction in the near-horizon region of a BTZ black hole. Our exploration involves the derivation of an exact solution for the covariant two-body Dirac equation, derived from quantum electrodynamics through the action principle. To commence our analysis, we formulate the relevant equation, resulting in a 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} dimensional matrix equation that governs the relative motion of the fermion–antifermion pair. Notably, we demonstrate that this matrix equation results in an exactly solvable wave equation, enabling us to determine the relativistic frequency modes for this spinless static composite system. Our findings unveil a temporal decay in these modes, with the decay time explicitly dependent on both the inter-particle interaction and the spacetime parameters. Our comprehensive examination extends to a detailed analysis of the system’s evolution, shedding light on the influence of inter-particle interaction on the evolution of a fermion–antifermion pair in the near-horizon region of the BTZ black hole.

Study of explicit travelling wave solutions of nonlinear (2 + 1)-dimensional Zoomeron model in mathematical physics

Abstract This article studeis the nonlinear (2 + 1)-dimensional Zoomeron equation by utilizing the various prominent analytical approaches namely the unified method and the extended hyperbolic function approach. The analysis in the current paper demonstrates the presence of travelling wave solutions. The applied methods are utilized as powerful tools to investigate and solve the model. The results obtained through these analytical methods reveal insightful patterns in the behavior of the Zoomeron equation. The significance of our work lies in the uniqueness of the methods employed. The two methods are applied to systematically analyze the equation, revealing hidden patterns and structures within its solution space. This leads to the discovery of a collection of solitary wave solutions such as kink waves, singular kink waves, periodic waves and dark soliton using contour plots, 3D and 2D graphics. In this article, we definitely prove that as the free parameters change, the wave amplitude changes as well. It is shown that the applied strategies are more effective and may be implemented to a variety of contemporary nonlinear evolution models emerging in mathematical physics.

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.