Dimensional Equation Research Articles

Propagation of lump and travelling wave solutions to the (4+1)-dimensional Fokas equation arise in mathematical physics

Propagation of lump and travelling wave solutions to the (4+1)-dimensional Fokas equation arise in mathematical physics

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Soliton, lump and hybrid solutions of a generalized (2+1)-dimensional Benjamin-Ono equation in fluids

Soliton, lump and hybrid solutions of a generalized (2+1)-dimensional Benjamin-Ono equation in fluids

Darcy-Forchheimer magnetohydrodynamic flow of non-Newtonian Reiner-Rivlin hybrid nanofluid bounded by double-revolving disks

The applications of fluid flow and heat transfer between coaxial double-revolving disks are diverse and can be found in various engineering and scientific domains. In general, the utilization of engine oil ( EO) mixed with hybrid nanoparticles serves various purposes in engine cooling, such as maintaining viscosity-temperature properties, preventing corrosion, and achieving other cooling-related objectives. This investigation focuses on the Darcy-Forchheimer flow of a non-Newtonian Reiner-Rivlin hybrid nanofluid confined between double-revolving disks. The hybrid nanoparticles used include cadmium telluride ( CdTe) and titanium dioxide ( TiO2), combined with the base fluid engine oil. The dimensional flow equations are converted into dimensionless forms using suitable similarity transformations. Subsequently, a semi-analytical methodology known as the homotopy analysis method ( HAM) is utilized to tackle this problem. Graphs are used to determine the effects of the various flow parameters of the hybrid nanofluids on the velocities, temperature, skin friction coefficient, and the Nusselt number. After validating the findings against previous research, an attractive agreement was observed. Some major outcomes from this present problem are that axial, radial, and temperature profiles increase with the rise of the cross-viscosity parameter while the tangential velocity decreases. Radial velocity increased, and tangential velocity diminished with an enhanced magnetic field parameter. Radial velocity is increased for higher values of the dimensionless forced parameter. For higher values of the porosity parameter, the temperature profile improved. Also, the exploration of the Nusselt number and skin friction for the disks, incorporating the effects of a magnetic field and porous medium, reveals that skin friction for both disks increase with the rising Reynolds number and volume fraction of the cadmium telluride nanoparticles.

A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation

A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation

Data-driven prediction of vortex solitons and multipole solitons in whispering gallery mode microresonator

Deep learning incorporating physics knowledge has become a powerful tool for studying the dynamic behavior of high-dimensional nonlinear systems. In this paper, the two-stage mini-batch resampling of adaptive physics-informed neural network (TMA-PINN) method is proposed to solve the (2 + 1)-dimensional variable-coefficient Lugiato-Lefever equation (vLLE). The vortex soliton in the WGM microresonator with different external excitation is investigated by TMA-PINN. It is found that external excitation can cause the rotation of vortex solitons. In addition, the effect of topological charge and external excitation on the dynamical characteristics of spatial solitons including vortex solitons and multipole solitons are investigated. The results show that the final shape of the rotation of vortex solitons and the number of azimuth lobes of multipole solitons are controlled by topological charges. Compared with classical PINN, TMA-PINN can better handle the gradient balance of various loss terms in (2 + 1)-dimensional vLLE to reconstruct the dynamic behavior of WGM microresonator solitons, having potential applications in other nonlinear systems.

Novel (3 + 1)-Dimensional Variable-Coefficients Boussinesq-Type Equation: Exploring Integrability, Wronskian, and Grammian Solutions

Abstract In this paper, we introduce a novel (3+1)-dimensional variable-coefficients Boussinesq-type equation. We
analyze its integrability using the Painlev´e test and the N -soliton solution, demonstrating that both tests yield identical
conditions. Using the Hirota bilinear form of the equation, we derive Wronskian and Grammian determinant solutions
utilizing Pl¨ucker relations and the Jacobi identity for determinants. In particular, we use elementary transformation
and long wave limit to get the determinant expression of mth-order lump solutions from the 2mth-order Wronskian
determinant solutions. Furthermore, we reveal a variety of novel semi-rational solutions using the Hirota method and
Grammian determinant techniques.

Multi-kink solitons, resonant soliton molecules and multi-lumps solutions to the (3+1)-dimensional shallow water wave equation

The objective of this research is to seek some new exact solutions to the [Formula: see text]-dimensional shallow water wave equation (SWWE). By means of the Cole–Hopf transformation, the Hirota bilinear expression of the considered equation is developed. Then, taking advantage of the Hirota bilinear approach, the multi-kink solitons solutions are developed. Based on this, the resonant soliton molecules are derived via imposing certain resonant conditions (RCs). Furthermore, the new homoclinic technique is applied to explore the multi-lump solutions. To reveal the performances of the extracted solutions, the corresponding profiles are unveiled graphically through the Maple. For all we know, the derived solutions in this paper have not been probed in other literature and can expand the exact solutions of the considered equation.

Investigating novel optical soliton solutions for a generalized (3+1)-dimensional q-deformed equation

In this study, we investigate the (3+1) q-deformed tanh-Gordon equation due to its importance in the context of mathematical physics. It describes solitonic solutions in quantum field theory; it can sometimes be used in condensed matter physics to describe interactions between particles in magnetic materials or superconductors; it can model light propagation in nonlinear optical fibers or photonic crystals where the refractive index has a q-deformed structure; and it also can be applied in studying shock waves, turbulence and rogue waves where the deformation introduces corrections to classical wave phenomena. Utilizing the (G′ωG′+G+r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\frac{{\\mathfrak {G}}^{\\prime }}{\\omega {\\mathfrak {G}}^{\\prime }+{\\mathfrak {G}}+r})$$\\end{document}-expansion technique, we derive novel analytical solutions that enhance our understanding of the underlying dynamics. Additionally, we employ a finite element method (extended cubic B-spline method) to validate our analytical findings and explore the behavior of the q-deformed equation under different parameter regimes. Our results demonstrate the versatility of the q-deformed framework in generating rich optical phenomena, paving the way for future research in both theoretical and applied physics.

Index Concepts for Linear Differential-Algebraic Equations in Infinite Dimensions

Different index concepts for regular linear differential-algebraic equations are defined and compared in the general Banach space setting. For regular finite dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.

Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves.

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

Exploring Wave Dynamics: Symmetry Reductions and Similarity Solutions to the (2+1) Dimensional ZK-BBM Equation

Exploring Wave Dynamics: Symmetry Reductions and Similarity Solutions to the (2+1) Dimensional ZK-BBM Equation

MHD Free Convection Boundary Layer Flow on a Horizontal Circular Cylinder in a Williamson Hybrid Ferrofluid with Viscous Dissipation

The present research examined the flow of a horizontal circular cylinder immersed in a Williamson hybrid ferrofluid via free convection boundary layer flow. The fluid flow is modelled by means of governing partial differential equations that incorporate viscous dissipation and magnetohydrodynamic (MHD) effects. Prior to being converted into a more practical partial differential equation, these dimensional equations are reduced to a non-dimensional form by utilising appropriate dimensionless variables. The Keller-box approach is then used to solve these equations with fewer dependent variables numerically. Graphical representations are generated, and the impacts of different interested factors are analysed using MATLAB software. According to the results, the hybrid ferrofluid outperformed ferrofluid with the same particle volume fraction in terms of skin friction and convective heat transfer capabilities.

Periodic solutions, breather, lump and interaction solutions of a generalized (2+1)-dimensional Hirota bilinear equation via bilinear neural network method

Abstract The main focus of this paper is to address a generalized (2+1)-dimensional Hirota bilinear equation utilizing the bilinear neural network method. The paper presents the periodic solutions through a single-layer model of [3-4-1], followed by breather, lump and their interaction solutions by using double-layer models of [3-3-2-1] and [3-3-3-1], respectively. A significant innovation introduced in this work is the computation of periodic cross-rational solutions through the design of a novel [3-(2+2)-4-1] model, where a specific hidden layer is partitioned into two segments for subsequent operations. Three-dimensional and density figures of the solutions are given alongside an analysis of the dynamics of these solutions.

Multiple solitons, multiple lump solutions, and lump wave with solitons for a novel (2+1)-dimensional nonlinear partial differential equation

The primary aim of this paper is to explore exact solutions to a novel (2+1)-dimensional water wave equation that models oceanic wave phenomena. We begin by applying the Hirota bilinear transformation method to derive multi-soliton solutions, including 3-soliton and 4-soliton solutions. Then, utilizing the bilinear form of the equation and the long-wave limit method, we identify multiple lump solutions and interaction solutions between lumps and solitons. These include 1-lump, 2-lump, and 3-lump solutions, as well as interactions between a 1-lump and a 1-soliton, and between a 1-lump and 2-solitons. The physical dynamics of these solutions are visually represented, offering insight into the corresponding oceanic wave phenomena.

Double-loop importance sampling for McKean–Vlasov stochastic differential equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \ imes d$$\\end{document}-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of OTOLr-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \ extrm{TOL}_{\ extrm{r}}^{-4}\\right) $$\\end{document} with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document}. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document} compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

Novel complexiton, rational wave, multi-lumps and the kink solitary wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In this exploration, we aim to seek a number of new exact solutions to the new (3+1)-dimensional integrable fourth-order nonlinear equation, which is widely used to describe the shallow water waves. Employing the Cole-Hopf transformation, we develop its bilinear form. Then, taking advantage of the ansatz function method, a new functional form is utilized to probe the singular complexiton solutions. Based on which, the non-singular complexiton solutions are derived by imposing the constraint conditions. In addition, we find the rational wave solutions and multi-lumps solutions wielding the rational function method and new homoclinic method respectively. At the end, we investigate the kink solitary wave solutions using the variational approach that is based on the variational principle and Ritz method. Meanwhile, the Hamiltonian of the system is also elaborated. Correspondingly, the graphic descriptions of the extracted results are presented to unfold their dynamic behaviors through Maple. As we all know, the findings of this paper are firstly reported and can enlarge the exact solutions of the considered PDE.

Auto-Bäcklund transformation and exact solutions for a new integrable (2+1)-dimensional shallow water wave equation

Shallow water waves (SWWs) are often used to describe water flow and wave movement in shallow water areas. The article introduces a novel (2 + 1)-dimensional SWW equation. We prove that the equation is integrable and obtain an auto-Bäcklund transformation by truncating Painlevé expansion. Using the bilinear form of the equation, a new auto-Bäcklund transformation and some exact solutions are obtained. Besides, a convergent power series solution is derived using Lie symmetry method. These exact solutions can enrich mathematical modeling and help us understand nonlinear wave phenomena. Finally, conserved vectors are derived.