Burgers Equation Research Articles

Instability of periodic waves for the Korteweg–de Vries–Burgers equation with monostable source

In this paper, it is proved that the KdV–Burgers equation with a monostable source term of Fisher–KPP type has small-amplitude periodic traveling wave solutions with finite fundamental period. These solutions emerge from a subcritical local Hopf bifurcation around a critical value of the wave speed. Moreover, it is shown that these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. To that end, classical perturbation theory for linear operators is applied in order to prove that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution, which intersects the unstable complex half plane.

Extension of optimal auxiliary function method to nonlinear Sine Gordon differential equations

In this paper, the new semi-analytical method called optimal auxiliary function method has been extended for finding the approximate solution of the non-linear Sine Gordon and Burger equations. The parameters of the auxiliary function’s optimum values control convergence, and they were determined using the well-known collocation approach. The Modified Adomian Decomposition Technique, Variational Homotopy Perturbation Method, Variational Iteration Method, and the Adomian Decomposition Method are used to compare the results produced by the suggested technique. It has been demonstrated that the suggested approach is straightforward, quickly convergent, effective, and consistent. The proposed method can also be made more accurate when making higher-order approximations.

The Burgers-type equation driven by a stochastic measure

We study the one-dimensional equation driven by a stochastic measure μ \mu . For μ \mu we assume only σ \sigma -additivity in probability. Our results imply the global existence and uniqueness of the solution to the heat equation and the local existence and uniqueness of the solution to the Burgers equation. The averaging principle for such equation is studied.

Uncovering the multifractality of Lagrangian pair dispersion in shock-dominated turbulence

Lagrangian pair dispersion provides insights into mixing in turbulent flows. By direct numerical simulations (DNSs) we show that the statistics of pair dispersion in the randomly forced two-dimensional Burgers equation, which is a typical model of shock-dominated turbulence, is very different from its incompressible counterpart because Lagrangian particles get trapped in shocks. We develop a heuristic theoretical framework that accounts for this—a generalization of the multifractal model—whose prediction of the scaling of Lagrangian exit times agrees well with our DNS. Published by the American Physical Society 2024

Optimal decay estimate and asymptotic profile for solutions to the generalized Zakharov–Kuznetsov–Burgers equation in 2D

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov–Burgers equation in 2D. This is one of the nonlinear dispersive–dissipative equations, which has a spatial anisotropic dissipative term −μuxx. In this paper, we prove that the solution to this problem decays at the rate of t−34 in the L∞-sense, provided that the initial data u0(x,y) satisfies u0∈L1(R2) and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the L∞-norm of the solution. As a result, we prove that the given decay rate t−34 of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schrödinger equation, we derive the explicit asymptotic profile for the solution.

Exact Solutions for the Modified Burgers Equation with Additional Time-dependent Variable Coefficient

In this article, we investigated new travelling wave solutions for the modified Burgers equation with additional time-dependent variable coefficient via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions. All solutions of this equation have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. This method presents a wider applicability for handling nonlinear wave equations.

Theoretical examination of solitary waves for Sharma–Tasso–Olver Burger equation by stability and sensitivity analysis

Theoretical examination of solitary waves for Sharma–Tasso–Olver Burger equation by stability and sensitivity analysis

Nonlinear enhanced dissipation in viscous Burgers-type equations

We construct a class of infinite mass functions for which solutions of the viscous Burgers equation decay at a better rate than the solution of the heat equation for the same initial data in this class. In other words, we show an enhanced dissipation coming from a nonlinear transport term. We compute the asymptotic profile in this class for both equations. For the viscous Burgers equation, the main novelty is the construction and description of a time-dependent profile with a boundary layer, which enhances the dissipation. This profile will be stable up to a computable nonlinear correction depending on the perturbation. We also extend our results to other convection–diffusion equations.

Ermakov Equations can be Derived from Zel’dovich Pancake, and they are Cold and Nonlocal through using Neutrosophic Venn Diagram

As we argue in the previous article [3], the labyrinthine worlds of Jorge Luis Borges are more than captivating narratives; they are portals to a deeper understanding of existence. By weaving elements of science-fiction fantasy with philosophical and ethical inquiries, Borges's short stories bridge the seemingly disparate realms of physics and the humanities, offering fertile ground for contemporary physics research. The present-day universe consists of galaxies, galaxy clusters, one-dimensional filaments and two-dimensional sheets or pancakes, all of which combine to form the cosmic web. The so called ”Zeldovich pancakes”, are very difficult to observe, because their overdensity is only slightly greater than the average density of the universe. Falco et al. presented a method to identify Zeldovich pancakes in observational data, and the method were used as a tool for estimating the mass of galaxy clusters [2]. Here we provide an outline from Zel’dovich pancake to Burgers equations to represent cosmic turbulence, and then from Burgers equations to Ermakov dynamics systems, which in turn they plausibly lead to nonlocal current.

An invariance constrained deep learning network for partial differential equation discovery

The discovery of partial differential equations (PDEs) from datasets has attracted increased attention. However, the discovery of governing equations from sparse data with high noise is still very challenging due to the difficulty of derivatives computation and the disturbance of noise. Moreover, the selection principles for the candidate library to meet physical laws need to be further studied. The invariance is one of the fundamental laws for governing equations. In this study, we propose an invariance constrained deep learning network (ICNet) for the discovery of PDEs. Considering that temporal and spatial translation invariance (Galilean invariance) is a fundamental property of physical laws, we filter the candidates that cannot meet the requirement of the Galilean transformations. Subsequently, we embedded the fixed and possible terms into the loss function of neural network, significantly countering the effect of sparse data with high noise. Then, by filtering out redundant terms without fixing learnable parameters during the training process, the governing equations discovered by the ICNet method can effectively approximate the real governing equations. We select the one-dimensional (1D) Kuramoto–Sivashinsky equation, the two-dimensional (2D) Burgers equation, the equation of two-dimensional channel flow over an obstacle, and the equation of three-dimensional (3D) intracranial aneurysm as examples to verify the superiority of the ICNet for fluid mechanics. Furthermore, we extend similar invariance methods to the discovery of wave equation (Lorentz Invariance) and verify it through single and coupled Klein–Gordon equation. The results show that the ICNet method with physical constraints exhibits excellent performance in governing equations discovery from sparse and noisy data.

Universality in coupled stochastic Burgers systems with degenerate flux Jacobian

In our contribution we study stochastic models in one space dimension with two conservation laws. One model is the coupled continuum stochastic Burgers equation, for which each current is a sum of quadratic nonlinearities, linear diffusion, and spacetime white noise. The second model is a two-lane stochastic lattice gas. As distinct from previous studies, the two conserved densities are tuned such that the flux Jacobian, a 2×2 matrix, has coinciding eigenvalues. In the steady state, spacetime correlations of the conserved fields and the time-integrated currents at the origin are investigated. For a particular choice of couplings, the dynamical exponent 3/2 is confirmed. Furthermore, at these couplings, the continuum stochastic Burgers equation and lattice gas are demonstrated to be in the same universality class.

Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions

Abstract The fractional-order differential equations (FO-DEs) faithfully capture both physical and biological phenomena making them useful for describing nature. This work presents the stable and more effective closed-form traveling-wave solutions for the well-known nonlinear space–time fractional-order Burgers equation and Lonngren-wave equation with additional terms using the exp ( − Φ ( ξ ) ) (-\Phi (\xi )) expansion method. The main advantage of this method over other methods is that it provides more accuracy of the FO-DEs with less computational work. The fractional-order derivative operator is the Caputo sense. The transformation is used to reduce the space–time fractional differential equations (FDEs) into a standard ordinary differential equation. By putting the suggested strategy into practice, the new closed-form traveling-wave solutions for various values of parameters were obtained. The generated 3D graphical soliton wave solutions demonstrate the superiority and simplicity of the suggested method for the nonlinear space–time FDEs.

Effect of trapping of electrons and positrons on the evolution of shock wave in magnetized plasma: A complex trapped K-dV burgers’ equation

Our research demonstrates that positive and negative ions, along with trapped electrons and positrons, propagate as small-amplitude electrostatic shock waves in a magnetized collisions-less plasma. The trapped Korteweg-de Vries Burger's (T-KdVB) equation was derived by the widely recognized standard reductive perturbation technique (RPT), employing two separate scales of standard coordinates. The study examines the impact of temperature ratio on the shock wave characteristics of positive and negative ions, as well as positrons with electrons. The analysis and discussion focus on the impacts of negative ion number densities, as well as the number densities of electrons and positrons. The T-KdVB equations are responsible for governing rarefactive shock waves. Additionally, it is important to acknowledge that nonlinearity and dissipation play a key role in controlling the amplitude of the shock.

Generalized variational structures of the (3+1)-dimensional Zakharov–Kuznetsov–Burgers equation in dusty plasma

The center of this paper is to establish the generalized variational structure (GVS) of the [Formula: see text]-dimensional Zakharov–Kuznetsov–Burgers equation (ZKBe) by taking advantage of the Semi-inverse method (SIM). Two different GVSs are extracted and the derivation process is presented in detail. The extracted GVSs reveal the energy conservation law and can offer some new insights on the study of the variational method.

Nonclassical symmetries, optimal classification, and dynamical behavior of similarity solutions of (3+1)-dimensional Burgers equation

This paper is devoted to the study of symmetry group classification, and optimal subalgebras of the three-dimensional Burgers equation, which describes nonlinear wave motion exhibits nonlinear effects and enhanced dispersion in the context of nonlinear dynamics. The equation undergoes a Painlevé analysis to assess its integrability properties to verify the possibility that it passes the Painlevé test, followed by the derivation of nonclassical symmetries. Utilizing the invariance property of Lie groups, desirable infinitesimal symmetries of Lie algebra have been outlined for the equation. Relying on the invariance characteristic of adjoint transformation along with the newly generated similarity variables, an intensive and systematically organized approach is employed to achieve a one-dimensional optimal system. The entire set of group invariant solutions for each of the associated subalgebras has been established. The dynamical behavior of derived exact solutions is examined using numerical simulations, and numerous intriguing occurrences are discovered, such as flat sheet, periodic wave, doubly soliton, multisoliton, bright-dark soliton, breather soliton, line soliton type, which offer valuable insights into the dynamics of the system and can predict the behavior of waves in real-world scenarios.

Gaussian Fluctuations for the Stochastic Burgers Equation in Dimension d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/PhysRevLett.54.2026). In both the critical d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and super-critical d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 3$$\\end{document} cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 3$$\\end{document} the scaling adopted is the classical diffusive one, while in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.

Analytical study of solitons for the (2+1)-dimensional Painlevé integrable Burgers equation by using a unified method

Analytical study of solitons for the (2+1)-dimensional Painlevé integrable Burgers equation by using a unified method

Bi-objective and hierarchical control for the Burgers equation

Bi-objective and hierarchical control for the Burgers equation

Riemann problem for a general variable coefficient Burgers equation with time-dependent damping

In this paper, we study the Riemann problem for a general variable coefficient Burgers equation with time-dependent damping. We use a nonlinear time-dependent viscosity equation with a similarity variable. Thus, when the viscosity goes to zero, we obtain Riemann solutions to the general variable coefficient Burgers equation with time-dependent damping. Moreover, we use the Lax–Friedrichs scheme to obtain numerical evidence of the Riemann solutions.

Unveiling new insights: taming complex local fractional Burger equations with the local fractional Elzaki transform decomposition method

This study aims to address the difficulties in solving coupled generalized non-linear Burger equations using local fractional calculus as a framework. The methodology used in this work, particularly in the area of local fractional calculus, combines the Elzaki transform with the Adomian decomposition method. This combination has proven to be a highly effective strategy for addressing non-linear partial differential equations within the local fractional context, which finds numerous practical applications. The proposed method offers a systematic and easily understandable procedure for tackling both linear and non-linear partial differential equations (PDEs). It provides an easy-to-follow path to solve these problems. We offer a real-world example that exhibits the method's successful use in resolving issues to corroborate its efficacy. The obtained solution is visually represented to illustrate the practical utility of this approach.2010 Mathematics Subject Classification34A34, 65M06, 26A33.