Burgers-type Equations Research Articles

Random attractors for locally monotone stochastic partial differential equations

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-α model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic generalized porous media equations and stochastic generalized p-Laplace equations.

Geometric Optics for Surface Waves in Nonlinear Elasticity

This work is devoted to analysis of high frequency solutions to equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in general class of isotropic hyperelastic models, which includes in particular Saint Venant-Kirchhoff system. Work has been done by a number of authors since 1980s on formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives leading term of an \emph{approximate} Rayleigh wave solution to underlying elasticity equations. This evolution which we refer to as the amplitude equation, is an integrodifferential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of theory of amplitude equation. The remainder of paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^\eps$ to nonlinear elasticity equations exist on a fixed time interval independent of wavelength $\eps$, and that approximate Rayleigh wave solution provided by analysis of amplitude equation is indeed close in a precise sense to $u^\eps$ on a time interval independent of $\eps$. The paper focuses mainly on case of Rayleigh waves that are \emph{pulses}, which have profiles with continuous Fourier spectrum, but our method applies equally well to case of wavetrains, whose Fourier spectrum is discrete.

Small time asymptotics for SPDEs with locally monotone coefficients

<p style='text-indent:20px;'>This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

Stochastic Entropy Solutions for Stochastic Scalar Balance Laws

We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations.

Conserved currents and T overline{mathrm{T}} s irrelevant deformations of 2D integrable field theories

The T overline{mathrm{T}} deformation of 2-dimensional QFTs is closely-related to Jackiw- Teitelboim gravity. It has been shown that, at the classical level, this perturbation induces an interaction between the stress-energy tensor and space-time and the equations of motion of the deformed theory map onto the original ones through a field-dependent coordinate transformation. At the quantum level, instead, the perturbation is induced by a modification of the original S-matrix by a specific CDD factor and, correspondingly, the quantised energy levels evolve according to a Burgers-type equation. In this paper, we point out that, in the framework of integrable field theories, there exist infinite families of perturbations characterised by a coupling between space-time and local conserved currents, labelled by the Lorentz spin. Similarly to the T overline{mathrm{T}} case, the deformed models emerge through a field-dependent coordinate transformation involving conserved currents with higher Lorentz spin. Furthermore, using a geometric construction, we present a general method to derive the integrable hierarchy of the corresponding deformed models. The resulting expressions of the conserved currents turn out to be essential for the identification of the scattering phase factors which generate the deformations of the S-matrix, at the quantum level. Finally, the effect of the perturbations on the finite-volume spectrum is investigated using a non-linear integral equation. Exact spectral flow equations are derived, and links with previous literature, in particular on the J overline{mathrm{T}} model, are discussed. While the classical setup is very general, the sine-Gordon model and its CFT limit are used as illustrative quantum examples. Most of the final equations and considerations are, however, of broader validity, or easily generalisable to more complicated systems.

Modeling nonlinear fractional diffusion-wave equations with Burgers-type evolution equations

Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions under which the FPDE can be modeled accurately with a Burgers-type evolution equation for progressive wave motion. The method of lines in combination with a general Runge-Kutta method for forward integration is used for numerical analysis. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions ...

Stochastic Burgers type equations with reflection: Existence, uniqueness

In this paper, we establish the existence and uniqueness of solutions of reflected stochastic Burgers-type equations. The main difficulty is to deal with the highly non-linear coefficients and the singularities caused by the reflection.

Traveling Waves of Some Symmetric Planar Flows of Non-Newtonian Fluids

We present some variants of Burgers-type equations for incompressible and isothermal planar flow of viscous non-Newtonian fluids based on the Cross, the Carreau and the power-law rheology models, and on a symmetry assumption on the flow. We numerically solve the associated traveling wave equations by using industrial data and in order to validate the models we prove existence and uniqueness of solutions to the equations. We also provide numerical estimates of the shock thickness as well as the maximum stress associated with the traveling waves.

Application of Asymptotic Analysis for Solving the Inverse Problem of Determining the Coefficient of Linear Amplification in Burgers’ Equation

Asymptotic analysis of a singularly perturbed reaction—diffusion—advection equation, which is called a Burgers-type equation in applications and has a solution with a sharp transition layer, is applied to solve the coefficient inverse problem of determining the coefficient of linear amplification from known information on the observed solution of the direct problem at the final moment of time. The efficiency of the approach proposed in this study is shown using a series of model numerical experiments.

Numerical Solutions of Fractional Burgers’ Type Equations with Conformable Derivative

In this article, we introduce the residual power series method (RPSM) for finding approximate solutions of the time-fractional Burgers type equations using the conformable fractional derivative definition. This definition is simple and effective in the solution procedure of the fractional differential equations that have complicated solutions with classical fractional derivative definitions like Caputo and Riemann-Liouville. The results indicate the proposed method gives significant and reliable solutions.

The existence and asymptotic stability of periodic solutions with an interior layer of Burgers type equations with modular advection

We consider a new class of singularly perturbed parabolic periodic boundary value problems for reaction-advection-diffusion equations: Burgers type equations with modular advection. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. The asymptotic stability of this solution is also established.

The generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Burgers type

The nonlinear space–time fractional differential equations (FDE) of Burgers’ type play an important role for describing many phenomena in applied sciences. This paper develops a generalized version of the modified Kudryashov method to obtain the exact solutions for FDE of Burgers’ type. The FDE are firstly reduced to a set of ordinary differential equations by means of a fractional complex transformation and, afterward, they are solved by applying the generalized Kudryashov method (GKM). Several kinds of traveling wave solutions such as kink and singular kink waves are illustrated with various figures and discussed. The results demonstrate the efficiency of the GKM in tackling space–time partial FDE.

Limit Shapes of the Stochastic Six Vertex Model

It is shown that limit shapes for the stochastic 6-vertex model on a cylinder with the uniform boundary state on one end are solutions to the Burger type equation. Solutions to these equations are studied for step initial conditions. When the circumference goes to infinity the solution corresponding to critical initial densities coincides with the one found by Borodin, Corwin, and Gorin.

Wave Resonance in Media with Modular, Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed.

Riccati–Ermakov systems and explicit solutions for variable coefficient reaction–diffusion equations

We present several families of nonlinear reaction–diffusion equations with variable coefficients including generalizations of Fisher–KPP and Burgers type equations. Special exact solutions such as traveling wave, rational, triangular wave and N-wave type solutions are shown. By means of similarity transformations the variable coefficients are conditioned to satisfy Riccati or Ermakov systems of equations. When the Riccati system is used, conditions are established so that finite-time singularities might occur. We explore solution dynamics across multi-parameters. In the supplementary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions.

Unconditional Convergence in Maximum-Norm of a Second-Order Linearized Scheme for a Time-Fractional Burgers-Type Equation

We study linearized finite difference scheme for a time-fractional Burgers-type equation in this paper. A linearized scheme with second-order accuracy in time and space is proposed. The advantage of the scheme is that iterative method is not required for finding the approximated solutions. Nonlinearity involving derivatives causes difficulties in analysis. By refined estimates of our previous study, we show that the scheme unconditionally converges with second-order in maximum-norm. The theoretical results are justified by numerical tests.

On Front Motion in a Burgers-Type Equation with Quadratic and Modular Nonlinearity and Nonlinear Amplification

A singularly perturbed initial–boundary value problem for a parabolic equation known in applications as a Burgers-type or reaction–diffusion–advection equation is considered. An asymptotic approximation of solutions with a moving front is constructed in the case of modular and quadratic nonlinearity and nonlinear amplification. The influence exerted by nonlinear amplification on front propagation and blowing- up is determined. The front localization and the blowing-up time are estimated.

Numerical solution for space and time fractional order Burger type equation

In this paper, we study the fractional differential operators thereby considering the space and time order Burger type equation with initial condition. The extended finite difference method which is based on shifted Grünwald, Caputo and Riemann formulas are used. The fractional terms are approximated using two fractional difference schemes. Time Order Burger type Equation, Space Order Burger type Equation and Space-Time Order Burger type Equation are discussed with an example and error estimates obtained for the Finite difference method (FDM) and Generalized Taylor Series Methods (GTSM). The numerical methods have been applied to solve a numerical example, and results are compared with the exact solutions. By these methods, the numerical solutions of space and time order Burger type equation are obtained with the help of Wolfram Mathematica 11 software package. We present the tables of the obtained numerical results.

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.

Dust-acoustic shock excitations in κ-nonthermal electron depleted dusty plasmas

A theoretical investigation on the characteristics of dust-acoustic shock waves (DASHWs) in an unmagnetized multi-ion dusty plasma has been made both theoretically and analytically. The plasma medium is assumed to compose of arbitrarily charged inertial dusts, Boltzmann distributed negatively charged heavy ions, positively charged light ions, and superthermal (excess energetic) electrons (where superthermality is modelled via the κ-distribution). The reductive perturbation technique is employed in order to derive the nonlinear time evolution Burgers type equation. The basic properties of DASHWs are analysed via the solution of Burgers equation. It is observed that different plasma parameters (electrons superthermality, dust temperature, ion temperature, dust kinematic viscosity, etc.) influence the propagation of DASHWs. Both polarity (positive and negative potential) shock waves are also found to exists. The study of our model under consideration is helpful for explaining the propagation of DASHWs in space observations, where superthermal electrons and Maxwellian ions are accountable.