Aspects of the stochastic Burgers equation and their connection with turbulence. Summary. We present results for the 1 dimensional stochastically forced Burgers equation when the spatial range of the forcing varies. As the range of forcing moves from small scales to large scales, the system goes from a chaotic, structureless state to a structured state dominated by shocks. This transition takes...

We analyze the unforced and deterministically forced Burgers equation in the framework of the (diffusive) interpolating dynamics that solves the so-called Schrödinger boundary data problem for the random matter transport. This entails an exploration of the consistency conditions that allow to interpret dispersion of passive contaminants in the Burgers flow as a Markovian diffusion process. In g...

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the const...

In this paper, we deal with both nonviscous and viscous Burgers type equations on a bounded interval. We study the global exact controllability of these equations when we have three controls: one control is the right member of the equation and is constant with respect to the space variable, the two others are the boundary values. In a first time, we are interested in nonviscous Burgers type equ...

The statistics of solutions to a family of one-dimensional random-force-driven advection-diffusion equations is studied using high resolution numerical simulations. The equation differs from the usual Burgers equation by the non-local form of the nonlinear interaction term mimicking the non-locality of the Navier–Stokes equation. It is shown that under an appropriate choice of random forcing th...

Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced in [4], we analyze a Lyapunov functional for such equation which gives the large deviations asymptotics of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functi...

In this paper an efficient unconditionally stable numerical scheme is proposed for solving one dimensional quasi linear Burgers’ equation. The proposed scheme comprises of semi discretization via method of lines for the space variable and backward differentiation formula of order two (BDF2) for the time variable. The method of lines reduces the quasi linear partial differential equation in to n...

We discuss the recent paper by Inan and Ugurlu [Inan I.E., Ugurlu Y., Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation, Appl. Math. Comp. 217 (2010) 1294 – 1299]. We demonstrate that all exact solutions of fifth order KdV equation and modified Burgers equation by Inan and Ugurlu are trivial solutions that are reduced to constants. Moreover, w...

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in t...

In this paper, an analytic solution is presented using differential transform method (DTM) for a class of wave equation. The emphasis is on the nonlinear two-dimensional wave equation. The procedures introduced in this paper are in recursive forms which can be used to obtain the closed form of the solutions, if they are required. The method is tested on various examples, and the results reveal ...