We observe that application of a recursion operator of Burgers equation does not produce the expected symmetries. This is explained by the incorrect assumption that D−1 x Dx = 1. We then proceed to give a method to compute the symmetries using the recursion operator as a first order approximation.

The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumps p > q to the right and left, respectively, and with initial product measure with densities ~ < 2 to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle ...

We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we...

1. On numerical spectra for the linearized Burgers’ equation The stability of a traveling wave depends on the spectrum of a differential operator L obtained by linearization about the wave profile. As a simple example, consider Burgers’ equation ut = uxx − 1 2 (u)x, x ∈ R, t ≥ 0, with stationary solution U(x) = − tanh x 2 . Linearization about U(x) leads to the spectral problem Lu ≡ uxx − (Uu)x...

In this paper, we employ the modified simple equation method to find the exact traveling wave solutions involving parameters of nonlinear evolution equations via the (1+1)dimensional generalized shallow water-wave equation and the(2+1)-dimensional KdV-Burgers equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave sol...

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503–3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singula...

The aim of this study is to provide the modeling and exact analytic solutions for hydromagnetic oscillatory rotating flows of an incompressible Burgers’ fluid bounded by a plate. The governing time-dependent equation for the Burgers’ fluid is different than those from the NavierStokes’ equation. The entire system is assumed to rotate around an axis normal to the plate. The governing equations f...