Here, we rigorously prove some of the results obtained with asymp-totic methods for a modiied Burgers' equation, obtained by replacing the ux term in Burgers' equation considered in its hyperbolic form by the Hilbert transform. We show that, under the asumptions ju x j 1 C ?1=3 , jH(u x)j 1 C ?1=3 , and R 1 0 jju x jj 2 dt C, H(u) being the Hilbert transform of u and C a constant independent of...

We study centred second-order in time and space discretizations of the inviscid Burgers equation. Although this equation in its continuum formulation supports non-smooth shock wave solutions, the discrete equation generically supports smooth solitary wave solutions. Using backward error analysis we derive the modified equation associated with the numerical scheme. We identify three different eq...

Lie point symmetries of time-fractional potential Burgers' equation are presented. Using these symmetries fractional potential Burgers' equation has been transformed into an ordinary differential equation of fractional order corresponding to the Erdélyi-Kober fractional derivative. Further, an analytic solution is furnished by means of the invariant subspace method. AMS subject classifications:...

We study the scalar conservation law with a noisy non-linear source, viz., ut +f(u)x = h(u; x; t) + g(u)W (t), where W (t) is white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This seque...

In this letter, a coupled system of viscous Burgers’ equations with zero Dirichlet boundary conditions and appropriate initial data is considered. For the well-known single viscous Burgers’ equation with zero Dirichlet boundary conditions, the zero equilibrium is the unique global exponential point attractor. A similar property is shown for the coupled Burgers’ equations, i.e., trajectories sta...

We have studied the phase dynamics in an oscillatory chemical system (Belousov-Zhabotinsky reaction). The phase distribution and isophase lines in the 2D phase space were reconstructed. The dynamics proved to be well described in terms of the Burgers equation. The phase dependence of the coefficients of the equation was taken into account to estimate the region of the applicability of the Burge...

random burgers equation, random Hamilton-Jacobi equation, uniqueness of global minimizers We discuss a dynamical system approach to a problem of Burgers turbulence. It is shown that there exists a unique stationary distribution for solutions to spatially periodic inviscid random forced Burgers equation in arbitrary dimension. The construction is based on analysis of minimizing orbits for time-d...

The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report...

A mixture of liquid and gas bubbles of the same size may be considered as an example of a classic nonlinear medium. In practice, analysis of propagation of the pressure waves in a liquid with gas bubbles is important problem. We know that there are solitary and periodic waves in a mixture of a liquid and gas bubbles and these waves can be described by nonlinear partial differential equations. A...

They are of first order, second order, third order 27 and fourth order, respectively. The third-order Burg28 ers equation, similar to the second-order ones, appear 29 in many physical and engineering fields [8–10], such 30 as the plasma physics and fluid mechanics. In addi31 tion, these equations play a key role in nonlinear the32 ory and mathematical physics, in particular in the inte33 grable...