In this paper we study the asymptotic behaviour of solutions of a system of N partial differential equations. When N = 1 the equation reduces to the Burgers equation and was studied by Hopf. We consider both the inviscid and viscous case and show a new feature in the asymptotic behaviour.

In this work, we construct the general solution to the Heat Equation (HE) and to many tensor structures associated to the Heat Equation, such as Symmetries, Lagrangians, Poisson Brackets (PB) and Lagrange Brackets (LB), using newly devised techniques that may be applied to any linear equation (e.g., Schrödinger Equation in field theory, or the small-oscillations problem in mechanics). In partic...

We explore a connection of the forced Burgers equation with the Schrödinger (diffusive) interpolating dynamics in the presence of deterministic external forces. 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 general, the usage of a continuity equation ∂ t ρ = −∇(vρ), wh...

Abstract: From the approximate symmetry point of view, the perturbed burgers equation is investigated. The symmetry of a system of the corresponding PDEs which approximates the perturbed burgers equation is constructed and the corresponding general approximates symmetry reduction is derived, which enables infinite series solutions and general formulae. Study shows that the similarity solutions ...

Burgers’ equations is one of the typical nonlinear evolutionary partial differential equations. In this paper, a mesh-free method is proposed to solve the Burgers’ equation numerically using the finite difference and collocation methods. After the temporal discretization of the equation using C-N Scheme, the solution is approximated spatially by Radial Basis Function (RBF). The numerical result...

In this paper, the reduced differential transform method (RDTM) is applied to various nonlinear evolution equations, Korteweg–de Vries Burgers' (KdVB) equation, Drinefel’d–Sokolov–Wilson equations, coupled Burgers equations and modified Boussinesq equation. Approximate solutions obtained by the RDTM are compared with the exact solutions. The present results are in good agreement with the exact ...

In this paper, we show the existence and uniqueness of the stationary solution u(t, ω) and stationary point Y (ω) of the differentiable random dynamical system U : R×L[0, 1]×Ω → L[0, 1] generated by the stochastic Burgers equation with L[0, 1]-noise and large viscosity, especially, u(t, ω) = U(t, Y (ω), ω) = Y (θ(t, ω)), and Y (ω) ∈ H[0, 1] is the unique solution of the following equation in L[...

We prove that the KdV-Burgers is globally well-posed in H−1(T) with a solution-map that is analytic fromH−1(T) to C([0, T ];H−1(T)) whereas it is ill-posed in Hs(T), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(T) to even D′(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dis...

In this paper we study the large time behavior for the viscous Burgers’ equation with initial data in L(R). In particular, after a time dependent scaling, we provide the optimal rate of convergence in relative entropy and Wasserstein metric, towards an equilibrium state corresponding to a positive diffusive wave. The main tool in our analysis is the reduction of the rescaled Burgers’ equation t...