Dynamical properties of generalized traveling waves of exactly solvable forced Burgers equations with variable coefficients

Abstract

The initial value problem for a generalized forced Burgers equation with variable coefficients Ut+(μ˙(t)/μ(t))U+UUx=(1/2μ(t))Uxx−a(t)Ux+b(t)(xU)x−ω2(t)x+f(t),x∈R,t>0, is solved using Cole-Hopf linearization and Wei-Norman Lie algebraic approach for finding the evolution operator of the associated linear diffusion type equation. As a result, solution of the initial value problem is obtained in terms of a corresponding linear second-order inhomogeneous ordinary differential equation and a standard Burgers model. Then, using the translation and Galilean invariance of standard Burgers equation, families of generalized nonlinear waves propagating according to a Newtonian type equation of motion are constructed. The influence of the damping, dilatation and forcing terms on the dynamics of shocks, multi-shocks, triangular and N-shaped generalized traveling waves and rational type solutions with moving singularities is investigated. Finally, exactly solvable models with concrete time-variable coefficients are introduced and dynamical properties of certain particular solutions are discussed.