Connections between the convective diffusion equation and the forced Burgers equation

The convective diffusion equation with drift b ( x ) and indefinite weight r ( x ) , ∂ ϕ ∂ t = ∂ ∂ x [ a ∂ ϕ ∂ x − b ( x ) ϕ ] + λ r ( x ) ϕ , ( 1 ) is introduced as a model for population dispersal. Strong connections between Equation (1) and the forced Burgers equation with positive frequency ( m ≥ 0 ) , ∂ u ∂ t = ∂ 2 u ∂ x 2 − u ∂ u ∂ x + m u + k ( x ) , ( 2 ) are established through the Hopf-Cole transformation. Equation (2) is a prime prototype of the large class of quasilinear parabolic equations given by ∂ u ∂ t = ∂ 2 u ∂ x 2 + ∂ ( f ( v ) ) ∂ x + g ( v ) + h ( x ) . ( 3 ) A compact attractor and an inertial manifold for the forced Burgers equation are shown to exist via the Kwak transformation. Consequently, existence of an inertial manifold for the convective diffusion equation is guaranteed. Equation (2) can be interpreted as the velocity field precursed by Equation (1). Therefore, the dynamics exhibited by the population density in Equation (1) show their effects on the velocity expressed in Equation (2). Numerical results illustrating some aspects of the previous connections are obtained by using a pseudospectral method.