A DIFFERENCE-DIFFERENTIAL ANALOG OF THE BURGERS-EQUATION - STABILITY OF THE 2-WAVE BEHAVIOR

We study the Cauchy problem for the difference-differential equation d F(n)/dt = phi(F-n)(F-n-1 - F-n), n epsilon Z, () where phi is some po sitive function on [0, 1], Z is a set of integer numbers, and F-n = F- n(t) are non-negative functions of time with values in [0, 1], F--prop ortional to(t) = 0, F-proportional to(t) = 1 for any fixed t. For non- increasing and non-constant phi it was shown [V. Polterovich and G. He nkin, Econom. Math. Methods, 24, 1988, pp. 1071-1083 (in Russian)] tha t the behavior of the trajectories of () is similar to the behavior o f a solution for the famous Burgers equation; namely, any trajectory o f () rapidly converging at the initial moment of time to zero as n -- > -infinity and to 1 as n --> infinity converges with the time uniform ly in n to a wave-train that moves with constant velocity. On the othe r hand, () is a variant of discretization for the shock-wave equation , and this variant differs from those previously examined by Lax and o thers. In this paper we study the asymptotic behavior of solutions of the Cauchy problem for the equation () with non-monotonic function ph i of a special form, considering this investigation as a step toward e laboration of the general case. We show that under certain conditions, trajectories of () with time convergence to the sum of two wave-trai ns with different overfalls moving with different velocities. The velo city of the front wave is greater, so that the distance between wave-t rains increases linearly. The investigation of () with non-monotonic phi may have important consequences for studying the Schumpeterian evo lution of industries (G. Henkin and V. Polterovich, J. Math. Econom., 20, 1991, 551-590). In the framework of this economic problem, F-n(t) is interpreted as the proportion of industrial capacities that have ef ficiency levels no greater than n at moment t.