A difference-differential analogue of the burgers equation: Stability of the two-wave behavior

Abstract

We study the Cauchy problem for the difference-differential equation (*) $$\frac{{dF_n }}{{dt}} = \varphi \left( {F_n } \right)\left( {F_{n - 1} - F_n } \right),n \in \mathbb{Z},$$ whereϕ is some positive function on [0, 1], ℤ is a set of integer numbers, andF n=Fn(t) are non-negative functions of time with values in [0, 1],F –∞(t)=0,F ∞(t)=1 for any fixedt. For non-increasing the non-constantϕ it was shown [V. Polterovich and G. Henkin,Econom. Math. Methods, 24, 1988, pp. 1071–1083 (in Russian)] that the behavior of the trajectories of (*) is similar to the behavior of a solution for the famous Burgers equation; namely, any trajectory of (*) rapidly converging at the initial moment of time to zero asn → −8 and to 1 asn → ∞ converges with the time uniformly inn to a wave-train that moves with constant velocity. On the other hand, (*) is a variant of discretization for the shock-wave equation, and this variant differs from those previously examined by Lax and others. In this paper we study the asymptotic behavior of solutions of the Cauchy problem for the equation (*) with non-monotonic functionϕ of a special form, considering this investigation as a step toward elaboration of the general case. We show that under certain conditions, trajectories of (*) with time convergence to the sum of two wave-trains with different overfalls moving with different velocities. The velocity of the front wave is greater, so that the distance between wave-trains increases linearly. The investigation of (*) with non-monotonicϕ may have important consequences for studying the Schumpeterian evolution 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 efficiency levels no greater thann at momentt.