Author(s)

P. Barrera & T. Brugarino

Abstract

We perform the Painlev´e test to a system of two coupled Burgers-type equations which fails to satisfy the Painlev´e test. In order to obtain a class of solutions, we use a slightly modified version of the test. These solutions are expressed in terms of the Airy functions. We also give the travelling wave solutions, expressed in terms of the trigonometric and hyperbolic functions. 1 Introduction The nonlinear diffusion-convection equations  ut(x, t) = uxx(x, t) + µu(x, t)ux(x, t) + λ11v(x, t)ux(x, t) + λ12u(x, t)vx(x, t) vt(x, t) = vxx(x, t) + νv(x, t)vx(x, t) + λ21v(x, t)ux(x, t) + λ22u(x, t)vx(x, t) (1) have a lot of applications in physics, chemistry and biology [1-3], particularly in the study of porous media [4], in polydispersive sedimentation [5], in dynamic of growing interfaces [6] and in the study of integrable coupled Burgers-type equations [7], [8]. The paper is organized as follows: in sect. 2 we show that system (1), for arbitrary coefficients, is not integrable in Painlev´e sense; in sect. 3 a slightly modified version of the truncated Painlev´e test is used to obtain analytic solutions for particular values of the coefficients; in sect. 4 we determine some exact solutions of

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