ИСТИНА

We show that an asymptotic representation of smooth solutions to the Cauchy problem for any genuinely nonlinear hyperbolic system of equations written in the Riemann invariants can be obtained by a method of stochastic perturbation of the associated Langevin system [1]. Then we find relations between the system of Riemann invariants with additional property of potentiality and a specially constructed Hamilton-Jacobi equation. The Hamilton-Jacobi equation can be associated with a quasilinear system of equations of the first order, which can reduced to the vectorial Hopf equation for potential vector field [2]. This allows to apply the Cole-Hopf transform to the viscous regularization of the Hopf equation and obtain asymptotic formula that gives both smooth and shock wave solution as parameter of viscosity tends to zero. The shock wave solution is based on conservation laws implied by the Hamilton-Jacobi equation. The result is illustrated with important examples of the system of isentropic gas dynamics and the equations of chromatography. [1] Rozanova O.S. {\it Stochastic perturbations method for a system of Riemann invariants}// Mathematical Communications, 19 (2014), 573--580. [2] Rozanova O.S. {\it On the connection of the Hamilton–Jacobi equation with some systems of quasilinear equations}// Proceedings of the Steklov Institute of Mathematics. Supplement, 2015, in press.