Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems

The objective of this paper is to construct Riemann $k$-wave solutions the general form first-order quasilinear hyperbolic systems partial differential equations geometrically. To end, we adapt and combine elements two approaches construction $k$-waves, namely, symmetry reduction method generalized characteristics. We formulate a geometrical setting for problem discuss in detail conditions existence solutions. An auxiliary result concerning Frobenius theorem established. use it obtain formulae describing closed form. Our theoretical considerations are illustrated by examples hydrodynamic type including Brownian motion equation.