The objective of this paper is to construct Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations geometrically. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely, the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.
"Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems." Adv. Differential Equations 28 (1/2) 73 - 112, January/February 2023. https://doi.org/10.57262/ade028-0102-73