$$\delta '''$$ -shock wave solution to a nonstrictly hyperbolic system of conservation laws using weak asymptotic method

This article is concerned with the existence of a weak asymptotic solution for a \(5\times 5\) system of nonstrictly hyperbolic conservation laws. We provide additional weak asymptotic expansions within the framework of the weak asymptotic approach. Then, with the aid of these weak asymptotic expansions, we establish sufficient conditions for the existence of a weak asymptotic solution for the \(5\times 5\) system with initial data of Riemann type. Combining the Riemann problems allow us to form a weak asymptotic solution for a more general type of initial data.