Monotone decomposition of the Cauchy problem for a hyperbolic equation based on transport equations

For the Cauchy problem for a hyperbolic equation, a multiplicative approach is developed: a monotone decomposition of the problem is constructed since the hyperbolic operator can be represented by a product of transport operators. The problem for the hyperbolic equation is reduced to a system of problems for transport equations–transport in the direction of the axis $x$ and transport in the opposite direction of the axis $x$. Conditions for the monotonicity of each problem for the transport equations and for the entire multiplicative problem are found. Such a decomposition of the Cauchy problem based on transport problems solved one after the other significantly simplifies the solution of the hyperbolic equation, and the problems for the transport equations are monotone thus ensuring the monotonicity of the decomposition of the Cauchy problem for the hyperbolic equation.