Transformation of systems of partial differential equations to systems of quasilinear and linear differential equations. Their reduction and unification

The paper deals with first-order PDE systems. The purpose of this paper is to investigate some general properties of first-order PDE systems and the possibility of their simplification (reduction), using the results obtained earlier by the authors. In previous works of the authors, the possibility of reducing the dimensionality of overdetermined systems of differential equations was shown. The task was to obtain, as much as possible, and as better as possible, the overrides of the broad classes of PDE. Earlier, overdeterminations of the equations of hydrodynamics and ODE were obtained, and an assumption was made about the possibility of overriding of any PDE systems. In the first half of our work, we give a new way to override any PDEs of the first order and in doing so we try to take into account that the general solutions of this extended system of equations contain only solutions to a pre-defined Cauchy problem. This is advantageous in the sense that then our method of diminishing the dimensionality theoretically can reduce the dimension of these equations up to a complete solution of the Cauchy problem, which can be represented explicitly. In addition, we also establish a link between Euler's hydrodynamic equations and arbitrary first-order PDE systems.
In the second part of this paper, we consider the reduction of PDE systems to just one quasi-linear (even universal) evolution equation of the second order from the one unknown. This increases the number of variables, and the new problems arise for the study. It shows that the Cauchy problem for these systems of equations can be reduced to the Cauchy problem for a second-order quasilinear equation, but with a larger dimension. The question of the existence and uniqueness of the solution of such a Cauchy problem is not solved.
It has long been well known that there is a general way of transforming the PDE systems to systems of first-order quasilinear differential equations. This fact is used in the proof of the Cauchy–Kovalevskaya theorem, the main theorem of the theory of partial differential equations. In our work, further progress is made in the study of this issue. We study the unification of the first-order PDE systems using the parameterization of the Cauchy problem.