The proof of a local solvability theorem for a quasi-linear first-order partial differential equation of the common type with initial data in Cartesian coordinates on an infinite length line

The Cauchy problem for a quasi-linear first order partial differential equation in case when initial data is given on an infinite length smooth line with non-vertical gradient is reduced to a system in 15 integral equations. The local solvability of this system of integral equations is proved. Connections between unknown functions of the integral equations and the unknown function and its derivatives of the primary Cauchy problem is established, because at a deriving of the system of integral equations the partial derivatives of a seeking function were taking as new unknown functions, so the inverse passage is necessary and not trivial. As a result, the local solvability theorem is proved. The local solvability conditions do not include conditional assumptions about behavior of the characteristic lines.