The method of characteristics solves a single first-order PDE by reducing it to ODEs along characteristic curves, but a system in which each unknown carries its own characteristic family admits no such reduction, since many characteristics leave each point at once. An alternative is to bracket the solution instead of iterating toward it: slice the strip above the initial hyperplane into dyadically many parallel hyperplanes, and on each slice construct upper and lower bound functions - defined from the data alone, not from any assumed solution - between which every solution must lie. One recursion in the slice index controls the gap between the bounds and a second controls the Lipschitz constants of the bounds themselves; once those constants are shown to stay bounded near the initial hyperplane, the gap closes like a geometric sequence. The bound regions are nested, so their intersection is the graph of a single function, and existence and uniqueness are established in the same step rather than separately - which is also the method's limitation, since it needs a Lipschitz hypothesis and cannot deliver existence alone under mere continuity. The argument yields explicit constants: how far the solution extends transversally, and the Lipschitz constant of the solution, both in terms of the sup-norm and Lipschitz bounds of the coefficients and the initial data.
Shalchian proves a local existence-and-uniqueness theorem for a system of first-order quasilinear PDEs in which each unknown function's partial derivatives appear in its own equation: for each i, a s…