The solvability conditions of the system of long waves in a water rectangular channel, the depth of which varies along the axis.

Nonlocal solvability of the Cauchy problem in physical variables is proved for the system of long waves in a water rectangular channel with the depth varying along its axis. Most often this system of quasi-linear equations is called as the Shallow water system. The starting system is transformed to the system of symmetric quasi-linear equations with help of Riemann invariants. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantees existence of a global classical solution continued from a local solution. The existence of the local solutions, the smoothness of which is not lower than the smoothness of the initial conditions, is also proven. The investigation of the considered problem is based on the method of an additional argument. The proof of the nonlocal solvability relies on original global estimates.