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We are concerned with hyperbolic systems of order-one linear PDEs originatedon non-characteristic manifolds. We put forward a simple but effective methodof transforming such initial conditions to standard initial conditions (i.e.when the solution is specified at an initial moment of time). We then show howour method applies in fluid mechanics. More specifically, we present a completesolution to the problem of long waves run-up in inclined bays of arbitraryshape with nonzero initial velocity.

Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations