In the regime of weakly transverse long waves, given long-wave initial data, we prove that the nondimensionalized water wave system in an infinite strip under influence of gravity and surface tension on the upper free interface has a unique solution on $[0,{ T}/\eps]$ for some $\eps$ independent of constant $T.$ We shall prove in the subsequent paper \cite{ MZZ2} that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev-Petviashvili (KP) equations.
Categories: math.AP