On the Symmetry and Recovery of Steady Continuously Stratified Periodic Water Waves

This paper considers two-dimensional steady continuously stratified periodic water waves. We first use the moving plane method to establish that the stratified water waves must be symmetric provided streamlines in a neighborhood of the trough line are monotone. Further, we prove that all streamlines are real analytic (including the free surface) by applying the Schauder estimates on the uniform oblique derivative problem. Finally we provide an analytic expansion method to recover the water waves from relative horizontal velocity on the axis of symmetry and wave height. All results in this paper hold for stratified waves of large (or small) amplitude.