Gravity Perturbed Crapper Waves

Crapper waves are a family of exact periodic traveling wave solutions of the freesurface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity-capillary waves nearby to the Crapper waves. This result implies the existence of traveling gravity-capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated which depicts the existence of single-valued and multivalued traveling waves in the gravity-amplitude plane. A new largest water wave is computed which is composed of a string of bubbles at the interface.