On Linear Instability of 2D Solitary Water Waves

We consider instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are known to not be attained at the highest wave that has a 120 angle at the crest. Under the assumption of nonexistence of secondary bifurcation, which had been confirmed numerically, we prove linear instability of solitary waves, which are higher than the wave of maximal energy and lower than the wave of maximal travel speed. It is also shown that there exist a sequence of unstable solitary waves approaching the highest wave, which suggests the instability of the highest wave. These unstable waves are of large amplitude and therefore this type of instability cannot be captured by the approximate models derived under small-amplitude assumptions. For the proof, we introduce a family of nonlocal dispersion operators to relate the linear instability problem with the elliptic nature of solitary wave problem. A continuity argument with a moving kernel formula is used to study these dispersion operators and yield the instability criteria.