Instability of large solitary water waves

We consider the linearized instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are not obtained at the highest wave, which has the 120 degree angle at the crest. Under the assumption of non-existence of secondary bifurcation which is con…rmed 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 unstable solitary waves approaching the highest wave. The unstable waves are of large amplitude and therefore this type of instability is not captured by the approximate models derived under small amplitude assumptions. For the proof, we introduce a family of nonlocal dispersion operators, which are related to the bifurcation of solitary waves. A continuity argument with a moving kernel formula is used to study these dispersion opeartors to yield the instability criteria. 1 Introduction Preliminaries. The water-wave problem in its simplest form concerns twodimensional motion of an incompressible inviscid liquid with a free surface, acted on only by gravity. Suppose, for de…niteness, that in the (x; y)-Cartesian coordinates gravity acts in the negative y-direction and that the liquid at time t occupies the region bounded from above by the free surface y = (t;x) and from below by the ‡at bottom y = h, where h > 0 is the water depth. In the ‡uid region f(x; y) : h < y < (t;x)g, the velocity …eld (u(t;x; y); v(t;x; y)) satis…es the incompressibility condition