The rate of decay of solitary waves of finite amplitude

Abstract

This paper is concerned with the problem of a solitary wave moving with constant form and constant velocity c on the surface of an incompressible, inviseid fluid over a horizontal bottom. The metion is assumed to be two-dimensional and irrotational, and if h is the depth of the fluid at infinity and g the acceleration due to gravity, then the Froude number F is defined by F² = C² /gh. It has been shown that necessarily F > 1, and Amick and Toland have recently conjectured that, if H(X) is the depth of the fluid at a horizontal distance x from the crest, then

where ∝ is the smallest positive root of the equation

and A is a positive number. The object of the paper is to prove this conjecture.

AMS (MOS) Subject Classification:

• 35R35
• 45G10
• 45MO5
• 76Bl5
• 76B25