SPATIAL ANALYTICITY PROPERTIES OF NONLINEAR WAVES

Abstract

In this paper, we study spatial analyticity properties of two classes of equations modeling unidirectional waves in nonlinear, dispersive media, namely KdV-type equations and BBM-type equations. The commentary begins with KdV-type equations and the observation that, for a class of such equations, boundedness of a solution suffices to maintain analyticity and so loss of analyticity detects loss of L∞-regularity. For a larger class of KdV-type equations, the same conclusion is valid provided that L∞-boundedness of a solution is replaced by [Formula: see text]-boundedness. It is also shown that these nonlinear dispersive wave equations are amenable to Gevrey-class analysis based on the boundedness of a Sobolev norm. This analysis yields an explicit lower bound on the possible rate of decrease in time of the uniform radius of analyticity of a solution in terms of the assumed Sobolev bound and the Gevrey-norm of the initial data. Attention is then shifted to BBM-type equations. It is shown that, regardless of the strength of the nonlinearity, a solution starting in a Gevrey space remains in this class for all time. Moreover, a lower bound on the possible rate of decrease in time of the uniform analyticity radius has temporal asymptotics that are independent of the degree of the nonlinearity, and so apparently determined in the main by the dispersion.