Asymptotics of solutions of a modified Whitham equation with surface tension

Abstract

We study the large-time behaviour of solutions of the Cauchy problem for a modified Whitham equation, where the pseudodifferential operator is given by the symbol with a parameter . This symbol corresponds to the total dispersion relation for water waves taking surface tension into account. Assuming that the total mass of the initial data is equal to zero () and the initial data are small in the norm of , , we prove the existence of a global-in-time solution and describe its large-time asymptotic behaviour.