Type: Article

Asymptotics of solutions of a modified whitham equation with surface tension

Naumkin, Pavel I.;

Abstract:

We study the large-time behaviour of solutions of the Cauchy problem for a modified Whitham equation, {ut+i?u??xu3=0,(t,x)?R2, u(0,x)=u0(x),x?R, where the pseudodifferential operator ???(?i?x)=F?1[?(?)F] is given by the symbol ?(?)=a?1/2?((1+a2?2)tanha?a?????????????????1) with a parameter a>0. 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 (?Ru0(x)dx=0) and the initial data u0 are small in the norm of H?(R)?H0,1(R), ??22, we prove the existence of a global-in-time solution and describe its large-time asymptotic behaviour