Solitons and cavitons in a nonlocal Whitham equation

Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations studied. The fourth order differential equation traveling waves a parameter in front of derivative is reduced to reversible Hamiltonian system defined on two-sheeted four-dimensional space. Solutions which stay one sheet represent smooth but those perform transitions through branching plane jumps. They correspond singularities – breaks first third derivatives continuous even derivatives. can have two types equilibria different sheets, they be saddle-centers or saddle-foci. Using analytic numerical methods we found many homoclinic orbits these both monotone asymptotics oscillating ones. solitons initial equation. When deal saddle-center, values second (physical wave speed) discrete case saddle-focus continuous. presence multiplicity such displays very complicated dynamics system.