Mixed nonlinearity and double shocks in superfluid helium

Abstract

We examine weak second-sound waves in He II at temperatures and pressures near one of the zeros of the Khalatnikov steepening parameter Γ. An extension of the reductive perturbation scheme of Taniuti & Wei is employed to derive the cubic Burgers’ equation governing these waves. It is shown that mixed nonlinearity may occur in disturbances in which the local value of Γ remains strictly positive or strictly negative. Further new results include expressions for the shock speed, shock structure and the conditions under which the shock thickness increases, rather than decreases, with strength. The fundamental existence conditions for temperature shocks are also delineated and related to the shock disintegration process observed in experimental studies.