Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-α model of single-phase lossless fluids

A one-dimensional weakly-nonlinear model equation based on a Lagrangian-averaged Euler-α model of compressible flow in lossless fluids is presented. Traveling wave solutions (TWS)s, in the form of a topological soliton (or kink), admitted by this fourth-order partial differential equation are derived and analyzed. An implicit finite-difference scheme with internal iterations is constructed in order to study soliton collisions. It is shown that, for certain parameters, the TWSs interact as solitons, i.e., they retain their “identity” after a collision. Kink-like solutions with an oscillatory tail are found to emerge in a signaling-type initial-boundary-value problem for the linearized equation of motion. Additionally, connections are drawn to related weakly-nonlinear acoustic models and the Korteweg–de Vries equation from shallow-water wave theory.

Research highlights
► Weakly-nonlinear Lagrangian-averaged model of lossless compressible flow is derived.
► Equation of motion is valid for both liquids and gasses.
► Spatially-localized traveling wave solutions are found analytically.
► Numerical simulations show these localized waves can interact as solitons.