Display Abstract

Title Dissipative acoustic solitons

Name Ivan C Christov
Country USA
Email christov@princeton.edu
Submit Time 2012-03-14 13:57:03
Special Session 4: Nonlinear PDEs and Control Theory with Applications
Lagrangian-averaged models for compressible flow have recently been proposed [Bhat & Fetecau, DCDS-B 6 (2006) 979--1000]. Like their counterparts in turbulence, these models introduce a minimal (``cut-off'') length scale beyond which energy dissipation cannot occur. When applied to weakly-nonlinear acoustic phenomena in inviscid, lossless single-phase fluids, the Lagrangian-averaged model represents a higher-order dispersive regularization of the governing equation, which exhibits nonlinear dissipation as well [Keiffer et al., Wave Motion 48 (2011) 782--790]. Kink-type solitary wave solutions are derived analytically, and an implicit finite-difference scheme with internal iterations is constructed in order to study their collisions. It is shown that two kinks can interact and retain their identity after a collision, meaning that these waves represent dissipative acoustic solitons. For a different choice of parameters, finite-time blow-up can be observed numerically. Finally, while the classical equations of nonlinear acoustics can be reduced to Burgers' equation, we show that the present model reduces to the Korteweg--de Vries equation. This work is, in part, a collaboration with R.S. Keiffer, R. McNorton and P.M. Jordan from the Naval Research Laboratory, Stennis Space Center.