An analytical study of Kuznetsov's equation: diffusive solitons, shock formation, and solution bifurcation

The exact traveling wave solution (TWS), which assumes the form of a diffusive soliton, is determined for Kuznetsov's equation, a PDE arising in nonlinear acoustics. It is shown (a) that a TWS exists iff the Mach number is less than or equal to a critical value; (b) that the wave speed suffers a bifurcation and is always supersonic; and (c) that a shock develops as the Reynolds number --> infinity. Lastly, some shortcomings associated with approximating Kuznetsov's equation with Burgers' equation are noted. (C) 2004 Elsevier B.V. All rights reserved.