Nonlinear diffraction and caustic formation

Abstract

This paper reports a new family of symmetries to the Zabolotskaya-Khokhlov, dissipative Zabolotskaya-Khokhlov, and Kadomtsev-Petviashvili equations. It also reports the details of the corresponding set of exact similarity solutions to the Zabolotskaya-Khokhlov equation, and the corresponding reduction of the dissipative Zabolotskaya-Khokhlov equation onto the generalized Burgers’ equation, and implies that of the Kadomtsev-Petviashvili equation onto a simpler equation. The bearing that the symmetries and exact solutions have on other work is discussed. The first non-trivial smooth global solutions to the Zabolotskaya-Khokhlov equation are presented, answering a conjecture as to the existence of such. The formation of line caustics is examined, using the exact solutions quasi-statically, giving new results.