The Cauchy Problem for the Generalized Hyperbolic Novikov-Veselov Equation via the Moutard Symmetries

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov-Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(xi) which in turn serves as a solution to the ordinary differential equation d(2)z/d xi(2)=xi z. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov-Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation d(n-1)z/d xi(n-1)=xi z.