The Cauchy problem for the generalized hyperbolic Novikov-Veselov equation via the Moutard symmetries

TL;DR

This paper introduces a new method using Moutard symmetries and Airy functions to construct exact solutions for the hyperbolic Novikov-Veselov equation and its generalizations, extending to higher-order cases.

Contribution

It presents a novel solution construction procedure based on Moutard symmetries applicable to the Novikov-Veselov equation and its higher-order generalizations.

Findings

New exact solutions for the hyperbolic Novikov-Veselov equation.

Extension of the solution method to higher-order generalizations.

Utilization of Airy functions and their higher-order analogs in the solution process.

Abstract

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 $\frac{d^2 z}{d \xi^2} = \xi z$. 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 $\frac{d^{n-1} z}{d \xi^{n-1}} = \xi z$.