The Cauchy problem for the generalized hyperbolic Novikov-Veselov equation

TL;DR

This paper introduces a new method for constructing exact solutions to the hyperbolic Novikov-Veselov equation's Cauchy problem using Airy functions, extending to higher-order generalizations with analogous solutions.

Contribution

A novel solution construction procedure for the Novikov-Veselov equation and its higher-order generalizations using special functions related to differential equations.

Findings

Method successfully constructs exact solutions for the hyperbolic Novikov-Veselov equation.

Procedure extends to higher-order equations with appropriate special functions.

Demonstrates the versatility of Airy functions and their generalizations in solving nonlinear PDEs.

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. The procedure shown therein utilizes the well-known Airy function $\text{Ai}(\xi)$ 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 $\frac{d^{n-1} z}{d \xi^{n-1}} = \xi z$.