# Analytic self-similar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise term

## TL;DR

This paper derives and analyzes self-similar solutions to the 1D KPZ equation with various noise terms, revealing complex mathematical structures and solutions expressed through special functions.

## Contribution

It introduces new analytic solutions for the KPZ equation with different noise distributions, expanding understanding of interface growth dynamics.

## Key findings

- Solutions expressed with special functions like Kummer, Heun, Whittaker, error functions
- Rich mathematical structure with common characteristics across cases
- Analytic solutions for Gaussian, Lorentzian, white, and pink noise

## Abstract

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar Ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions were evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics.