Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms

TL;DR

This paper derives and analyzes exact traveling-wave solutions of the 1D KPZ interface growth equation with various noise distributions, revealing complex mathematical structures and comparing with previous self-similar solutions.

Contribution

It introduces new analytic solutions for the KPZ equation with different noise types using traveling-wave Ansatz, expanding understanding of its mathematical structure.

Findings

Solutions expressed with Mathieu, Bessel, Airy, and Whittaker functions

Different noise types lead to diverse solution behaviors

Comparison with previous self-similar solutions highlights similarities and differences

Abstract

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the traveling-wave 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 are evaluated and analyzed for all cases. All results are expressible with various special functions Mathieu, Bessel, Airy or Whittaker functions showing a very rich mathematical structure with some common general characteristics. This study is the continuation of our former work, where the same physical phenomena was investigated with the self-similar Ansatz. The differences and similarities among the various solutions are enlightened.