Generalizations of the KPZ equation (minor changes to enable easy latexing)

TL;DR

This paper extends the KPZ equation to an O(3) multi-component model, analyzing its behavior across different dimensions and component numbers, revealing how the dynamic exponent varies and providing numerical insights.

Contribution

It introduces a generalized multi-component KPZ model and derives analytical and numerical results on its dynamic exponents across dimensions.

Findings

Dynamic exponent z increases from 3/2 to 2 as dimension d increases from 1 to approximately 3.6.

For d=1, z=3/2 holds for all component numbers j.

Numerical integration for j=2 in d=2 provides specific insights into the model's behavior.

Abstract

We generalize the KPZ equation to an O(3) $N=2j+1$ component model. In the limit $N \to \infty$ we show that the mode coupling equations become exact. Solving these approximately we find that the dynamic exponent $z$ increases from $3/2$ for $d=1$ to $2$ at the dimension $d\approx3.6$. For $d=1$ it can be shown analytically that $z=3/2$ for all $j$. The case $j=2$ for $d=2$ is investigated by numerical integration of the KPZ equation.