# Numerical study on diverging probability density function of flat-top solitons in an extended Korteweg-de Vries equation

## TL;DR

This paper numerically investigates the statistical behavior of flat-top solitons in an extended KdV equation with weak disorder, revealing a diverging probability density function near the maximum amplitude parameter.

## Contribution

It introduces a numerical analysis of the probability density function divergence of flat-top solitons in an extended KdV equation with disorder, highlighting a loglognormal divergence near the maximum amplitude.

## Key findings

- Probability density function of soliton parameter diverges near maximum amplitude.
- Divergence follows a loglognormal distribution.
- Weak disorder causes amplitude fluctuations in solitons.

## Abstract

We consider an extended Korteweg-de Vries (eKdV) equation, the usual Korteweg-de Vries equation with inclusion of an additional cubic nonlinearity. We investigate the statistical behaviour of flat-top solitary waves described by an eKdV equation in the presence of weak dissipative disorder in the linear growth/damping term. With the weak disorder in the system, the amplitude of solitary wave randomly fluctuates during evolution. We demonstrate numerically that the probability density function of a solitary wave parameter $\kappa$ which characterizes the soliton amplitude exhibits loglognormal divergence near the maximum possible $\kappa$ value.