Stochastic Analysis of Fifth-Order KdV Soliton in Damping Regime and Reduction to Painlev\'e Second Equation

TL;DR

This paper analyzes the stochastic behavior of fifth-order KdV solitons under damping, deriving explicit momentum representations, interpreting propagation modes statistically, and reducing the nonlinear evolution to the Painlevé II equation for deeper insight.

Contribution

It provides a novel stochastic framework for fifth-order KdV solitons in damping regimes, including explicit momentum formulas and reduction to a classical integrable equation.

Findings

Explicit soliton momentum representation derived.

Statistical analysis of amplitude fluctuations conducted.

Reduction to Painlevé II equation demonstrated.

Abstract

This work presents a stochastic analysis of fifth-order KdV soliton momentum distribution in a damping regime. An explicit representation of the soliton momentum associated with amplitude variation is derived in terms of a random time function in the presence of dissipation. Statistical interpretations of soliton propagation modes, amplitude fluctuations, and amplification are analyzed within a $\delta$-correlated Gaussian random framework. Graphical results obtained using Python illustrate the physical insight into amplitude fluctuation and energy flow. Finally, under a dominant approximation, the nonlinear momentum evolution equation is shown to reduce to the Painlev\'e second equation, a well-known integrable model appearing in diverse physical systems.