Stochastic Analysis of Fifth-Order KdV Soliton in Damping Regime and Reduction to Painlevé Second Equation

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 $δ$-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 Painlevé second equation, a well-known integrable model appearing in diverse physical systems.

Figures (5)

• Figure 1: (a) Three dimensional profile of single soliton. (b) Two dimensional perspective of single soliton solution. (c) contour representation of single soliton.
• Figure 2: (a) Three dimensional profile of single soliton. (b) Two dimensional perspective of single soliton solution.(c) contour representation of single soliton.
• Figure 3: Here graph shows the variations in amplitudes on time scale for different values of $\sigma^2$ while pulse propagating through damping medium.
• Figure 4: Here graph shows the variations in amplitudes on time scale for different values of $\sigma^2$ while pulse propagating with zero dissipation.
• Figure 5: Here, the graph shows the variations in amplitudes for $\sigma^2 t<1$ along the straight lines in the damping regime on the time scale.