Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

Abstract

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

Cauchy Processes, Dissipative Benjamin–Ono Dynamics and Fat-Tail Decaying Solitons

“…With A = 0, B = 0, and C = 0, Equation ( 3) is the Kortweg de Vries (KDV) and admits symmetric positive definite solitons [4]. Finally, for non-vanishing A, B, C, the BKDV is solved by positive asymmetric solitons [5][6][7][8]. All BKDV soliton-type densities exhibit rapidly decreasing tails and arbitrarily high moments.…”

“…This paper is organised as follows: In Section 2, we calculate a single decaying soliton solution solving the dissipative BO Equation (8). In Section 3, we expose the multi-agent modelling for agents driven by Cauchy jump processes.…”