Solitons in fluctuating hydrodynamics of diffusive processes

Abstract

We demonstrate that fluid mechanical systems arising from large fluctuations of one-dimensional statistical processes generically exhibit solitons and nonlinear waves. We derive the explicit form of these solutions and examine their properties for the specific cases of the Kipnis-Marchioro-Presutti model (KMP) and the Symmetric Exclusion Process (SEP). We show that the two fluid systems are related by a nonlinear transformation but still have markedly different properties. In particular, the KMP fluid has a nontrivial sound wave spectrum exhibiting birefringence, while sound waves for the SEP fluid are essentially trivial. The appearance of sound waves and soliton configurations in the KMP model is related to the onset of instabilities.

Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process

“…The transformation (10), (11) unveils the integrability of SEP at the hydrodynamic level (as was foreseen in [54] by finding solitons in the MFT equations). In the low density limit, obtained by writing ρ := αρ with α → 0, this change of variable reduces to the canonical Cole-Hopf transformation, i.e., (u, v) → (∂ x ρe −H , −∂ x e H ) and the AKNS equations decouple into two diffusion equations, evolving forward and backward in time, that were used to investigate reflecting Brownian motions [26,37].…”

Controllable soliton transition and interaction in nonlocal nonlinear media

“…Solitons, as localized wave structures, have been largely investigated in a plethora of materials and several nonlinear mechanisms, such as fluids [1], plasma physics, cold Rydberg atomic gas [2], Bose-Einstein condensates [3], and optics [4][5][6]. In the field of nonlinear optical, solitons arise from the result of the balance between dispersion (or diffraction) and nonlinearity.…”