Anomalous coalescence from a nonlinear Schroedinger equation with a quintic term: interpretation through Thompson's approach

Inspired by models for A+A→A(0) reactions with non-Brownian diffusion, we suggest a possible analytical explanation for the phenomena of anomalous coalescence of bubbles found in one-dimension (1d) by Josserand and Rica through numerical work [Phys. Rev. Letters 78 (1997) 1215]. The explanation firstly requires an exponent γ, which is sometimes used to describe anomalous diffusion. Here it displays an explicit dependence on the dimensionality (γ=γ(d)=4/d for d⩽2). So we have dc=2, coinciding with the upper critical dimension of A+A→A(0) reactions (Mod. Phys. Lett. B 13 (1999) 829; Mod. Phys. Lett. B 15(26) (2001) 1205) with Brownian diffusion condition (γ=2). Thus anomalous coalescence emerges, only below the critical dimension (d<2). We show that the typical size of the structures (bubbles) grows as R(t)∼t1/4 in 1d. An alternative explanation could also be thought as a diffusion constant D which depends on the average concentration (〈n〉), namely D=D0〈n〉α. It is introduced into an effective action for A+A→A(0) reactions. Therefore we are also able to reproduce the anomalous behavior for n(t) and R(t) in 1d, being α=0 for d⩾2 (mean field behavior) and α=2(2−d)/d2 for d⩽2.