Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

TL;DR

This paper constructs a new type of concentrated vortex solutions for a sinh-Poisson equation with asymmetric vortex intensities, revealing how asymmetry affects vortex concentration in 2D turbulence.

Contribution

It introduces a tower of singular Liouville bubbles with different degeneracy exponents for asymmetric nonlinearities, extending previous symmetric results.

Findings

Constructed solutions with alternating vortex orientations

Analyzed effects of asymmetry parameter on vortex concentration

Extended known symmetric results to asymmetric cases

Abstract

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $\gamma\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma\neq1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena.