On the Moutard transformation singularity for the Davey--Stewartson II equation
Yi C. Huang
TL;DR
The paper analyzes singularity formation for the Davey–Stewartson II equation through explicit Moutard-transform constructions. It derives a two-parameter family of DS II solutions from trivial data using holomorphic inputs $f,g,h$ that satisfy evolution and compatibility conditions, and shows that singularities are encoded in the zero set of $|g|^2+|h|^2$. A concrete two-point indeterminacy scenario at $t=0$ demonstrates a mass drop of magnitude $2\pi$, and the method extends to higher-order data $f_m,g_n$ yielding generalized mass behaviors described by $(m+n-1)\pi$ away from singular times with $n\pi$ drops at the singular instant. These results provide explicit mechanisms for singularity formation in DS II and connect to geometric interpretations as DS II deformations of surfaces in $\mathbb{R}^4$.
Abstract
We indicate explicitly how to obtain the Moutard transformation singularity for the Davey--Stewartson II equation with two indeterminancy points. We also justify an arbitrary order ``mass drop" phenomenon via this singularity formation scheme.