Variable separation solutions obtained from Darboux Transformations for the asymmetric Nizhnik–Novikov–Veselov system

The use of a seed solution with some arbitrary functions for the asymmetric Nizhnik–Novikov–Veselov system in the first step Darboux transformation yields the variable separable solutions with two space-variable separated functions. The more variable separated functions which are not arbitrary can be introduced by using the Darboux transformation repeatedly. TheNth step Darboux transformation (for arbitrary N) with arbitrary number of space–variable separated functions is explicitly written down by means of the Pfaffian. The ”universal” variable separation formula which is valid for a diversity of (2 + 1)-dimensional integrable systems can be obtained from a particular reduction of the solutions constructed from the second step Darboux transformation. A new saddle-type ring soliton solution with completely elastic interaction and nonzero phase shifts is also studied in this paper.