Darboux Transformation and Variable Separation Approach: theNizhnik-Novikov-Veselov Equation

Darboux transformation (DT) is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the one-step DT yields the variable separable solutions, which can be obtained from the truncated Painlevé analysis, and the two-step DT leads to some new variable separable solutions, which are the generalization of the known results obtained by using a guess ansatz to solve the generalized trilinear equation.