Searching for Higher Dimensional Integrable Models from Lower Ones via Painlevé Analysis

Extending the Painlev\'e approach to a more general form, one can get infinitely many new integrable models under the meanings that they possess conformal invariance and the Painlev\'e property in any space dimensions from a given lower dimensional integrable model. Using the Kadomtsev-Petviashvili, nonlinear Schr\"odinger, and Schwarz Korteweg--de Vries equations as simple examples, some explicit $(3+1)$-dimensional integrable models are given.