Higher Dimensional Integrable Models from Lower Ones via Miura Type Deformation Relation

To find nontrivial high dimensional integrable models (especially in (3+l)-dimensions) is one of the most important problems in nonlinear physics. A systematic method to find some nontrivial high dimensional integrable models is established by means of noninvertible deformation relations. Starting f rom a noninvertible Miura type transformation, we find that the (l + l)-dimensional sinhGordon model appearing in many physical fields is a deformation of the (0+l)-dimensional Riccati equation. A high dimensional Miura type deformation (including two different (3+l)-dimensional reductions) of the heat conduction equation is proved to be Painleve integrable. Some special types of explicit exact solutions, like multi-plane and/or multi-camber soliton solutions, multi-dromion solutions and multiple ring soliton solutions, are obtained.