Harmonics on hyperspheres,separation of variables and the bethe ansatz

A class of integrable evolution systems in the spacetimeR ²ⁿ⁺¹ (n 2) based on the generalized self-dual Yang-Mills equations are constructed. It is proved that the Darboux matrix method is applicable to these systems and a lot of explicit solutions are obtained. Starting with the trivial solutions, single soliton solutions and multi-soliton solutions are constructed. They are almost localized and the interaction between solitons is almost elastic.Supported by Chinese national basic research project Nonlinear Science and Japanese Inoue Scientific Foundation.