Non Abelian Nonlinear Evolution Equations and Recursion operators: Some new results

The results presented are based on results obtasined in Joint work with M. Lo Schiavo and C. Schiebold. Structural Properties of Non Abelian Nonlinear Evolution Equations are studied. Non Abelian Nonlinear Evolution Equations, such as potential Korteweg deVries (pKdV), Korteweg deVries (KdV) as well as different versions of modified Korteweg deVries (mKdV), and the Korteweg deVries interacting soliton and Korteweg deVries singuarity manifold equations and properties deduced link, via Baecklund transformations among them are studied [1,2,3]. A Baecklund Chart which shows the connections, via Baecklund Transformations, among all these equations is given. The analogies as well as discrepancies between this Baecklund Chart and that one in [4] where the case of Abelian nonlinear evolution equations is considered. Notably, also in the non Abelian case, a hereditary recursion operator is obtained for all the studied equations. Hence the corresponding hierarchies are constructed. Results on non Abelian Burgers equations and hierarchies are also mentioned [5,6]. References [1] S. Carillo and C. Schiebold. Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Non-commutative soliton solutions. J. Math. Phys. 52, 053507 (2011). [2] S. Carillo and C. Schiebold, Non-commutative KdV and mKdV hierarchies via recursion methods. J. Math. Phys. 50, 073510 (2009). [3] S. Carillo, M. Lo Schiavo and C. Schiebold, submitted 2015 [4] Fuchssteiner B., S. Carillo, Soliton structure versus singularity analysis:third order completely integrable non linear differential equations in 1+1-dimensions, Physica A, 154, (1989), 467-510. [5] S. Carillo, M. Lo Schiavo and C. Schiebold, submitted 2016