Higher dimensional Lax pairs of lower dimensional chaos and turbulence systems

In this letter, a definition of the higher dimensional Lax pair for a lower dimensional system which may be a chaotic system is given. A special concrete (2+1)-dimensional Lax pair for a general (1+1)-dimensional three order autonomous partial differential equation is studied. The result shows that any (1+1)-dimensional three order semi-linear autonomous system (no matter it is integrable or not) possesses infinitely many (2+1)-dimensional Lax pairs. Especially, every solution of the KdV equation and the Harry-Dym equation with their space variable being replaced by the field variable can be used to obtain a (2+1)-dimensional Lax pair of any three order (1+1)-dimensional semi-linear equation.Comment: 8 page

Symmetries and integrable systems

Symmetry plays key roles in modern physics especially in the study of integrable systems because of the existence of infinitely many local and nonlocal generalized symmetries. In addition to the fundamental role to find exact group invariant solutions via Lie point symmetries, some important new developments on symmetries and conservation laws are reviewed. The recursion operator method is important to find infinitely many local and nonlocal symmetries of (1+1)-dimensional integrable systems. In this paper, it is pointed out that a recursion operator may be obtained from one key symmetry, say, a residual symmetry. For (2+1)-dimensional integrable systems, the master-symmetry approach and the formal series symmetry method are reviewed. For the discrete systems, the symmetry related discrete KP hierarchy and the BKP hierarchy are also discussed. One believes that all the solutions of integrable models may be obtained by means of symmetry approach because the Darboux transformations and algebro-geometric solutions can be obtained from the localization of nonlocal symmetries and the symmetry constraint approach. The conservation laws are used to find higher dimensional integrable system from lower dimensional ones via a deformation algorithm. The ren variable, an extension of the Grassmann variable, are introduced to find novel aspect on integrable theory. The super-integrable theory and super-symmetric integrable theory are extended to ren integrable and ren-symmetric integrable theories