Higher dimensional peculiar Lax pairs for 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 Korteweg de-Vries (KdV) equation and the Harry–Dym (HD) 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.