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. In the nonlinear mathematical physics, if an (n+1)-dimensional nonlinear system can be considered as a consistent condition of an (n+1)-dimensional linear system, then many types of interesting properties of the nonlinear system can be obtained by analyzing the linear system. The linear system is called as the Lax pair of the original nonlinear system[1]. In this letter, we use the concept of the Lax pair under a more general meaning. If the compatible condition [L1, L2]ψ ≡ (L1L2 − L2L1)ψ = 0, (1) of a pair of linear equations L1(x1, x2, ..., xn, t, u(x1, x2, ..., xm, t))ψ(x1, x2, ..., xn, t) ≡ L1ψ = 0, (2) L2(x1, x2, ..., xn, t, u(x1, x2, ..., xm, t))ψ(x1, x2, ..., xn, t) ≡ L2ψ = 0, m < n, (3) ∗Email: [email protected]