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 no...

The gauge-invariant description of zero-curvature representations of evolution equations is applied to the problem of how to distinguish the fake Lax pairs from the true Lax pairs. The main difference between the true Lax pairs and the fake ones is found in the structure of their cyclic bases. In this note, we use the gauge-invariant description of zero-curvature representations of evolution eq...

We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg–deVries equation and show that it can be consistently extended multidimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series of q-discrete Painlevé equations, and Lax pairs for the reduced equations are found. A 2× 2 Lax pair is given f...

The tensorial form of the Lax pair equations are given in a compact and geometrically transparent form in the presence of Cartan’s torsion tensor. Three dimensional spacetimes admitting Lax tensors are analyzed in detail. Solutions to Lax tensor equations include interesting examples as separable coordinates and the Toda lattice.

The non-isospectral problem (Lax pair) associated with a hierarchy in 2+1 dimensions that generalizes the well known Camassa-Holm is presented. Here, we have investigated non-classical Lie symmetries of this Lax pair when spectral parameter considered as field. These can be written terms five arbitrary constants and three functions. Different similarity reductions these been derived. Of particu...

In paper by I.T. Habibullin and our joint the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on requirement existence an infinite set Darboux reductions notion characteristic Lie-Rinehart algebras. The applied cases different subclasses $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$ special forms. Under this approac...

The tensorial form of the Lax pair equations are given in a compact and geometrically transparent form in the presence of Cartan's torsion ten-sor. Three dimensional spacetimes admitting Lax tensors are analyzed in detail. Solutions to Lax tensor equations include interesting examples as separable coordinates and the Toda lattice.

The Lax pair of the Ruijsenaars-Schneider model with interaction potential of trigonometric type based on Dn Lie algebra is presented. We give a general form for the Lax pair and prove partial results for small n. Liouville integrability of the corresponding system follows a series of involutive Hamiltonians generated by the characteristic polynomial of the Lax matrix. The rational case appears...