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The recursion operator of a new modified KdV equation and its inverse are explicitly given. Acting the recursion operator and its inverse on the trivial symmetry 0 related to the identity transformation, the infinitely many local and nonlocal symmetries are obtained. Using a closed finite dimensional symmetry algebra with both local and nonlocal symmetries of the original model, some symmetry reductions and exact solutions are found.

The symmetry study plays an important role in almost all the scientific fields, especially, in mathematical physics. To find infinitely many symmetries of a given integrable model, one of the best ways is to construct a recursion operator of the studied model. Recently, Lou pointed out that the infinitely many nonlocal symmetries can be simply constructed via acting the inverse recursion operator on the identity transformation [

The mKdV equation,

In this paper, we investigate some integrable properties and exact solutions of the NMKdV equation (

To find the relation between the mKdV equation (

The usual tanh function expansion method, with

Substituting (

A symmetry,

From the transformation relation (

It is known that for a (

From the symmetry transformation relation (

One recursion operator of the mKdV has also been given in the literature [

Using the above results, we get the recursion operator

To give out some sets of infinitely many symmetries of (

Applying the recursion operator

It is not difficult to find that the

In this section, we discuss the possible group invariant solutions via symmetry reductions by using the Lie point symmetry algebra

To look for the group invariant solutions under the symmetry algebra

Solving the symmetry constraint condition (

Substituting (

The group invariant solution for the field

Obviously, the solution of (

In summary, we have studied the symmetries and symmetry reduction solutions of a new modified KdV equation proposed by Lou. The model is a combination form of the Schwarzian KdV and the potential modified KdV equation.

The recursion operator and the inverse recursion operator are explicitly given. Applying the recursion operator and its inverse to the trivial seed solution, 0, related to the identity transformation, we find one (half) set of local symmetries

For the usual KdV and the modified KdV equations, to find the interaction solutions between one soliton and another kind of waves such as the elliptic periodic waves, one has to use the nonlocal symmetries related to the Darboux transformation [

The work was supported by the Special Funds of the National Natural Science Foundation of China (Grant no. 11141003). The authors are indebted to Professor S. Y. Lou for his helpful discussions.