Repeated application of the recursion operator for a new hierarchy of negative-order integrable KdV equations

In this work we use the repeated application of the recursion operator to establish a new hierarchy of negative-order integrable KdV equations of higher orders. The concept of the inverse recursion operator allows us to develop this new hierarchy. The complete integrability of each equation is guaranteed via the use of the recursion operator. We show that the dispersion relations of this hierarchy follow an infinite geometric series. Multiple soliton solutions for each equation of the hierarchy are obtained.