Negative order KdV equation with both solitons and kink wave solutions

Abstract: In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u}){t}=2uu{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge transform. The Lax pair of the equation is derived to guarantee its integrability, and furthermore the equation is shown to have classical solitons, periodic soliton and kink solutions.