**Yoritaka Iwata**

Faculty of Chemistry, Materials and Bioengineering, Kansai University, Osaka 564-8680, Japan; iwata\_phys@08.alumni.u-tokyo.ac.jp

† This article has been presented in ICRAAM 2020 Conference, Kuala Lumper, Malaysia, 4–6 February 2020.

Received: 10 March 2020; Accepted: 29 April 2020; Published: 8 May 2020

**Abstract:** Miura transform is known as the transformation between Korweg de-Vries equation and modified Korweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type transformations as an origin of a certain kind of nonlinearity in the soliton equations. In this article, based on the logarithmic representation of operators in infinite-dimensional Banach spaces, a structure common to both Miura and Cole-Hopf transforms is discussed. In conclusion, the Miura transform is generalized as the transform in abstract Banach spaces, and it is applied to the higher order abstract evolution equations.

**Keywords:** Miura transform; soliton equations; logarithm
