**1. Introduction**

The Korteweg-de-Vries equation (KdV equation, for short) and the modified Korweg de-Vries equation (mKdV equation, for short) are known as nonlinear equations holding the soliton solutions. Let *u* and *v* be the solutions of the KdV equation and mKdV equation, respectively. Let functions *u* and *v* be the general solutions that satisfy

$$\begin{aligned} \text{[KdV]} & \quad \partial\_t u - 6u \partial\_x u + \partial\_x^3 u = 0, \\ \text{[mKdV]} & \quad \partial\_t v - 6v^2 \partial\_x v + \partial\_x^3 v = 0. \end{aligned}$$

without identifying the details such as the initial and boundary conditions of the mixed problem. For a recent result associated with the well-posedness of the KdV equations, the existence and uniqueness of the solution of semilinear KdV equations in non-parabolic domain is obtained in [1] by using the parabolic regularization method, the Faedo-Galerkin method, and the approximation of a non-parabolic domain by a sequence of regularizable subdomains. Meanwhile, interesting studies on the family of KdV-type equations have been recently carried out in [2]. Let a set of all the real numbers be denoted by <sup>R</sup>. Although *<sup>u</sup>* and *<sup>v</sup>* are functions of *<sup>t</sup>* <sup>∈</sup> <sup>R</sup> and *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>, they are not apparently shown if there is no confusion. The Miura transform [3] M : *u* → *v* reads

$$
\mu = \partial\_x v + v^2,\tag{1}
$$

which is formally the same as the Riccati's differential equation of a variable *x* if *u* is assumed to be a known function. In this article, the Miura transform is generalized as the transform in the abstract spaces. The essence of several nonlinear transforms are pined downed within the theory of abstract equations defined in a general Banach spaces. In conclusion, the structure of the general solutions of second order abstract evolution equations are presented in association with the Miura transform.

#### **2. Operator Logarithm as Nonlinear Transform**

#### *2.1. Nonlinear Transform Associated with the Riccati's Equation*

Following the method for solving the Riccati's equation, the logarithmic type transform appears as

$$
\psi = \psi^{-1} \partial\_x \psi,\tag{2}
$$

which corresponds to *v* = *∂<sup>x</sup>* log *ψ* if log *ψ* and its derivative are well-defined. This is formally the same as the Cole-Hopf transform [4–6]. By applying this transform to the Miura transform, the Miura transform is written by

$$\begin{aligned} \mu &= \partial\_x \upsilon + \upsilon^2 \\ \nu &= \partial\_x (\psi^{-1} \partial\_x \psi) + (\psi^{-1} \partial\_x \psi)^2 \\ \nu &= -\psi^{-2} (\partial\_x \psi)^2 + \psi^{-1} (\partial\_x^2 \psi) + \psi^{-1} (\partial\_x \psi) \psi^{-1} (\partial\_x \psi) . \end{aligned} \tag{3}$$

If *ψ* and *∂xψ* commute,

$$\begin{aligned} u &= - (\partial\_x \psi)^2 \psi^{-2} + (\partial\_x^2 \psi) \psi^{-1} + (\partial\_x \psi)^2 \psi^{-2} = (\partial\_x^2 \psi) \psi^{-1} \\ \Leftrightarrow \quad \partial\_x^2 \psi &= u \psi. \end{aligned} \tag{4}$$

This is the second order evolution equation in which *u* plays a role of infinitesimal generator, and the evolution direction is fixed to *x*. It is remarkable that, after the combination with the Cole-Hopf transform, the Miura transform M : *u* → *ψ* is a transform between nonlinear KdV equation and linear equation. In other words, it provides the transform between the evolution operator and its infinitesimal generator. In the following the obtained transform from *u* to *ψ* is called the combined Miura transform.

#### *2.2. Miura Transform and Cole-Hopf Transform*

It is worth differentiating Equation (2) for clarifying the identity of the Miura transform. Under the commutation assumption, the formal calculation without taking the differentiability into account leads to

$$
\partial\_x^2(\log \psi) := \partial\_x(\psi^{-1} \partial\_x \psi) = (\partial\_x^2 \psi) \psi^{-1} - (\psi^{-1} \partial\_x \psi)^2,\tag{5}
$$

where the first term of the right hand side corresponds to the combined Miura transform, and the second term of the right hand side is the square of the Cole-Hopf transform. It simply means that *∂*2 *<sup>x</sup>*(log *ψ*) being defined by the right hand side of (5) can be defined by the combined Miura transform and Cole-Hopf transform simultaneously. As is well known in the theory of integrable systems, *∂*2 *<sup>x</sup>*(log *ψ*) corresponds to one typical type of Hirota's methods [7], thus a typical type of linear to nonlinear transformation. This type is known to be associated with the Bäcklund transform and KP theory (for a textbook, see [8]). That is, the second order derivative can be represented by the two transforms.
