*3.1. Generalization of Miura Transform*

Let X be a Banach space, and Y be a dense Banach subspace of X . By focusing on establishing the definition of infinitesimal generator, the discussion is limited to the autonomous case. The Cauchy problem for the second order abstract evolution equation of hyperbolic type (for example, see [17]) is defined by

$$\begin{cases} d^2 u(t) / dt^2 - \mathcal{A}(t) u(t) = 0, & t \in [0, T] \\\\ u(0) = u\_0 \end{cases} \tag{8}$$

in X and A(*t*) : Y→X . Let the Cauchy problem be solvable; i.e., it admits the well-defined evolution operator U(*t*,*s*) ∈ *B*(X ) satisfying the strong continuity and the semigroup property:

$$\mathcal{U}(t,s) = \mathcal{U}(t,r)\mathcal{U}(r,s)$$

for 0 ≤ *s* ≤ *r* ≤ *t* < *T*. The solution is represented by *u*(*t*) = U(*t*,*s*)*us* with *us* ∈ X for a certain 0 ≤ *s* ≤ *T*. For the second order equation, U(*t*,*s*) is not equal to the abstraction of exp( - A(*t*)*dt*), so that A(*t*) is not the infinitesimal generator of U(*t*,*s*), and A(*t*) is the infinitesimal generator of exp( - A(*t*)*dt*) instead. In the following, the combined Miura transform is shown to be equivalent to the logarithmic representation for the infinitesimal generator of the second order abstract evolution equations.

The master equation of (8) is also written as a system of equations:

$$\begin{cases} \, \left. \operatorname{d} u(t)/dt - v(t) = 0, \\ \right|\_{\, \left. \operatorname{d} v(t)/dt \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right]}}}} \\ \, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right]}}}} \\ \end{cases} \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right]}}} \\ \} \\ \, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right]}}} \\ \} \\ \, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right]}}} \\ \} \\ \end{cases} \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right|\_{\, \left. \operatorname{d} t \right]}}} $$

where, by focusing on the representation of *v*(*t*), *v*(*t*) is formally represented by *v*(*t*) = *∂t*U(*t*,*s*)*vs* for a certain *vs* ∈ *D*(*∂t*U(*t*,*s*)) that is compatible with the original *u*(*t*) = U(*t*,*s*)*us* for a certain *us* ∈ *D*(U(*t*,*s*)) = X .

**Lemma 2** (Logarithmic representation of the derivative)**.** *Let κ be a certain complex number. For the evolution operator of Equation* (8)*, let* <sup>U</sup>(*t*,*s*) *be included in the <sup>C</sup>*<sup>1</sup> *class in terms of variables t and s, and the first order derivative* V(*t*,*s*) := *∂t*U(*t*,*s*) *be further assumed to be bounded on* X *and strongly continuous for* 0 ≤ *t*,*s* ≤ *T. Then, for* V(*t*,*s*)*,*

$$\partial\_t \text{Log}\mathcal{V}(t, \mathbf{s}) := (I - \kappa e^{-\mathbb{A}(t, \mathbf{s})})^{-1} \,\partial\_t \mathbb{A}(t, \mathbf{s}) \tag{10}$$

*is well defined if* V(*t*,*s*) *and ∂t*V(*t*,*s*) *commute, where a*ˆ(*t*,*s*) : *∂t*U(*t*,*s*) → X *is an operator defined by α*ˆ(*t*,*s*) = Log(V(*t*,*s*) + *κI*)*.*

**Proof.** The statement follows from Lemma 1.

On the other hand, another logarithmic representation

$$\partial\_t \text{Log} \mathcal{U}(t, \mathbf{s}) := (I - \kappa e^{-a(t, \mathbf{s})})^{-1} \, \partial\_t a(t, \mathbf{s}) \tag{11}$$

is trivially well-defined by the assumption. According to the representations (10) and (11), the abstract version of the Miura transform is obtained as the product of two logarithmic representations.

**Theorem 1** (Abstract formulation of the Miura transform)**.** *Let t and s satisfy* 0 ≤ *t*,*s* ≤ *T, κ be a certain complex number, and* Y *and* Y *be a dense subspace of* X *. The operator ∂tα*(*t*,*s*) *is assumed to be a closed operator from* Y *to* X *, and ∂tα*ˆ(*t*,*s*) *is assumed to be a closed operator from* Y *to* X *. If* U(*t*,*s*)*, ∂t*U(*t*,*s*)*, and ∂*<sup>2</sup> *<sup>t</sup>* U(*t*,*s*) *commute with each other within a properly given domain space, the operators* {A(*t*)}0≤*t*≤*<sup>T</sup> are represented by means of the logarithm function; there exists a certain complex number κ* = 0 *such that*

$$\mathcal{A}(t) \, u = (I - \kappa e^{-\hbar \left(t, s\right)})^{-1} \partial\_l \mathring{u}(t, s) \, (I - \kappa e^{-a\left(t, s\right)})^{-1} \partial\_l u(t, s) \, u,\tag{12}$$

*for an element u of a dense subspace* {*<sup>u</sup>* ∈ Y; (*<sup>I</sup>* <sup>−</sup> *<sup>κ</sup>e*−*α*(*t*,*s*))−1*∂tα*(*t*,*s*)*<sup>u</sup>* <sup>⊂</sup> *<sup>D</sup>*(*∂tα*ˆ(*t*,*s*))} *of* <sup>X</sup> *, and*

$$\mathcal{A}(t) \text{ ä } = (I - \kappa e^{-\kappa(t,s)})^{-1} \partial\_l a(t,s) (I - \kappa e^{-\hbar(t,s)})^{-1} \partial\_l \pounds(t,s) \text{ är},\tag{13}$$

*for an element u of a dense subspace* ˆ {*u*ˆ ∈ Y ; (*<sup>I</sup>* <sup>−</sup> *<sup>κ</sup>e*−*α*ˆ(*t*,*s*))−1*∂tα*ˆ(*t*,*s*)*u*<sup>ˆ</sup> <sup>⊂</sup> *<sup>D</sup>*(*∂tα*(*t*,*s*))} *of* <sup>X</sup> *.*

**Proof.** The autonomous Equation (8), which corresponds to the abstract form of the combined Miura transform (4), is written by

$$
\partial\_t^2 \mathcal{U}(t, s)u = \mathcal{A}(t)\mathcal{U}(t, s)u
$$

for any *<sup>u</sup>* ∈ X , so that it follows that *<sup>∂</sup>*<sup>2</sup> *<sup>t</sup>* U(*t*,*s*) = AU(*t*,*s*) is valid as an operator equation. Under the assumption of commutative property between A and U(*t*,*s*), the operator A is represented by

$$\begin{aligned} \mathcal{A}(t) &= \mathcal{U}(t, \mathbf{s})^{-1} \partial\_t^2 \mathcal{U}(t, \mathbf{s}) \\ &= \mathcal{U}(t, \mathbf{s})^{-1} \partial\_t \mathcal{U}(t, \mathbf{s}) \left( \partial\_t \mathcal{U}(t, \mathbf{s}) \right)^{-1} \partial\_t^2 \mathcal{U}(t, \mathbf{s}), \end{aligned}$$

where the former part <sup>U</sup>(*t*,*s*)−1*∂t*U(*t*,*s*) and the latter part (*∂t*U(*t*,*s*))−1*∂*<sup>2</sup> *<sup>t</sup>* U(*t*,*s*) of the right hand side correspond to the logarithmic representation (*<sup>I</sup>* <sup>−</sup> *<sup>κ</sup>e*−*α*(*t*,*s*))−1*∂tα*(*t*,*s*) and (*<sup>I</sup>* <sup>−</sup> *<sup>κ</sup>e*−*α*ˆ(*t*,*s*))−1*∂tα*ˆ(*t*,*s*) respectively. In this equation <sup>U</sup>(*t*,*s*), *<sup>∂</sup>t*U(*t*,*s*), and *<sup>∂</sup>*<sup>2</sup> *<sup>t</sup>* U(*t*,*s*) are assumed to commute with each other, so that the logarithmic representation of A(*t*) follows.

In particular, for the commutation between two generally-unbounded operators, the intermediate domain space can be different depending on the order of operators. Here is a reason why two different orders of representations (12) and (13) are obtained.
