Recurrence formula for some higher order evolution equations

According to the preceding work [9, 10, 18] dealing with the logarithmic representation of operators, the evolution operator $U(t,s)$ is assumed to be generated by $A_{1}(t)$. Let $\psi$ satisfy Eq. (2.2). The solution $\psi$ is generally represented by $\psi(t)=U(t,s)u_{s}$ for a certain $u_{s}\in X$, so that the operator equality is obtained by identifying $\psi(t)$ with $U(t,s)$, that is, identifying a function with an operator. For instance, it is practical to imagine that $\psi(t)$ is represented by

(3.1)

$$\psi(t)=\exp\left(\int_{s}^{t}A_{1}(\tau)d\tau\right).$$

Here the integral representation (3.1) is valid at least if $A_{1}(t)$ is $t$-independent, whose validity in $t$-independent cases is shown in Appendix I. Let $\kappa$ be a certain complex number. Using the Riesz-Dunford integral [5], the infinitesimal generator $A_{1}(t)$ of the first order evolution equation is written by

(3.2)

$$A_{1}(t)=\psi^{-1}\partial_{t}\psi=(I+\kappa U(s,t))\partial_{t}{\rm Log}(U(t,s)+\kappa I)$$

under the commutation, where ${\rm Log}$ means the principal branch of logarithm, and $U(t,s)$ is temporarily assumed to be a group (i.e., existence of $U(s,t)=U(t,s)^{-1}$ is temporarily assumed to be valid for any $0\leq s\leq t\leq T$), and not only a semigroup. The validity of (3.2) is confirmed formally by

$$\begin{array}[]{ll}(I+\kappa U(s,t))\partial_{t}{\rm Log}(U(t,s)+\kappa I)\vspace{1.5mm}\\ =U(s,t)(U(t,s)+\kappa I)\partial_{t}U(t,s)(U(t,s)+\kappa I)^{-1}\vspace{1.5mm}\\ =U(s,t)\partial_{t}U(t,s)\vspace{1.5mm}\\ =U(s,t)A_{1}(t)U(t,s)=A_{1}(t)\end{array}$$

under the commutation assumption. This relation is associated with the abstract form of the Cole-Hopf transform [13]. Indeed the correspondence between $\partial_{t}\log\psi$ and $A_{1}(t)$ can be understood by $U(s,t)\partial_{t}U(t,s)=A_{1}(t)$ shown above. Indeed,

$$\psi^{-1}\partial_{t}\psi=\partial_{t}\log\psi\quad\Rightarrow\quad\partial_{t}\psi=(\partial_{t}\log\psi)\psi$$

is valid under the commutation assumption.

By introducing alternative infinitesimal generator $a(t,s)$ [10] satisfying

$$e^{a(t,s)}:=U(t,s)+\kappa I,$$

a generalized version of the logarithmic representation

(3.3)

$$A_{1}(t)=\psi^{-1}\partial_{t}\psi=(I-\kappa e^{-a(t,s)})^{-1}\partial_{t}a(t,s)$$

is obtained, where $U(t,s)$ is assumed to be only a semigroup. The right hand side of Eq. (3.3) is actually a generalization of (3.2); indeed, by only assuming $U(t,s)$ as a semigroup defined on $X$, $e^{-a(t,s)}$ is always well defined by a convergent power series, and there is no need to have a temporal assumption for the existence of $U(s,t)=U(t,s)^{-1}$. It is remarkable for $e^{a(t,s)}$ being generated by $a(t,s)$ that $e^{-a(t,s)}=e^{a(s,t)}$ is not necessarily satisfied [10]. The validity of Eq. (3.3) is briefly seen in the following. Using the generalized representation,

$$\begin{array}[]{ll}e^{a(t,s)}=U(t,s)+\kappa I\quad\Rightarrow\quad a(t,s)={\rm Log}(U(t,s)+\kappa I)\vspace{1.5mm}\\ \qquad\Rightarrow\quad\partial_{t}a(t,s)=\left[\partial_{t}U(t,s)\right](U(t,s)+\kappa I)^{-1}=A_{1}(t)U(t,s)(U(t,s)+\kappa I)^{-1}\end{array}$$

is obtained under the commutation assumption between $A_{1}(t)$ and $U(t,s)$. It leads to

$$\begin{array}[]{ll}(I-\kappa e^{-a(t,s)})^{-1}\partial_{t}a(t,s)=(I-\kappa e^{-a(t,s)})^{-1}A_{1}(t)U(t,s)(U(t,s)+\kappa I)^{-1},\end{array}$$

and therefore

$$\begin{array}[]{ll}A_{1}(t)=(U(t,s)+\kappa I)U(t,s)^{-1}A_{1}(t)U(t,s)(U(t,s)+\kappa I)^{-1}\vspace{1.5mm}\\ \quad\Leftrightarrow\quad A_{1}(t)=(I-\kappa e^{-a(t,s)})^{-1}A_{1}(t)(I-\kappa e^{-a(t,s)})\end{array}$$

is valid under the commutation assumption. It simply shows the consistency of representations using the alternative infinitesimal generator $a(t,s)$. In the following, the logarithmic representation (3.3) is definitely used, and the original representation (3.2) appears if it is necessary. It is notable for bounded/unbounded operators holding the above representation that, based on ordinary and generalized logarithmic representations, the algebraic property of set of infinitesimal generator is known [11, 17].

The Miura transform, which holds the same form as the Riccati’s differential equation, is represented by

$$u=\partial_{x}v+v^{2},$$

where $u$ is a solution of the modified Korteweg-de Vries equation (mKdV equation), and $v$ is a solution of the Korteweg-de Vries equation (KdV equation). For the Riccati’s differential equation, $u$ is a function standing for an inhomogeneous term, and $v$ is unknown function. It provides a representation of the infinitesimal generator of second order differential equation. Indeed, if the Miura transform is combined with the Cole-Hopf transform $v=\psi^{-1}\partial_{x}\psi$, it is written by

(3.4)

$$\begin{array}[]{ll}u=\partial_{x}(\psi^{-1}\partial_{x}\psi)+(\psi^{-1}\partial_{x}\psi)^{2}\vspace{2.5mm}\\ =-\psi^{-2}(\partial_{x}\psi)^{2}+\psi^{-1}\partial_{x}(\partial_{x}\psi)+\psi^{-2}(\partial_{x}\psi)^{2}\vspace{2.5mm}\\ =\psi^{-1}\partial_{x}^{2}\psi,\end{array}$$

where the commutation between $\psi$, $\partial_{x}\psi$ and $\partial_{x}^{2}\psi$ is assumed. This issue should be carefully treated in operator situation; i.e., in the standard theory of abstract evolution equations of hyperbolic type [21, 22], $x$-dependent infinitesimal generators (corresponding to $\partial_{x}\psi$ or $\partial_{x}^{2}\psi$ respectively) do not generally commute with the evolution operator (corresponding to $\psi$ or $\partial_{x}\psi$ respectively), although such commutations are always true in $x$-independent infinitesimal generators. For sufficiently smooth $\psi$ in the $x$-direction, one of the implication here is that the function $\psi$ satisfies both the second order equation

$$\begin{array}[]{ll}u=\psi^{-1}\partial_{x}^{2}\psi\quad\Leftrightarrow\quad\partial_{x}^{2}\psi=u\psi\end{array}$$

and the first order equation

$$\begin{array}[]{ll}v=\psi^{-1}\partial_{x}\psi\quad\Leftrightarrow\quad\partial_{x}\psi=v\psi,\end{array}$$

at the same time under the commutation assumption. The combined use of Miura transform and Cole-Hopf transform is called the combined Miura transform in [16] (for the combined use in the inverse scattering theory, e.g., see [1]). Provided the solvable first order autonomous differential equation (i.e., the Cole-Hopf transform) with its solution $\psi$, the Miura transform shows a way to find the second order autonomous differential equation to be satisfied by exactly the same $\psi$.

The combined Miura transform $\partial_{x}^{2}\psi=u\psi$ can be generalized as the second order abstract equation $\partial_{t}^{2}\psi=A_{2}(t)\psi$ in finite or infinite dimensional Banach spaces by taking $u$ as a closed operator $A_{2}(t):D(A_{2})\to X$ in a Banach space $X$, where the index $2$ denotes the order of differential equations, and the notation of variable is chosen as $t$. The solution $\psi$ is generally represented by $\psi(t)=U(t,s)u_{s}$ for a certain $u_{s}\in X$, so that the operator version of the combined Miura transform is formally obtained by assuming $\psi(t)=U(t,s)$.

(3.5)

$$\begin{array}[]{ll}A_{1}(t)=\psi^{-1}\partial_{t}\psi\vspace{1.5mm}\\ \qquad=\partial_{t}\log\psi=\partial_{t}\log U(t,s)\vspace{1.5mm}\\ \qquad=(I-\kappa e^{-a(t,s)})^{-1}\partial_{t}a(t,s)\end{array}$$

is valid under the commutation assumption between $\psi=U(t,s)$ and $\partial_{t}\psi=\partial_{t}U(t,s)$. According to the spectral structure of $U(t,s)$, its logarithm “$\log U(t,s)$” cannot necessarily be defined by the Riesz-Dunford integral. However, the logarithm “$\log e^{a(t,s)}$” is necessarily well defined by the Riesz-Dunford integral, because $a(t,s)$ with a certain $\kappa$ is always bounded on $X$ regardless of the spectral structure of bounded operator $U(t,s)$. That is, it is necessary to introduce a translation (i.e., a certain nonzero complex number $\kappa$) for defining logarithm functions of operator. Here it is not necessary to calculate $\log(U(t,s))$ at the intermediate stage, and only the most left hand side and the most right hand side of Eq. (3.5) make sense.

In terms of applying nonlinear transforms such as the Miura transform and the Cole-Hopf transform, it is necessary to identify the functions with the operators, which is mathematically equivalent to identify elements in $X$ with elements in $B(X)$. In other words, it is also equivalent to regard a set of evolution operators as a set of infinitesimal generators. The operator representation of the infinitesimal generator $A_{2}(t)$ of second order evolution equations has been obtained in [16]. Under the commutation assumption between $\psi$ and $\partial_{t}\psi$ and that between $\partial_{t}\psi$ and $\partial_{t}^{2}\psi$, the representation of infinitesimal generator $A_{2}(t)$ in Eq. (3.9) is formally obtained by

(3.6)

$$\begin{array}[]{ll}A_{2}(t)=\psi^{-1}\partial_{t}^{2}\psi=\left[\psi^{-1}\partial_{t}\psi\right]~{}\left[(\partial_{t}\psi)^{-1}\partial_{t}^{2}\psi\right]=\left[\partial_{t}\log\psi\right]~{}\left[\partial_{t}\log(\partial_{t}\psi)\right]\vspace{1.5mm}\\ \qquad=\left[\partial_{t}\log U(t,s)\right]~{}\left[\partial_{t}\log(\partial_{t}U(t,s))\right]\vspace{1.5mm}\\ \qquad=(I-\kappa e^{-a(t,s)})^{-1}\partial_{t}a_{1}(t,s)~{}(I-\kappa e^{{ua}(s,t)})^{-1}\partial_{t}a_{2}(t,s)\end{array}$$

in $X$, where alternative infinitesimal generators $a_{1}(t,s)$ and $a_{2}(t,s)$, which are defined by

$$e^{a_{1}(t,s)}=U(t,s)+\kappa I,\quad e^{a_{2}(t,s)}=\partial_{t}U(t,s)+\kappa I,$$

and therefore by

$$a_{1}(t,s)={\rm Log}(U(t,s)+\kappa I),\quad a_{2}(t,s)={\rm Log}(\partial_{t}U(t,s)+\kappa I),$$

generate $e^{a_{1}(t,s)}$ and $e^{a_{2}(t,s)}$, respectively. For $\psi(t)=U(t,s)u_{s}\in C^{0}([0,T];X)$, let generally unbounded operator $\partial_{t}U(t,s)$ be further assumed to be continuous with respect to $t$; for the definition of logarithm of unbounded evolution operators by means of the doubly-implemented resolvent approximation, see [18]. Note that the continuous and unbounded setting for $\partial_{t}U(t,s)$ is reasonable with respect to $C^{0}$-semigroup theory, since both $\psi(t)=U(t,s)u_{s}$ and $\partial_{t}U(t,s)u_{s}$ are the two main components of solution orbit defined in the infinite-dimensional dynamical systems. Similar to Eq. (3.5), it is necessary to introduce a translation to define $\log U(t,s)$. Consequently the most right hand side of Eq. (3.6) is a mathematically valid representation for $A_{2}(t)$, where it is not necessary to calculate $\log U(t,s)$ and $\log\partial_{t}U(t,s)$ at the intermediate stage. In this manner, the infinitesimal generators of second order evolution equations are factorized as the product of two logarithmic representations of operators: $\partial_{t}\log U(t,s)$ and $\partial_{t}\log\partial_{t}U(t,s)$ (for the details arising from the operator treatment, see [16]). The order of $\partial_{t}\log U(t,s)$ and $\partial_{t}\log\partial_{t}U(t,s)$ can be changed independent of the boundedness/unboundedness of operator, as it is confirmed by the commutation assumption

$$\psi^{-1}\partial_{t}^{2}\psi=(\partial_{t}^{2}\psi)\psi^{-1}=(\partial_{t}^{2}\psi)(\partial_{t}\psi)^{-1}(\partial_{t}\psi)\psi^{-1}=[(\partial_{t}\psi)^{-1}\partial_{t}^{2}\psi][\psi^{-1}\partial_{t}\psi].$$

This provides the operator representation of the combined Miura transform. In unbounded operator situations, the domain space of infinitesimal generators should be carefully discussed. Indeed, the domain space of $A_{2}(t)$, which must be a dense subspace of $X$, is expected to satisfy

(3.7)

$$\begin{array}[]{ll}D(A_{2}(t))=\left\{u\in X;~{}\{\partial_{t}{\hat{a}}(t,s)u\}\subset D(\partial_{t}a(t,s))\right\}\subset X,\vspace{1.5mm}\\ D(\partial_{t}a(t,s))=\left\{u\in X;~{}\{\partial_{t}a(t,s)u\}\subset X\right\}\subset X\end{array}$$

or

(3.8)

$$\begin{array}[]{ll}D(A_{2}(t))=\left\{u\in X;~{}\{\partial_{t}a(t,s)u\}\subset D(\partial_{t}{\hat{a}}(t,s))\right\}\subset X,\vspace{1.5mm}\\ D(\partial_{t}{\hat{a}}(t,s))=\left\{u\in X;~{}\{\partial_{t}{\hat{a}}(t,s)u\}\subset X\right\}\subset X\end{array}$$

depending on the order of product, where note that both $a(t,s)$ and $\partial_{t}a(t,s)$ depend on $t,s\in[0,T]$.

Consequently, for solutions $\psi(t)=U(t,s)u_{s}$ satisfying $\partial_{t}\psi=A_{1}(t)\psi$, the second order evolution equation

(3.9)

$$\begin{array}[]{ll}\partial_{t}^{2}\psi=A_{2}(t)\psi\end{array}$$

is also satisfied by setting the infinitesimal generator $A_{2}(t)$ as defined by the combined Miura transform (3.4). It means that $A_{2}(t)$ is automatically determined by a given operator $A_{1}(t)$. In the following, beginning with the second order formalism (i.e., the combined Miura transform), the relation is generalized as the recurrence formula for defining the higher order operator $A_{k}$ ($k\geq 3$).

For exactly the same $\psi(t)=U(t,s)$ satisfying $\partial_{t}\psi=A_{1}(t)\psi$, a generally-unbounded operator $\partial_{t}^{k}A$ be continuous with respect to $t$ ($k\geq 1$: integer). Let the infinitesimal generator of $k$-th order evolution equations be defined by

$$A_{k}=\psi^{-1}\partial_{t}^{k}\psi$$

under the commutation assumption. Among limited numbers of preceding works, a lecture note summarizing the theory of higher order abstract equations, see [32]. For defining the higher-order infinitesimal generator in an abstract manner, a recurrence formula is introduced. In this section, the infinitesimal generators $A_{k}(t)$ are assumed to be general ones holding the time dependence.

Using Eq. (4.3), the operator version of Riccati’s differential equation is obtained if $n=2$ is applied. The resulting equation

$$\begin{array}[]{ll}A_{2}(t)=\partial_{t}A_{1}(t)+[A_{1}(t)]^{2}\vspace{1.5mm}\\ \end{array}$$

is the Riccati type differential equation to be valid in the sense of operator. The domain space of $A_{2}(t)$ is determined by $A_{1}(t)$, and not necessarily equal to the domain space of $A_{1}(t)$. Consequently, the Riccati type nonlinear differential equation is generalized to the operator equation in finite/infinite-dimensional abstract Banach spaces. The obtained equation

$$A_{n}(t)=(\partial_{t}+A_{1}(t))A_{n-1}(t)$$

itself is nonlinear if $n=2$, while linear if $n\neq 2$. Its dense domain, which is also recursively determined, is assumed to satisfy

$$\begin{array}[]{ll}D(A_{n}(t))=\left\{u\in X;~{}\{A_{n-1}u\}\subset D(A_{1}(t))\right\}\subset X,\end{array}$$

for $n\geq 2$, with

$$\begin{array}[]{ll}D(A_{1}(t))=\left\{u\in X;~{}\{A_{1}u\}\subset X\right\}\subset X.\end{array}$$

In the following examples, concrete higher order evolution equations are shown. In each case, the recurrence formula plays a role of transform between the first order equation and $n$th order equation.
Example 1. [2nd order evolution equation].  The second order evolution equations, which is satisfied by a solution $\psi$ of first order evolution equation $\partial_{t}\psi=\partial_{x}^{k}\psi$, is obtained. The operator $A_{1}$ is given by $t$-independent operator $\partial_{x}^{k}$ ($k$ is a positive integer).

$$\begin{array}[]{ll}A_{2}=(\partial_{t}+\partial_{x}^{k})\partial_{x}^{k}=\partial_{t}\partial_{x}^{k}+\partial_{x}^{2k},\vspace{1.5mm}\\ \end{array}$$

and then, by applying Eq. (4.3), the second order evolution equation

(4.4)

$$\begin{array}[]{ll}\partial_{t}^{2}u=\partial_{t}\partial_{x}^{k}u+\partial_{x}^{2k}u\end{array}$$

is obtained. Indeed, let $u$ be a general or special solution of $\partial_{t}u=\partial_{x}^{k}u$, the validity of statement

$$\begin{array}[]{ll}\partial_{t}u=\partial_{x}^{k}u\qquad\Rightarrow\qquad\partial_{t}^{2}u=\partial_{t}\partial_{x}^{k}u+\partial_{x}^{2k}u\end{array}$$

is confirmed by substituting the operator equality $\partial_{x}^{k}=u^{-1}\partial_{t}u$ and the associated equality

$$\begin{array}[]{ll}\partial_{t}\partial_{x}^{k}=\partial_{t}(u^{-1}\partial_{t}u)=-(u^{-1}\partial_{t}u)^{2}+(\partial_{t}^{2}u)u^{-1},\vspace{2.5mm}\\ \partial_{x}^{2k}=(u^{-1}\partial_{t}u)^{2}\\ \end{array}$$

to the right hand side of Eq. (4.4), where the commutation between $u$, $\partial_{t}u$ and $\partial_{t}^{2}u$ is assumed to be vald. Equation (4.4) is a kind of wave equation in case of $k=1$. Note that the obtained equation is a linear equation.
Example 2. [3rd order evolution equation]  The third order evolution equations, which is satisfied by a solution $\psi$ of first order evolution equation $\partial_{t}\psi=\partial_{x}^{k}\psi$, is obtained in the same manner. The operator $A_{1}$ is given by time-independent operator $\partial_{x}^{k}$ ($k$ is a positive integer).

$$\begin{array}[]{ll}A_{3}=(\partial_{t}+\partial_{x}^{k})(\partial_{t}\partial_{x}^{k}+\partial_{x}^{2k})=\partial_{t}(\partial_{t}\partial_{x}^{k}+\partial_{x}^{2k})+\partial_{x}^{k}(\partial_{t}\partial_{x}^{k}+\partial_{x}^{2k}),\vspace{1.5mm}\\ =\partial_{t}^{2}\partial_{x}^{k}+2\partial_{t}\partial_{x}^{2k}+\partial_{x}^{3k},\vspace{1.5mm}\\ \end{array}$$

and the third order evolution equation

$$\begin{array}[]{ll}\partial_{t}^{3}u=\partial_{t}^{2}\partial_{x}^{k}u+2\partial_{t}\partial_{x}^{2k}u+\partial_{x}^{3k}u\end{array}$$

is obtained. The validity is confirmed by substituting the operator equality $\partial_{x}^{k}=u^{-1}\partial_{t}u$.

Although the relation between $A_{n}$ and a given $A_{1}$ can be understood by Theorem 1, those representations and the resulting representations of evolution operator ($C_{0}$-semigroup) are not understood at this point. In this section, utilizing the logarithmic representation of the infinitesimal generator, the representation of infinitesimal generator for the high order evolution equation is obtained. Since the logarithmic representation has been known to be associated essentially with the first- and second-order evolution equations, the discussion in the present section clarifies a universal role of logarithm of operators, independent of the order of evolution equations.

Although the commutation assumption in Theorems 4.1 and 4.2 is restrictive to the possible applications, all the $t$-independent infinitesimal generators satisfy this property. That is, the present results are applicable to linear/nonlinear heat equations, linear/nonlinear wave equations, and linear/nonlinear Schrödinger equations. Consequently, a new path is introduced to the higher order evolution equation in which the concept of ”higher order” is reduced to the concept of ”operator product” in the theory of abstract evolution equations. Using the alternative infinitesimal generator being defined by $e^{\alpha_{k}(t,s)}={\mathcal{U}}_{k}(t,s)+\kappa I$, ${\mathcal{U}}_{k}(t,s)$ can be replaced with $e^{\alpha_{k}(t,s)}$, and the following corollary is valid.