数理解析研究所講究録 1261 巻 2002 年 115-122

115
Solvability of non-linear totally characteristic partial
differential equations in the complex domain-when resonances occur-

Hidetoshi TAHARA (田原秀敏).

Department of Mathematics, Sophia University (上智大 理工)

Abstract
Let us consider the following non-linear singular partial differential equationin the complex domain. When theauthor has proved with H. Chen intheequation is of totally characteristic type,[2] the existence of the unique holomorphic solution provided that the equation
satisfies the Poincar\"e condition and that no resonances occur. In this PaPer, hewill solve the same equation in the case where some resonances occur.
$(t\partial_{t})^{m}u=F(\mathrm{t} , \{(t\partial_{t})^{\mathrm{j}}\partial_{x}^{\alpha}u\}_{j+\alpha\leq m,j<m})$\S 1. Introduction.

. Let

, set
complex$N=\#\{(j, \alpha)\in \mathrm{N}\cross \mathrm{N};j+\alpha\leq m,j<m\}(=m(m+3)/2)$ , and write theNotations:$(t, x)\in \mathbb{C}_{t}\cross \mathbb{C}_{x}$,$\mathrm{N}=\{0$, 1, 2,$\ldots$

$\}$

, and

$\mathrm{N}^{*}=\{1,2, \ldots\}$$m\in \mathrm{N}^{*}$.
variableIn this paper we will consider the following non-linear partial differential equation:$z=\{zj,\alpha\}_{j+\alpha<m,j<m}\in \mathbb{C}^{N}$(E)$(t \frac{\partial}{\partial t})^{m}u=F(t,$ $x$,$\{(t\frac{\partial}{\partial t})^{j}(\frac{\partial}{\partial x})^{\alpha}u\}_{j<m}j+\alpha\leq m)$,ofwhere $F(t, x, z)$ is afunction in the variables $(t,x, z)$ defined in aneighborhood$\Delta_{0}=\Delta\cap\{t=$, and $u=u(t, x)$ is the unknown function. Set
the origin of, $z=0\}$ . We impose the following conditions on $F(t, x, z)$ :$\Delta$$\mathbb{C}_{t}\cross \mathbb{C}_{x}\cross \mathbb{C}_{z}^{N}$$0$$\mathrm{A}_{1})F(t, x, z)$is aholomorphic function on$\mathrm{A}_{2})F(0,x,0)\equiv 0$

on$\Delta_{0}$$\Delta$;.

Set $I_{m}=\{(j, \alpha)\in \mathrm{N}\cross \mathrm{N} ; j+\alpha\leq m,j<m\}$ and $I_{m}(+)=\{(j, \alpha)\in I_{m} ; \alpha>0\}$ .
Then the situation is divided into the following three cases:

Case 1:$\frac{\partial F}{\partial z_{j,\alpha}}(0, x,0)\equiv 0$

Case 2:$\frac{\partial F}{\partial z_{j,\alpha}}(0,0,0)\neq 0$on$\Delta_{0}$for all $(j, \alpha)\in I_{m}(+)$ ;for some $(j, \alpha)\in I_{m}(+)$ ;Case 3: the other case.
In the case 1, equation (E) is called anon-linear Fuchsian type partial differential
by G\’erard-Tahara $[3][4]$ . In the case 2, equation
equation and it was studied quite$\mathrm{w}\mathrm{e}\mathrm{U}$116(E) is called aspacially non-degenerate type partial differential equationand it gives
us akind of Grousat problem: G\’erard-Tahara [5] discussed aparticular class ofthecase 2and proved the existence of holomorphic solutions and also singular solutions of(E). In the case 3, equation (E) is called anon-linear totally
characteristic type partial
differential equation. The main thema of this paper is to discuss the case 3under thefollowing condition:$\mathrm{A}_{3})$$\frac{\partial F}{\partial z_{\mathrm{j},\alpha}}(0,x, 0)=O(x^{a})$$(\mathfrak{B}$X$arrow 0)$

for all (j,$\alpha)\in I_{m}(+)$ .

\S 2. Review of the result of Chen-Tahara [2].
Under the condition A3), Chen-Ihhara [2] has proved the existence of the uniqueholomorphic solution provided that the equation satisfies both non-resonanceconditionand the Poincare’ condition. We will recall this result now.
By the condition ) we have $(\partial F/\partial zj,a)(0,x, 0)=x^{a}cj,a(x)$ for some holomorphic
functions. Set$\mathrm{A}_{3}$$c_{j,a}(x)$(2.1)$L(\lambda,\rho)=\lambda^{m}-j+$$j<m \sum_{a\leq m},c_{j,a}(0)\lambda^{j}\rho(\rho-1)\cdots(\rho-\alpha+1)$.

Then equation (E) is rewritten in the form
(2.2)

$L(t \frac{\partial}{\partial t},x\frac{\partial}{\partial x})u$$=x \sum_{(j,a)\in I_{m}}S(c_{j,a})(x)(t\frac{\partial}{\partial t})^{j}(x\frac{\partial}{\partial x})(x\frac{\partial}{\partial x}-1)\cdots(x\frac{\partial}{\partial x}-\alpha+1)u$$+a(x)t+R_{2}(t,x,$$\{(t\frac{\partial}{\partial t})^{j}(\frac{\partial}{\partial x})^{a}u\}_{(j,\alpha)\in I_{m}})$,where $S(cj,a)(x)=(c_{j,a}(x)-\mathrm{C};,\mathrm{a}(0))/\mathrm{x}$, $a(x)$ is aholomorphic function on Ao, andis a holomorphic fimction whoae Rylor expansion in $(t,z)$ consists of theterms with degree greater than or equal to 2(with respect to ( , )). Therefore, it iseasy to see that if $L(k, l)\neq 0$ holds for any
the equation (2.2) has aunique formal solution of the form
$R_{2}(t,x, z)$

$t$

$z$

$(k, l)\in \mathrm{N}^{*}\cross \mathrm{N}$(2.3)$u(t,x)= \sum_{k\geq 1,l\geq 0}u_{k,l}t^{k}x^{l}$.

Next, let us consider the convergence of this formal solution. Denote by
the roots of the folowing equation in $X$ :$c_{\dot{g},a}(0)X^{j}=0$$X^{m}-\mathrm{j}+$$j<m \sum_{a=m},$.

$c_{1}$,$\ldots$,$c_{m}$

117
Then, if we factorize(2.4)

into the form

$L(\lambda, l)$$L(\lambda, l)=(\lambda-\xi_{1}(l))\cdots(\lambda-\xi_{m}(l))$

by renumbering the subscript

$i$

of

$\xi_{i}(l)$for$l\in \mathrm{N}$,suitably we have.

$\lim\underline{\xi_{i}(l)}=\mathrm{q}$for $i=1$ , $\ldots,m$ .

$l$$larrow\infty$

we can find a $\sigma>0$ such that $|L(k, l)|\geq\sigma(k+l)^{m}$,
Therefore, if ,with $k+l$ being sufficiently large. This condition leads us
holds for any
to the convergence of the formal solution.
Thus, set$c_{1}$$\ldots$$c_{m}\in \mathbb{C}\backslash [0, \infty)$$(k, l)\in \mathrm{N}^{*}\cross \mathrm{N}$(N)(non-resonance)$L(k, l)\neq 0$(P)(Poincare’ condition)holds for any$c_{\dot{*}}\in \mathbb{C}\backslash [0, \infty)$$(k, l)\in \mathrm{N}^{*}\cross \mathrm{N}$for $i=1$ ,$\ldots,$

$m$

,

;

and we have:). Then, if the), A2) andAssumeTheorem 1(Chen-Tahara [2]).
conditions (P) and (N) are satisfied, equation (E) has a unique holomorphic solution
satisfying $u(0, x)\equiv 0$ near $x=0$ .
$u(t, x)$ in a neighborhood of$A_{3}$$A_{1}$$(0, 0)\in \mathbb{C}_{t}\cross \mathbb{C}_{x}$The purpose of this paper is to solve the equation (E) in the case where the Poincar\’econdition (P) is satisfied but the non-resonance condition (N) is not satisfied.

\S 3. When resonances occur.
Let$L(\lambda, \rho)$be the polynomial in (2.1), and let $\xi_{i}(l)(i=1, \ldots, m)$ be as in (2.4).

Set$\mathcal{M}=\{(k, l)\in \mathrm{N}^{*}\cross \mathrm{N};L(k, l)=0\}$,$\mathcal{M}:=\{(k, l)\in \mathrm{N}^{*}\cross \mathrm{N};k-\xi_{\dot{l}}(l)=0\}$$(i=1, \ldots, m)$.

is equivalent to the non-resonance. Note that
We have
we note:condition (N). In the caseLemma 1.
If the Poincari condition (P) is satisfied, we have the following
is a finite set; (2) there is a $\sigma>0$ such that $|k-\xi:(l)|\geq\sigma(k+l)$properties: (1).
holds for any
and we say that $\mu(k, l)$ is thewe setFor$\mathcal{M}=\emptyset$$\mathcal{M}=\mathcal{M}_{1}\cup\cdots\cup \mathcal{M}_{m}$$\mathcal{M}\neq\emptyset$$\mathcal{M}$$(k, l)\in(\mathrm{N}^{*}\cross \mathrm{N})\backslash \mathcal{M}_{i}(i=1, \ldots, m)$$(k, l)\in \mathcal{M}$$\mathrm{u}(\mathrm{t},\mathrm{x})=\#\{i;\xi_{i}(l)=k\}$multiplicity of resonance ofmultiplicities of resonances of$L(\lambda, \rho)$(3.1)

at $(k, l)$ . We denote by, that is,$L(\lambda, \rho)$l).

$\mu=\sum_{(k,l)\in\lambda 4}\mu(k,$The following is the main result of this paper$\mu$

the total number of the

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Theorem 2(when resonances occur). Assume
equation (E) has a solution $u(t,$ x) of the form
(3.2)

$u(t, x)=W(t, t(\log t),$ $t(\log t)^{2}$ ,$\ldots$,$A_{1}$),$A_{2}$$t(\log t)^{\mu}$), As) and (P). Then,, $x)$

where is the number in (3.1) andis a holomorphic function inneighborhood of $(t_{0}, t_{1}, \ldots, t_{\mu},x)=(0,0, \ldots, 0, 0)$ satisfying $W(0,0, \ldots, 0, x)\equiv 0$ near$W(t_{0}, t_{1}, \ldots, t_{\mu}, x)$

$\mu$

$a$

$x=0$ .

Sketch of the proof. We set$t_{0}=t$, $t_{1}=t(\log t)$ ,$\ldots$, $t_{\mu}=t(\log t)^{\mu}$and set $u(t,x)=W(t_{0},t_{1}, \ldots, t_{\mu},x)$ . Then we have$t \frac{\partial u}{\partial t}=\tau W$

where$\tau=\sum_{\dot{l}=0}^{\mu}t:\frac{\partial}{\partial t}.\cdot+\sum_{\dot{|}=1}^{\mu}it:-1\frac{\partial}{\partial t_{\dot{1}}}$and our equation (E) is written in the form

(3.3)

where

$C(x,$$\mathrm{W}$(to,$t_{1}$,$\tau$$\ldots$,,$x \frac{\partial}{\partial x})W=a(x)t+R_{2}(t_{0},x$$t_{\mu},$$x$,$\{\tau^{j}(\frac{\partial}{\partial x})^{\alpha}W\}_{(j,a)\in I_{m}})$,) is the new unknown function and$C(x, \lambda,\rho)=L(\lambda,\rho)-x\sum_{(j,a)\in I_{m}}S(c_{j,a})(x)\lambda^{j}\rho(\rho-1)\cdots(\rho-\alpha+1)$(Step 1) Construction of a formal solution of (3.3).
Denote by
be the set of all the homogeneous polynomials of degree infor aformal solution $W$ of the form
$k$

(3.4)
with wkyl(3.5).

$H_{k}[t_{0}, t_{1}, \ldots, t_{\mu}]$$(t_{0}, t_{1}, \ldots, t_{\mu})$. Let us look$W(t_{0}, t_{1}, \ldots,t_{\mu},x)=\sum_{k\geq 1,l\geq 0}w_{k,l}(t_{0}, t_{1}, \ldots, t_{\mu})x^{l}$$\in H_{k}[t_{0}, t_{1},$$t_{\mu}]$\ldots ,(for k$\geq 1$). Set$t_{\mu})x^{l}\in H_{k}$$w_{k}= \sum_{l\geq 0}w_{k,l}(t_{0},$\ldots ,[to,\ldots ,$t_{\mu}$] $[[x]]$(k$\geq 1)$;

we have $W= \sum_{k>1}w_{k}$ . By substituting this $W$ into (3.3) and by comparing thehomogeneous part of degree with respect to to, ,, in both sides of (3.3) we seethat equation (3.3) is decomposed into the following recursive family:$k$$(3.6)_{k}$$C(x,$$\tau$,$x \frac{\partial}{\partial x})w_{k}=f_{k}(t_{0},x$k $=1,$ 2,\ldots ,$t_{1}$,$\{D_{j,a}w_{p}$$\ldots$$t_{\mu}$; $1\leq p\leq k$ -1, (j,$\alpha)\in I_{m}\})$ ,119
where $f1=a(x)t_{0}$ and$I_{m}\}$(for $k\geq 2$ ) is apolynomial of $\{Dj,\alpha w;p1\leq p\leq k-1$ ,$f_{k}$$(j, \alpha)\in$.

Moreover, by substituting (3.5) into $(3.6)_{k}$ and by comparing the homogeneous partof degree with respect to we see that equation $(3.6)_{k}$ is decomposed into the following
recursive family:
$l$

$x$

$L(\tau, l)w_{k,l}=g_{k,l}$$(3.7)_{k,l}$,$l=0,1,2$ ,$\ldots$

with

,

(3.8)

$g_{k}, \iota=\sum_{(j,\alpha)\in I_{m}}\sum_{h=0}^{l-1}c_{j,\alpha,l-h}\tau^{j}[h]_{\alpha}w_{k,h}+\phi_{k,l}$,
are the coefficients of the Taylor expansion
are the coefficients of $f_{k}=$
, and
for
, $wk-1$
which are determined by ,, ,are of the form (3.5).
provided that

where
,
$1$

$cj, \alpha(x)=\sum_{l\geq 0^{\mathrm{C}}j,\alpha,l}x^{l}$$c_{j,a,l}$$\alpha\geq 1$$[\lambda]_{\alpha}=\lambda(\lambda-1)\cdots(\lambda-\alpha+1)$$\sum_{l\geq 0}\phi_{k,l}(t_{0}, t_{1}, \ldots, t_{\mu})x^{l}\in H_{k}[t_{0}, t_{1}, \ldots, t_{\mu}][[x]]$$w_{1}$$\ldots$$[\lambda]_{0}=$$\phi_{k,l}$$w_{1}$$\ldots$$w_{k-1}$

Thus, to get aformal solution $W$ in the form (3.4) it is enough to solve $(3.7)_{k,l}$inductively on $(k, l)$ in the folowing way: 1) first we solve $(3.7)_{1,0}$ , then we solve$(3.5)_{1,l}$ inductively on, $w_{k-1}$ are already constructed, weand obtain ; 2) if ,$(3.7)_{k,l}$; 3) repeating the$(3.7)_{k,0}$
and obtainon
inductively, then we solve
solve
same procedure, we can obtain aformal solution of (3.3).
we can get
Therefore, if the equation $(3.7)_{k,l}$ is always solvable in$(3.7)_{k,l}$ isequationtheThough,of the form (3.4).
aformal solution
in aresonant case, and so in this case we must change
not solvable in$l$$w_{1}$$w_{1}$$\ldots$$l$$w_{k}$$H_{k}[t_{0}, t_{1}, \ldots, t_{\mu}]$$W(t_{0}, t_{1}, \ldots, t_{\mu}, x)$$H_{k}[t_{0},t_{1}, \ldots, t_{\mu}]$

our idea: we will consider the equation$L(\tau,l)w_{k,l}\equiv g_{k,l}$$(3.9)_{k,l}$where$(3.7)_{k,l}$$\mathcal{R}_{0}=\mathcal{R}_{1}=\{0\}$in amodulo class$(\mathrm{m}\mathrm{o}\mathrm{d}.R_{k})$,and for $k\geq 2$$\mathcal{R}_{k}=\cup H_{k-2}[t_{0},t_{1}, \ldots,t_{\mu}]i+j=p+q\cross(t_{i}t_{j}-t_{p}t_{q})$.

, $t(\log t)^{\mu})\equiv 0$This causes no troubles in the solution of (3.2), because $f(t, t(\log t)$ ,$(3.9)_{k,l}$$(3.7)_{k,l}$) has the(or
. Moreover we note that
holds for any
form $(\tau-\xi_{1}(l))\cdots(\tau-\xi_{m}(l))w_{k,l}=g_{k,l}$ , and that if resonances occur we have $\xi_{j}(l)=k$
for some $j\in\{1,2, \ldots, m\}$ . The following lemma guarantees the solvability of $(3.7)_{k,l}$(or $(3.9)_{k,l}$ ).
$\ldots$$f(t_{0},t_{1}, \ldots, t_{\mu})\in \mathcal{R}_{k}$(with $0\leq d\leq\mu$)(1) If $\xi_{j}(l)\neq k$ , for any
Lemma 2.
, , ]. (2) If$(\tau-\xi_{j}(l))w=g$)
uniquesolutionhas athe equation$\xi_{j}(l)=k$ ,(with $0\leq d\leq\mu-1$ ) we can find a function
for any).
which satisfies $(\tau-\xi_{j}(l))w\equiv g$ (mod.
$g\in H_{k}[t_{0}, t_{1}, \ldots, t_{d}]$$w\in H_{k}\mathrm{t}\mathrm{f}\mathrm{i}$$t_{1}$$\ldots$$t_{d}$$g\in H_{k}[t_{0},t_{1}, \ldots, t_{d}]$$w\in H_{k}[t_{0}, t_{1}, \ldots, t_{d}, t_{d+1}]$$R_{k}$

Thus, our way of solving the equation $(3.7)_{k,l}$ (or $(3.9)_{k,l}$ ) is as follows: if aresonancedoes not occur at (k, l) we use (1) of lemma 2to solve $(3.7)_{k,l}$ ;while if some resonances

120
occur at $(k,l)$ we use (2) of Lemma 2 to solve. Note that anew variable
is introduced whenever aresonance occurs. Since araee)mnce occurs -times, we must
introduce anew variable also -times. Our starting point is $g_{1,0}=a(0)t_{0}\in H_{1}[t_{0}]$ .
Hence, finally we obtain aformal solution
at most in.
$(3.9)_{k,l}$$t_{d+1}$

$\mu$

$\mu$

$H_{k}[t_{0},t_{1}, \ldots,t_{\mu}]$$w_{k,l}$Summing up, we have constructed aformal solution of the form (3.4) which satisfies$\tau^{m}W-F$($t_{0},x$,$\{\tau^{\mathrm{j}}(\frac{\partial}{\partial x})^{a}W\}_{(j,a)\in I_{m}}$)$\in\sum_{(k,l)\in \mathcal{M}}\prime \mathcal{R}_{k}x^{l}$.

(Step ) Convergence of the formal solution (3.4). For
we write $|\vec{k}|=k0+k_{1}+\cdots+k_{\mu}$ and ( $k\vec{\rangle}=k_{1}+2k_{2}+\cdots+\mu k_{\mu}$ . For $c>0$ and$P$$\vec{k}=(k_{0},k_{1}, \ldots,k_{\mu})\in \mathrm{N}\mu+1$$w= \sum|\vec{k}|=kwt^{k_{0}}t^{k_{1}}\vec{k}01\cdots$ $t_{\mu}^{k_{\mu}}\in H_{k}[t0,t1, \ldots,t_{\mu}]$

we define the norm$|w|_{c}= \sum\frac{|w_{\tilde{k}}|}{c^{\{\vec{k})}}$$|w|_{c}$by.

$|\vec{k}|=k$

It is easy to see

Lemma 3.

For any w $\in H_{k}[t0,$tl,$t_{\mu}]$\ldots ,we have.

$|\tau w|_{c}\leq(1+c\mu)k|w|_{c}$For $c>0$ , $\rho>0$ and afunction $f= \sum_{l\geq 0}f_{k,l}(t_{0},t_{1}, \ldots,t_{d})d$
we define the norm
(or the formal) by$||f||_{\mathfrak{g}\rho}$$\in H_{k}[t_{0},t_{1}, \ldots,t_{d}][[x]]$$\mathrm{n}\mathrm{o}\mathrm{m}||f||_{c,\rho}$I.

$f||_{c,\rho}= \sum_{l\geq 0}|f_{k,l}|_{c}\rho^{l}$

Similarly, for $\rho>0$ and $f(x)= \sum_{l>0}f_{l}x^{l}\in \mathbb{C}[[x]]$ (the ring of formal power series inx) we define the norm$||f||_{\rho}$(or$\mathrm{t}\mathrm{h}\mathrm{e}\overline{\mathrm{f}}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{m}||f||_{\rho}$$||f||_{\rho}= \sum_{l\geq 0}|f_{l}|\rho^{l}$

) by

.

In G\’erard-Ihhara [4], we haveamethod to prove the convergence of formal solutions of non-linear partial differential equations. We will be able toapply this method to this case and obtain the convergence of the formal solution $W$ in(3.4), if we prove the folowing proposition:$\infty \mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{h}\mathrm{d}$$\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{i}_{1}1$

Proposition 1.
Suppose the Poincari condition (P). Then there are positive
constants c $>0$ , C $>0$ and R $>0$ such that the following estimate holds for any k:$\geq 1$

(3.10)$||w_{k}||_{\mathfrak{g}\rho} \leq\frac{C}{k^{m}}|1f_{k}1\mathrm{I}_{c,\rho}$

for any $0<\rho\leq R$ .

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Let us show this now. First we note the following basic lemma.
Lemma 4. If the Poincari condition (P) is satisfied, there are positive constants$c>0$ and $A>0$ which satisfy the following property. In Lemma 2we can choose
solution $wk,l$ (in both cases (1) and (2)) so that the following estimate holds for any:$a$$(k, l)\in \mathrm{N}^{*}\cross \mathrm{N}$(3.11)$|wk,l|_{c} \leq\frac{A}{(k+l)^{m}}|g_{k,l}|_{c}$.

If we admit this lemma, Proposition 1is proved in thefollowing way. Let $c>0$ and $A>0$ be as in Lemma 4, and take $R>0$ sufficiently
small so that $0<R\leq 1$ andProof of proposition 1.

(3.12)$A(1+c \mu)^{m-1}R\sum_{(j,\alpha)\in I_{m}}||S(cj,\alpha)||_{R}\leq\frac{1}{2}$.

By (3.8) and Lemma 3we have$|g_{k,l}|_{c} \leq\sum_{(j,\alpha)\in I_{m}}\sum_{h=0}^{l-1}|c_{j,\alpha,l-h}|(1+c\mu)^{j}k^{j}l^{\alpha}|w_{k,h}|_{c}+|\phi_{k,l}|_{c}$and therefore$\frac{A}{(k+l)^{m}}|g_{k,l}|_{c}$$\leq A(1+c\mu)^{m-1}\sum_{(j,\alpha)\in I_{m}}\sum_{h=0}^{l-1}|c_{j,\alpha,l-h}||w_{k,h}|_{c}+\frac{A}{k^{m}}|\phi_{k,l}|_{c}$.

Combining this with (3.11) and (3.12) we have$||w_{k}||_{c,\rho}= \sum_{l\geq 0}|w_{k,l}|_{c}\rho^{l}\leq\sum_{l\geq 0}\frac{A}{(k+l)^{m}}|g_{k,l}|_{c}\rho^{l}$$\leq A(1+c\mu)^{m-1}\rho$$\sum$$||S(c_{j,\alpha})||_{\rho}||w_{k}||_{c,\rho}+ \frac{A}{k^{m}}||f_{k}||_{c,\rho}$$(j,\alpha)\in I_{m}$$\leq\frac{1}{2}||w_{k}||_{c,\rho}+\frac{A}{k^{m}}||f_{k}||_{c,\rho}$.

Thus, by setting $C=2A$ we obtain the estimate (3.10).

$\square$

References[1] H. Chen and Z. Luo, On the holomorphic solution of nonlinear totaly characteristic
equations with several space variables, Preprint[2] H. Chen and H. Tahara: On totally characteristic type non-linear partial differential equations in the complex domain, Publ. RIMS, Kyoto Univ., 35 (1999), 621-

636.

122
[3] R. G\’erard and H. Tahara :Solutions holomorphes et singulieres d’iquations aux
der ivees partielles singulieres non liniaires, Publ. RIMS, Kyoto Univ., 29 (1993),

121-151.

[4] R. Gerard and H. Tahara :Singular nonlinear partial differential equations, Aspects of Mathematics, E28, Vieweg, 1996.
[5] R. G\’erard and H. Tahara :Holomorphic and singular solutions of non-linear
singular partial differential equations, II. “ Structure of differential equations,, 1995 ”(edited by MorimotO-Kawai), 135-150, World Scientific,$\mathrm{K}\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{a}/\mathrm{K}\mathrm{y}\mathrm{o}\mathrm{t}\mathrm{o}$1996.

[6] A. Shirai, Convergence offormal solutions of singular first order nonlinear partial
differential equations of totally characteristic type, preprint