2. Algebraic and Analytic Aspects of the Theory of Singular Perturbations
Let be a complete topological commutative algebra with unit e and let be a sequence of open sets . Let us denote by the spaces of functions continuous on the sets respectively with their values in . Let us formulate the block I of necessary conditions:
- (1°)
If the sequence
is a bounded set [
3] in
, then the series
converges at
.
- (2°)
If the sequence
is such that the series
converges on each set
, where
T is an arbitrary compact from
X;
is an arbitrary compact from
, ⋯;
is an arbitrary compact set from
in some neighborhood of the value
, the function
and it can be extended to all
, then we have
and the last row with coefficients from
is convergent.
Definition 1. The function represented by (1), is called ε-regular. - (3°)
If the system
with
-regular left-hand sides is uniquely solvable with respect to
for
in some neighborhood of the point
, then it is also uniquely solvable in some neighborhood of the same point and thus functions
are
-regular.
Remark 1. The conditions of the block I are satisfied if , are simply connected regions, are spaces of holomorphic functions on respectively.
In order to formulate the conditions of block II, we give some definitions.
Definition 2. s-product of tuples and is a function .
Definition 3. Let , , . The composition f and is determined by the formula as usual.
Block of conditions II:
- (1°)
All algebras contain constant functions and linear functions. We consider the embeddings together with topologies to be obvious.
- (2°)
On all spaces , a linear operation is defined such that , where p is a constant function, and if and does not depend on x. On each space , linear operations are defined and they comply with the following laws:
- (a)
, , where is a constant function;
- (b)
, ;
- (c)
if the function does not depend on , then for .
- (3°)
The operations form a commutative ring.
- (4°)
An operation d is introduced, and it satisfies the following rules:
- (a)
on ;
- (b)
;
- (c)
if , , , then , where , are tuples of length .
- (5°)
For every natural number k in the algebra , there exist a lot of tuples such that the operator , where , with a specially defined domain is surjective and has the inverse , which has the following property: for arbitrary compact sets there is a number such that, for an arbitrary function , the set is bounded in .
Let us consider the case
. We investigate the following equation:
in which
and
is a small complex parameter. The function
satisfying the initial condition
where
,
, is required to be found.
Definition 4. The invariant of Equation (2) is the function , which turns into a constant on the solution of this equation. Theorem 1. When the blocks of conditions I and II are satisfied, then Equation (2) has ε-regular invariants. Proof of Theorem 1. If
is an invariant of the Equation (
2), then, as it follows from Definition 4, we have
where
.
We seek a solution of Equation (
4) in the form of a series in powers of
:
for the coefficients of the equation above the following series of equations holds:
As a solution to the first equation of this series, we take an arbitrary function . To satisfy the condition () of block II, we assume that the domain of the surjective operator consists of functions from that vanish when , and the inverse operator is such that, for any compact sets , , there exists a number such that, for an arbitrary function set is limited in .
As a result, all equations of the series (
6), starting from the second, are uniquely solvable:
and this series converges in some neighborhood of the value
on the set
. Theorem 1 is proved. □
Remark 2. As it comes out from the form of series (7), we can consider for each fixed ε as the image of the linear operator given by the formulawhere I is the identity operator. Thus, . Theorem 2. forms a ε-regular family for homomorphisms of the algebra into the algebra .
Proof of Theorem 2. Let
U and
V be invariants of the Equation (
2). Obviously, then there exists a function
such that
, and therefore
. If in this equality we put
, then
, therefore
The equality (
8) is called the commutation relation.
Now, let
; then,
where
is a homomorphism. Theorem 2 is proved. □
For the concepts given below, we need a definition introduced by S.A. Lomov for the notion of the essentially singular manifold [
4].
Definition 5. Let , , , let it allow continuation to all , and let be some compact from X containing the point . The set is called an essentially singular variety generated by the point . Moreover, we say that it has the correct structure ifwhere is an increasing compact system. We introduce the concept of -pseudoregularity necessary for studying the analytic properties of a solution of .
Definition 6. The solution to the problems (2), (3) is called ε-pseudoregular if , in which ; the function is ε-regular for all where is some compact set containing the point , G is some unlimited set from . Theorem 3. If the essentially singular manifold is a bounded set in and the equationhas a unique solution of the form such that the function coincides with the contraction to the set of some function from , then problems (2), (3) have a ε-pseudoregular solution. Proof of Theorem 3. For the invariant represented by the Formula (
7), we compose the equality
which defines the solution to the problems (
2), (
3). We apply the function
to its left-hand and right-hand sides and, using the condition (
) of block I, we obtain the following equality:
where
is some
-regular function.
Let the small parameter
in Equation (
2) be such that the following expression holds:
for some natural number
m (depending on
). In accordance with the theorem conditions and the condition (
) of block I, Equation (
10) is solvable in some neighborhood
of each point
and has a solution
that is
-regular in a neighborhood of
, where
and is determined by this neighborhood. From the cover
of the compact set
, we choose the finite subcover
. Then,
will be a
-regular function in the smallest neighborhood of the point
defined by a finite subcover; the function
will give a
-pseudoregular solution to the problem (
2), (
3) on the part
such that the set
. The theorem is proved. □
4. Concrete Implementations of the Theory
In this section of the article, we assume that , , , . We shall use the following denotations , , a polycircle of .
Let be the algebra of holomorphic functions in the circle of the variable z; let be the algebra of holomorphic functions in the bicircle of the variables , …; let be the algebra of holomorphic functions of the variables in the polycircle . It is clear that, if , , …, , then all the conditions of block I and the conditions ()— () of block II are satisfied. In the concepts given below, we show that the condition () also holds under fairly general assumptions.
Thus, we investigate the Cauchy problem for
:
where
,
for
.
From the nonlinear system (
16), we come to the linear equation of its integrals (invariants):
Here, is the linear partial differential operator of the first order in partial derivatives: , where is the system of independent integrals.
First of all, we present an integral method for solving inhomogeneous linear differential equations of the first order with partial derivatives [
5].
Let
be a holomorphically smooth surface in
and we need to solve the initial problem
Let us suppose that the surface
is given by the coordinates
and, namely,
,
, where
are functions holomorphic in some region
. Next, we compose the equation system for the characteristic equation
in which
is an independent variable, and
z acts as a parameter. Let
be a solution to the system (
19) with the initial condition
where
.
The existence and uniqueness theorem guarantees the unique solvability of the system
relative to
and
s:
,
. We denote the operator of replacing variables
by the variable
by
and the backward replacement operator is denoted by
:
Then, as you know, if the phase trajectories in the system of characteristics are transversal (not tangent) to the surface
, then a solution to the Cauchy problem (
17) exists, is unique, and is expressed by the following formula:
We return to Equation (
18). We have
and, as this takes place
As a solution to the first equation of this series, we take the vector function
,
for
. The solution to the second equation of the series (
22) is the vector function
where
are functionally independent solutions of the equation
and such that
,
. We find solutions to other equations using Formula (
20), assuming that
:
Next, to each natural
, we associate
concentric circles
where
and
.
These circles are situated at the same distance from each other:
We use the equalities (
23) with the Cauchy integral formula:
We represent
in the following form:
Let
be the norm in
; then, for all
and all
from some subregion
of the polycircle
, the following inequality takes place:
where
As we have
then
and from that the convergence of the series (
21) on any compact set from the set
follows.
Thus, it is proved that the components of the vector
form an independent system of integrals (invariants) and are holomorphic (
-regular) at the point
. It is also clear that there is a statement similar to Theorem 5 on the existence of a pseudoholomorphic (
-pseudoregular) solution of the Cauchy problem (
16). Without loss of generality, we assume that
.
Theorem 6. Let the entire functions and the functions which are holomorphic in the circle be such that the essentially singular manifolds created by the functions described above where is some segment of the real axis, the left end of which coincides with the origin and belongs to the circle , are sets bounded in ; and the system of equationsin which are independent solutions of the equation , has a solution of the formeach component of it is holomorphic on the set . Then, the solution of the initial problem (16) is pseudoholomorphic at the point (ε-pseudoregular). It should be noted [
6] that this solution can be continued in a pseudoholomorphic way for a fixed
from some segment
(see the end of the proof of Theorem 5) by segment
.