We seek an asymptotic solution of the problem (2.1)-(2.2) of the form

where
,
,
; and for
,
(
) are coefficients of regular terms;
(
) are coefficients of boundary layer terms at
;
(
) are coefficients of boundary layer terms at
; and
(
) are left and right coefficients of internal transition terms at
. Meanwhile, similar definitions are for
,
,
and
.

The position of a transition time

is unknown in advance. It needs being determined during the construction of an asymptotic solution. Suppose that

has also an asymptotic expression of the form

where
(
) are temporarily unknown at the moment and will be determined later.

where
(
) are all constants, independent of
, and
takes value between
and
. For example,
.

Then, we will determine the asymptotic solution (3.1) step by step using "a smooth connection method'' based on the boundary function method [

13] or [

5]. The smooth connection condition (2.11) can be written as

where
,
;
and
are all the known functions depending only on
.

Substituting (3.1) into (2.1)-(2.2) and equating separately the terms depending on
,
,
and
by the boundary function method, we can obtain the equations to determine
;
,
and
, respectively. The equations to determine the zero-order coefficients of regular terms
(
) are given by

It is clear to see that (3.5) coincides with the reduced system (2.11). Therefore, by

, (3.5) has the solution

The equations to determine

,

(

) are given by

Here the superscript
is omitted for the variables
and
in (3.7) for simplicity in notation. To understand
and
, we agree that they take
when
; while they take
when
. The terms
(
;
) are expressed in terms of
and
(
;
). Also
are known functions that take value at
, where
when
and
when
.

Since (3.7) is an algebraic linear system, the solution
(
;
) is uniquely solvable by
.

Next, we give the equations and their conditions for determining the zero-order coefficient of an internal transition layer

as follows:

We rewrite (3.8) in a different form by making the change of variables

Then, (3.8) is further written in these new variables as

From
, it yields that the equilibrium
of the autonomous system (3.10) is a hyperbolic saddle point. Therefore, there exists an unstable one-dimensional manifold
. For the existence of a solution of (3.10) satisfying (3.11), we need the following assumption.

Suppose that the hyperplane
intersects the manifold
in the phase space
, where
is a parameter.

Then,

(

) are known values after

being solved by

. We can get the equations and the corresponding boundary conditions to determine

as follows:

Introducing a similar transformation as doing for (3.8), we can get

To ensure that the existence of a solution of (3.14)-(3.15), we need the following assumption.

Suppose that the hypercurve
intersects the manifold
in the phase space
, where
is a parameter.

Here it should be emphasized that under the conditions of
and
, the solutions
(
) not only exist but also decay exponentially [13], or [5].

If the parameter
is determined,
(
) are completely known. To determine
, it is closely related to the existence of a heteroclinic orbit connecting
and
in the phase space.

By the given initial values (3.13) or (3.15), we have already obtained

If we show

, the smooth connection condition (2.11) for the zero-order is satisfied. By

and

, we have

Since

(

) and

) only depend on

, while

only depends on

, the necessary condition for existence of a heteroclinic orbit connecting

and

at

is given by

However (3.18) or (3.19) is the one to determine
. Then, by [
], there exists an
from (3.18) or (3.19). We can see that the process of determining
is the one of a smooth connection. Therefore, all the zero-order terms
have now been completely determined by the smooth connection for the zero-order coefficients of the asymptotic solution.

For the high-order terms

(

), we have the equations and their boundary conditions as follows:

where
represent known functions that take value at
;
are the known functions that only depend on those asymptotic terms whose subscript is
; and
and
are all the known functions. Since (3.20) are all linear boundary value problems, it is not difficult to prove the existence of solution and the exponential decaying of solution without imposing any extra condition.

As for boundary functions

and

, it is easy to obtain their constructions by using the normal boundary function method. So we would not discuss the details on them here [

13] or [

5]. However, it is worth mentioning that the coefficients

(

) in (2.3) will be determined by an equation as follows:

Then,
can be solved from (3.21) by
. Then, we have so far constructed the asymptotic expansion of a solution with an internal transition layer for the problem (2.1)-(2.2) and the asymptotic expansion of an internal transition time
.