2.1. Direct Extensions in
Let us first give a brief account of the invariant subspace method as presented in [
1]. Consider the general evolution equation:
where
F is a
k-th-order ordinary differential operator with respect to the variable
x and
is a given sufficiently smooth function of the indicated variables. Let
be a finite set of
linearly independent functions, and
denotes their linear span
. The subspace
is said to be invariant under the given operator
F, if
, and then operator
F is said to preserve or admit
, which means:
for any
, where
are the expansion coefficients of
in the basis
. It follows that if the linear subspace
is invariant with respect to
F, then Equation (
2) has solutions of the form:
where
satisfy the
n-dimensional dynamical system:
Moreover, assume that the invariant subspace
is defined as the space of solutions of the linear
n-th-order ODE:
If the operator
admits the invariant subspace
, then the invariant condition with respect to
F takes the form:
where
denotes the equation
and its differential consequences with respect to
x. The invariant condition leads to the following theorem on the maximal dimension of an invariant subspace preserved by the operator
F.
Theorem 1. [1] If a linear subspace determined by the space of solutions of linear Equation (3) is invariant under a nonlinear differential operator F of order k, then: It is inferred from Equation (
4) and the invariant criteria for conditional Lie–Bäcklund symmetry [
32,
33] that Equation (
2) admits the conditional Lie–Bäcklund symmetry:
To look for the exact solutions of the form:
of the two-dimensional nonlinear evolution equations:
we now introduce the linear subspace:
as an extension to
. Assume that
is a
k-th-order differential operator with respect to the variables
x and
y, and
is a finite set of
linearly independent functions of variable
y. It is easy to see that the space
is also a set of linearly independent functions. Let
denote the linear span of the set
, i.e.,
. Similarly, the space
is defined as the space of solutions of the linear
m-th-order ODE:
If
, then there exists a vector
, such that:
We rewrite
u as:
which means that:
On the other hand, if the function
satisfies the condition (
9), then
u has the form (
8). Indeed,
means that there exists a vector function
, such that:
while
means that:
Since
are linearly independent, the above equation leads to:
Hence, there exists a set of vectors
, such that:
As above, we are able to obtain the invariance condition of the subspace
with respect to
F, i.e.,
, which takes the form:
where
denotes
,
, and their differential consequences with respect to
x and
y. If
admits the invariant subspace
, then Equation (
6) has solutions (
5) and can be reduced to an
-dimensional dynamic system.
We next consider a special case of the function (
5). If
, then
in (
3) and
in (
7). Without loss of generality, we assume
and
. Note that the function of the form:
is a special case of (
5), which is a separable function with respect to spacial variables
x and
y. We denote:
which is a linear span of the set
. Clearly, if
, then:
On the other hand, if
, then the function
u has the form:
From
(notice that
), we obtain:
which means that there exists a vector
, such that:
Similarly,
leads to:
where
are functions of
t. We denote
. Hence,
if and only if
u satisfies the condition (
12). Then, we can obtain the invariance condition of the subspace
with respect to
F, i.e.,
, which takes the form:
where
denotes the set
, and their differential consequences with respect to
x and
y. In this case, Equation (
6) has the solution of the form (
11) and can be reduced to an
-dimensional dynamic system.
Assume that the
k-th-order differential operator
, including the term
, admits the invariant subspace
(or
), and note that the operator
can also be regarded as a differential operator only with respect to
x; the first identity in the condition (
10) (or (
13)) leads to the estimate
. The same estimate is also true for
m.
Remark 1. It is noted that the and demonstrate two special forms of invariant subspaces of the operator . The general form can be introduced as below, which will be used in the following sections.
Let be a finite set of linearly independent functions, and denote their linear span . The subspace is said to be invariant under the given operator , if , and then, operator is said to preserve or admit .