#### 2.1. Direct Extensions in ${\mathbb{R}}^{2}$

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

$F(\xb7)$ is a given sufficiently smooth function of the indicated variables. Let

$\{{f}_{i}(x),\phantom{\rule{0.277778em}{0ex}}i=1,\cdots ,n\}$ be a finite set of

$n\u2a7e1$ linearly independent functions, and

${W}_{n}^{x}$ denotes their linear span

${W}_{n}^{x}=\mathcal{L}\{{f}_{1}(x),\cdots ,{f}_{n}(x)\}$. The subspace

${W}_{n}^{x}$ is said to be invariant under the given operator

F, if

$F[{W}_{n}^{x}]\subseteq {W}_{n}^{x}$, and then operator

F is said to preserve or admit

${W}_{n}^{x}$, which means:

for any

$C(t)=({C}_{1}(t),\cdots ,{C}_{n}(t))\in {\mathbb{R}}^{n}$, where

${\mathsf{\Psi}}_{i}$ are the expansion coefficients of

$F[u]\in {W}_{n}^{x}$ in the basis

$\left\{{f}_{i}\right\}$. It follows that if the linear subspace

${W}_{n}^{x}$ is invariant with respect to

F, then Equation (

2) has solutions of the form:

where

${C}_{i}(t)$ satisfy the

n-dimensional dynamical system:

Moreover, assume that the invariant subspace

${W}_{n}^{x}$ is defined as the space of solutions of the linear

n-th-order ODE:

If the operator

$F[u]$ admits the invariant subspace

${W}_{n}^{x}$, then the invariant condition with respect to

F takes the form:

where

$[H]$ denotes the equation

${L}_{x}[u]=0$ 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 ${W}_{n}^{x}$ 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

${W}_{n}^{x}$. Assume that

$F[u]=F(x,y,u,{u}_{x},{u}_{y},{u}_{xx},{u}_{xy},{u}_{yy},\cdots ,{u}^{(k)})$ is a

k-th-order differential operator with respect to the variables

x and

y, and

$\{{g}_{j}(y),\phantom{\rule{0.277778em}{0ex}}j=1,\cdots ,m\}$ is a finite set of

$m\u2a7e1$ linearly independent functions of variable

y. It is easy to see that the space

$\{{f}_{i}(x){g}_{j}(y),\phantom{\rule{3.33333pt}{0ex}}i=1,\cdots ,n,\phantom{\rule{0.277778em}{0ex}}j=1,\cdots ,m\}$ is also a set of linearly independent functions. Let

${W}_{m}^{y}$ denote the linear span of the set

$\{{g}_{j}(y),\phantom{\rule{0.277778em}{0ex}}j=1,\cdots ,m\}$, i.e.,

${W}_{m}^{y}=\mathcal{L}\{{g}_{1}(y),\cdots ,{g}_{m}(y)\}$. Similarly, the space

${W}_{m}^{y}$ is defined as the space of solutions of the linear

m-th-order ODE:

If

$u\in {W}_{nm}^{xy}$, then there exists a vector

$({C}_{11}(t),\cdots ,{C}_{1m}(t),\cdots ,{C}_{n1}(t),\cdots ,{C}_{nm}(t))\in {\mathbb{R}}^{nm}$, such that:

We rewrite

u as:

which means that:

On the other hand, if the function

$u=u(t,x,y)$ satisfies the condition (

9), then

u has the form (

8). Indeed,

${L}_{x}[u]=0$ means that there exists a vector function

$({C}_{1}(t,y),\cdots ,{C}_{n}(t,y))$, such that:

while

${L}_{y}[u]=0$ means that:

Since

${f}_{i}(x)\phantom{\rule{0.277778em}{0ex}}(i=1,\cdots ,n)$ are linearly independent, the above equation leads to:

Hence, there exists a set of vectors

$({C}_{i1}(t),\cdots ,{C}_{im}(t))\in {\mathbb{R}}^{m}$, such that:

As above, we are able to obtain the invariance condition of the subspace

${W}_{nm}^{xy}$ with respect to

F, i.e.,

$F[{W}_{nm}^{xy}]\subseteq {W}_{nm}^{xy}$, which takes the form:

where

$[{H}_{x}]\cap [{H}_{y}]$ denotes

${L}_{x}[u]=0$,

${L}_{y}[u]=0$, and their differential consequences with respect to

x and

y. If

$F[u]$ admits the invariant subspace

${W}_{nm}^{xy}$, then Equation (

6) has solutions (

5) and can be reduced to an

$nm$-dimensional dynamic system.

We next consider a special case of the function (

5). If

$1\in {W}_{n}^{x}\cap {W}_{m}^{y}$, then

${a}_{0}(x)=0$ in (

3) and

${b}_{0}(y)=0$ in (

7). Without loss of generality, we assume

${f}_{1}(x)=1$ and

${g}_{1}(y)=1$. 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

$\{1,{f}_{i}(x),{g}_{j}(y),i=2,\cdots ,n,j=2,\cdots ,m\}$. Clearly, if

$u\in {W}_{n+m-1}^{xy}$, then:

On the other hand, if

${u}_{xy}=0$, then the function

u has the form:

From

${L}_{x}[u]=0$ (notice that

${a}_{0}(x)=0$), we obtain:

which means that there exists a vector

$({A}_{1}(t),{C}_{2}(t),\cdots ,{C}_{n}(t))$, such that:

Similarly,

${L}_{y}[u]=0$ leads to:

where

${B}_{j}(j=1,\cdots ,m)$ are functions of

t. We denote

${C}_{1}={A}_{1}+{B}_{1}$. Hence,

$u\in {W}_{n+m-1}^{xy}$ if and only if

u satisfies the condition (

12). Then, we can obtain the invariance condition of the subspace

${W}_{n+m-1}^{xy}$ with respect to

F, i.e.,

$F[{W}_{n+m-1}^{xy}]\subseteq {W}_{n+m-1}^{xy}$, which takes the form:

where

$[H]$ denotes the set

$\{{L}_{x}[u]=0\}\cap \{{L}_{y}[u]=0\}\cap \{{u}_{xy}=0\}$, 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

$(n+m-1)$-dimensional dynamic system.

Assume that the

k-th-order differential operator

$F[u]$, including the term

${\partial}^{k}u/\partial {x}^{k}$, admits the invariant subspace

${W}_{nm}^{xy}$ (or

${W}_{n+m-1}^{xy}$), and note that the operator

$F[u]$ 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

$n\u2a7d2k+1$. The same estimate is also true for

m.

**Remark** **1.** It is noted that the ${W}_{mn}^{xy}$ and ${W}_{n+m-1}^{xy}$ demonstrate two special forms of invariant subspaces of the operator $F[u]$. The general form can be introduced as below, which will be used in the following sections.

Let $\{{f}_{i}(x,y),\phantom{\rule{0.277778em}{0ex}}i=1,\cdots ,n\}$ be a finite set of $n\u2a7e1$ linearly independent functions, and ${W}_{n}$ denote their linear span ${W}_{n}=\mathcal{L}\{{f}_{1}(x,y),\cdots ,{f}_{n}(x,y)\}$. The subspace ${W}_{n}$ is said to be invariant under the given operator $F[u]$, if $F[{W}_{n}]\subseteq {W}_{n}$, and then, operator $F[u]$ is said to preserve or admit ${W}_{n}$.