† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).

The functionally generalized variable separation of the generalized nonlinear diffusion equations *u** _{t}* =

*A*(

*u*,

*u*

*)*

_{x}*u*

*+*

_{xx}*B*(

*u*,

*u*

*) is studied by using the conditional Lie–Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie–Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.*

_{x}It is well known that symmetry-related methods play important roles in studying symmetry reductions and constructing exact solutions to nonlinear partial differential equations (NPDEs). Up to now, a number of effective approaches have been presented. For instance, the Lie point symmetry method,^{[1–4]} also known as the classical symmetry method due to Sophus Lie in 1881, is regarded as the most popular technique. The nonclassical symmetry method,^{[5–12]} as an extension of Lie’s classical method, which was introduced by Bluman and Cole, can be applied to find more types of exact solutions to some NPDEs. In comparison with the symmetry method, the direct method^{[13–17]} proposed by Clarkson and Kruskal can be used to realize reductions of NPDEs without any reference to the Lie group theory. Here we especially emphasize the conditional Lie–Bäcklund symmetry (CLBS) method introduced by Fokas and Liu^{[18]} and Zhdanov,^{[19]} which is a natural extension of the nonclassical symmetry method. For that matter, the algorithm of the CLBS is similar to that of the nonclassical symmetry. From the point of view of the ansatz form of the solution, we now focus on the following two kinds of ansatzs. One is the functional separable solution (FSS)^{[20,21]}

^{[22,23]}

In this paper, we develop the CLBS method to study the functionally generalized separable solutions (FGSSs)

*A*(

*u*,

*u*) and

_{x}*B*(

*u*,

*u*

*) are smooth functions of the indicated variables to be settled. Obviously, the FGSS is an extension of the FSS. The FSSs to Eq. (*

_{x}^{[24]}Furthermore, seeking for the approximate FSSs of Eq. (

^{[25]}

The layout of this paper is as follows. In Section 2, we review some basic definitions and fundamental theorems on the CLBS method and the invariant subspace (IS) method. In Section 3, the classification of Eq. (

For convenience, we review some basic facts on the CLBS and the IS methods for the nonlinear evolution equations.^{[18–23]}

Consider the *k*-th order nonlinear evolution equation

*η*=

*η*(

*t*,

*x*,

*u*,

*u*

_{1},

*u*

_{2},…,

*u*

*) be its characteristic, where the total derivative operators are defined as*

_{n}*L*] is the set of all the differential consequences of Eq. (

*M*] denotes the set of all the differential consequences of equation

*η*= 0 with respect to

*x*, that is

*j*= 0,1,2,….

^{[18,19]} Equation (*S*(*t*,*x*,*u*,*η*) so that

*E*,

*η*] =

*E′η*−

*η′E*, the prime denotes the Fréchet derivative, and

*S*is an analytic function of

*t*,

*x*,

*u*,

*u*

_{x},…and

*η*,

*D*

_{x}*η*,

Following this proposition, equation (*η* if

*g*(

*u*)]

*=*

_{ix}*∂*(

^{i}*g*(

*u*))/

*∂x*.

^{i}If equation (

*u*, and

*u*=

*f*(

*v*) denotes the inverse function of

*v*=

*g*(

*u*). To proceed, the following proposition is necessary.

By Proposition 2, we conclude that equation (

*E*, where

*W*

_{n}is the solution space of

*η*=

*v*+

_{nx}*a*

_{1}(

*x*)

*v*

_{(n−1)x}+ … +

*a*(

_{n}*x*)

*v*= 0.

Thus it is easy to know that *E*[*W** _{n}*] ⊂

*W*

*, that is*

_{n}*C*(

_{i}*t*)} are determined by the first order dynamical system

To estimate the dimension of the IS admitted by the operator *E*, the following theorem is helpful.

^{[26]} If linear space *W*_{n} is invariant with respect to the nonlinear operator *E* of order *j*, then

In terms of this theorem, we investigate CLBSs (*n* ≤ 5 for Eq. (

We first consider the case of *n* = 2.

By proposition 2, equation (

*P*=

*P*(

*v*,

*v*

*),*

_{x}*Q*=

*Q*(

*v*,

*v*

*), and*

_{x}*a*=

_{i}*a*(

_{i}*x*) (

*i*= 1,2). The primes denote the derivatives with respect to

*x*, and the subscripts denote the partial derivatives with respect to the indicated variables.

Because not all solutions of Eq. (

In Table

*r*,

*s*,

*m*, and

*n*are arbitrary constants in their sets of definition when not specified.

In the same way, we can obtain the determining systems for *n* = 3,4,5. Since the determining systems are quite complicated, they are omitted here. Finally, a complete classification to CLBSs (*n* = 3,4,5 of Eq. (

In Table

*r*

_{i},

*c*

_{i}(

*i*∈ N) are arbitrary constants in their sets of definition when not specified.

In this section, we construct GSSs to the equations obtained in the last section. These solutions are determined explicitly with some finite-dimensional dynamical systems. We take several examples to illustrate the calculation procedure.

By the transformation *v* = *g*(*u*), we can obtain the corresponding FGSS of the equation

*ϕ*

_{1}(

*t*) and

*ϕ*

_{2}(

*t*) satisfy the following dynamical system:

By the appropriate transformation *v* = *g*(*u*), we can obtain the FGSS of the equation

*l*(

*u*,

*u*) =

_{x}*g′*(

*u*)

*u*+

_{x}*rg*(

*u*).

By the appropriate transformation *v* = *g*(*u*), we can obtain the FGSSs of the equation

By the appropriate transformation *v* = *g*(*u*), we can respectively obtain the FGSSs of the equation

By using the CLBS and IS methods, we have studied the problem of FGSSs for the generalized nonlinear diffusion equations. Those equations admitting the corresponding CLBSs with ISs have been obtained and displayed in tables. Finally, we have derived the FGSSs of some resulting equations. In general, these solutions cannot be obtained by other relevant approaches.

There are still two interesting questions to be answered in the future: (i) How to develop the CLBS method with IS to investigate the FGSSs of other types of evolution equations, such as higher-dimensional equations? (ii) How to solve the approximate FGSS of NPDEs with perturbation terms? We expect that certain other types of NPDEs could be investigated by means of the CLBS and the IS approaches, and some new results would be obtained sooner or later.

**Reference**