Titov–Galaktionov Method

S.4.6. Titov–Galaktionov Method

S.4.6-1. Description of the method. Linear subspaces invariant under a nonlinear operator. Consider the nonlinear evolution equation

∂w

F [w],

F [w] is a differential operator of the form

∂w

F [w] ≡ F w,

∂x

∂x n

k = ϕ 1 ( x), . . . , ϕ k ( x)

(82) formed by linear combinations of linearly independent functions ✢ ✢ ✢✢ ϕ 1 ( x), . . . , ϕ k ( x) is called invariant

under the operator

F if F [ k ] ⊆ k . This means that there exist functions f 1 , ...,f k such that

F C i ϕ i ( x) =

f i ( C 1 , ...,C k ) ϕ i ( x)

for arbitrary constants C 1 , ...,C k .

F . Then equation (80) possesses generalized separable solutions of the form

Let the linear subspace (82) be invariant under the operator

w(x, t) =

Here, the functions ψ 1 ( t), . . . , ψ k ( t) are described by the autonomous system of ordinary differential equations

(85) where the prime denotes a derivative with respect to t.

ψ ′ i = f i ( ψ 1 , ...,ψ k ),

i = 1, . . . , k,

The following example illustrates the scheme for constructing generalized separable solutions. Example 15. Consider the nonlinear second-order parabolic equation

Obviously, the nonlinear differential operator

F [w] = aw xx +( w x ) 2 +

1, cos( x √ k) kw

bw + c for k > 0 has a two-dimensional

invariant subspace

. Indeed, for arbitrary C 1 and C 2 we have

√ cos( x k) = k(C 2 1 + C 2 2 )+ bC 1 + c+C 2 (2 kC 1 − ak + b) cos(x k ).

Therefore, there is a generalized separable solution of the form

w(x, t) = ψ 1 ( t) + ψ 2 ( t) cos(x √ k ),

where the functions ψ 1 ( t) and ψ 2 ( t) are determined by the autonomous system of ordinary differential equations

Remark 1. For k > 0, the nonlinear differential operator F [w] has a three-dimensional invariant subspace 3 =

1, sin( x √ k ), cos(x √ k) .

Remark 2. For 3 =

1, sinh( x √

k < 0, the nonlinear differential operator F [w] has a three-dimensional invariant subspace

k ), cosh(x √ k) .

Remark 3. A more general equation (86), with a = a(t), b = b(t), and c = c(t) being arbitrary functions, and k = const < 0, also admits a generalized separable solution of the form (87), where the functions ψ 1 ( t) and ψ 2 ( t) are determined by the system of ordinary differential equations (88).

S.4.6-2. Some generalizations. Likewise, one can consider a more general equation of the form

L 1 [ w] = L 2 [ U ], U = F [w],

where L 1 [ w] and L 2 [ U ] are linear differential operators with respect to t,

L 1 [ w] ≡

a i ( t) i ,

L 2 [ U]≡

b j ( t)

∂t

∂t j

i=0

j=0 j=0

F , i.e., for arbitrary constants

C 1 , ...,C k the following relation holds:

F C i ϕ i ( x) =

f i ( t, C 1 , ...,C k ) ϕ i ( x).

Then equation (89) possesses generalized separable solutions of the form (84), where the functions ψ 1 ( t), . . . , ψ k ( t) are described by the system of ordinary differential equations

L 1 ψ i ( t) = L 2 f i ( t, ψ 1 , ...,ψ k ) ,

i = 1, . . . , k. (93)

Example 16. Consider the equation

which, in the special case of a 2 ( t) = k 2 and a 1 ( t) = k 1 /t, is used for describing transonic gas flows (where t plays the role of a spatial variable). Equation (94) is a special case of equation (89), where L 1 [ w] = a 2 ( t)w tt + a 1 ( t)w t , L 2 [ U ] = U , and F [w] = w x w xx . It can be shown that the nonlinear differential operator ✣✣

F [w] = w x w xx admits the three-dimensional invariant subspace 3 = 1, x 3 /2 , x 3 . Therefore, equation (94) possesses generalized separable solutions of the form

w(x, t) = ψ 3 1 (

t) + ψ 3 2 ( t)x /2 + ψ 3 ( t)x ,

where the functions ψ 1 ( t), ψ 2 ( t), and ψ 3 ( t) are described by the system of ordinary differential equations

Remark. The operator ✣✣ F [w] also has a four-dimensional invariant subspace 4 = 1, x, x 2 , x 3 , which corresponds

to a generalized separable solution of the form

1 2 3 2 + 4 w(x, t) = ψ 3 ( t) + ψ ( t)x + ψ ( t)x ψ ( t)x . See also Example 17 with a 0 ( t) = 0, k = 1, and n = 2.

Example 17. Consider the more general nth-order equation

2 = 1, ϕ(x) , where the function ϕ(x) is determined by the autonomous ordinary differential equation (ϕ ′ x ) k ϕ ( x n) = ϕ. Therefore, equation (95) possesses generalized separable solutions of the form

The nonlinear operator

F [w] = (w x ) k w ( x n) has a two-dimensional invariant subspace

w(x, t) = ψ 1 ( t) + ψ 2 ( t)ϕ(x),

where the functions ψ 1 ( t) and ψ 2 ( t) are described by two independent ordinary differential equations

a 2 ( t)ψ ′′ 1 + a 1 ( t)ψ ′ 1 + a 0 ( t)ψ 1 = 0, a 2 ( t)ψ ′′ 2 + a 1 ( t)ψ ′ 2 + a 0 ( t)ψ 2 = ψ 2 k+1 .

Many other examples of this type, as well as some modifications and generalizations of the method described here, can be found in the literature cited below. The basic difficulty of using the Titov–Galaktionov method for the construction of exact solutions of specific equations consists in finding linear subspaces which are invariant under a given nonlinear operator. Moreover, the original equation may be of a different type than the equations considered here (it is not always possible to single out a suitable nonlinear operator ✤✂✥

F [w]).

References for Subsection S.4.6: S. S. Titov (1988), V. A. Galaktionov and S. A. Posashkov (1994), V. A. Galaktionov (1995), V. A. Galaktionov, S. A. Posashkov, and S. R. Svirshchevskii (1995), S. R. Svirshchevskii (1995, 1996).