15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 689 2 ◦ . At the second stage, we successively substitute the Φ i (X)andΨ j (Y ) of (15.5.1.4) into all solutions (15.5.4.1) to obtain systems of ordinary differential equations* for the unknown functions ϕ p (x)andψ q (y). Solving these systems, we get generalized separable solutions of the form (15.5.1.1). Remark 1. It is important that, for fixed k, the bilinear functional equation (15.5.1.3) used in the splitting method is the same for different classes of original nonlinear mathematical physics equations. Remark 2. For fixed m, solution (15.5.4.1) contains m(k – m) arbitrary constants C i,j .Givenk,the solutions having the maximum number of arbitrary constants are defined by Solution number Number of arbitrary constants Conditions on k m = 1 2 k 1 4 k 2 if k is even, m = 1 2 (k 1) 1 4 (k 2 – 1)ifk is odd. It is these solutions of the bilinear functional equation that most frequently result in nontrivial generalized separable solution in nonlinear partial differential equations. Remark 3. The bilinear functional equation (15.5.1.3) and its solutions (15.5.4.1) play an important role in the method of functional separation of variables. For visualization, the main stages of constructing generalized separable solutions by the splitting method are displayed in Fig. 15.2. 15.5.4-2. Solutions of simple functional equations and their application. Below we give solutions to two simple bilinear functional equations of the form (15.5.1.3) that will be used subsequently for solving specific nonlinear partial differential equations. 1 ◦ . The functional equation Φ 1 Ψ 1 + Φ 2 Ψ 2 + Φ 3 Ψ 3 = 0,(15.5.4.2) where Φ i are all functions of thesame argument and Ψ i are all functions of anotherargument, has two solutions: Φ 1 = A 1 Φ 3 , Φ 2 = A 2 Φ 3 , Ψ 3 =–A 1 Ψ 1 – A 2 Ψ 2 ; Ψ 1 = A 1 Ψ 3 , Ψ 2 = A 2 Ψ 3 , Φ 3 =–A 1 Φ 1 – A 2 Φ 2 . (15.5.4.3) The arbitrary constants are renamed as follows: A 1 = C 1,1 and A 2 = C 2,1 in the first solution, and A 1 =–1/C 1,2 and A 2 = C 1,1 /C 1,2 in the second solution. The functions on the right-hand sides of the formulas in (15.5.4.3) are assumed to be arbitrary. 2 ◦ . The functional equation Φ 1 Ψ 1 + Φ 2 Ψ 2 + Φ 3 Ψ 3 + Φ 4 Ψ 4 = 0,(15.5.4.4) where Φ i are all functions of thesame argument and Ψ i are all functions of anotherargument, has a solution Φ 1 = A 1 Φ 3 + A 2 Φ 4 , Φ 2 = A 3 Φ 3 + A 4 Φ 4 , Ψ 3 =–A 1 Ψ 1 – A 3 Ψ 2 , Ψ 4 =–A 2 Ψ 1 – A 4 Ψ 2 (15.5.4.5) dependent on four arbitrary constants A 1 , , A 4 ; see solution (15.5.4.1) with k = 4, m = 2, C 1,1 = A 1 , C 1,2 = A 2 , C 2,1 = A 3 ,andC 2,2 = A 4 . The functions on the right-hand sides of the solutions in (15.5.4.3) are assumed to be arbitrary. * Such systems are usually overdetermined. 690 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Write out the functional differential equation Obtain: (i) functional equation, (ii) determining system of ODEs Solve the determining system of ordinary differential equations Write out generalized separable solution of original equation Search for generalized separable solutions Substitute into original equation Apply splitting procedure Treat functional equation (i) Figure 15.2. General scheme for constructing generalized separable solutions by the splitting method. Abbre- viation: ODE stands for ordinary differential equation. Equation (15.5.4.4) also has two other solutions: Φ 1 = A 1 Φ 4 , Φ 2 = A 2 Φ 4 , Φ 3 = A 3 Φ 4 , Ψ 4 =–A 1 Ψ 1 – A 2 Ψ 2 – A 3 Ψ 3 ; Ψ 1 = A 1 Ψ 4 , Ψ 2 = A 2 Ψ 4 , Ψ 3 = A 3 Ψ 4 , Φ 4 =–A 1 Φ 1 – A 2 Φ 2 – A 3 Φ 3 (15.5.4.6) involving three arbitrary constants. In the fi rst solution, A 1 = C 1,1 , A 2 = C 2,1 ,andA 3 = C 3,1 , and in the second solution, A 1 =–1/C 1,3 , A 2 = C 1,1 /C 1,3 ,andA 3 = C 1,2 /C 1,3 . Solutions (15.5.4.6) will sometimes be called degenerate, to emphasize the fact that they contain fewer arbitrary constants than solution (15.5.4.5). 3 ◦ . Solutions of the functional equation Φ 1 Ψ 1 + Φ 2 Ψ 2 + Φ 3 Ψ 3 + Φ 4 Ψ 4 + Φ 5 Ψ 5 = 0 (15.5.4.7) can be found by formulas (15.5.4.1) with k = 5. Below is a simple technique for fi nding solutions, which is quite useful in practice, based on equation (15.5.4.7) itself. Let us assume that Φ 1 , Φ 2 ,andΦ 3 are linear combinations of Φ 4 and Φ 5 : Φ 1 = A 1 Φ 4 + B 1 Φ 5 , Φ 2 = A 2 Φ 4 + B 2 Φ 5 , Φ 3 = A 3 Φ 4 + B 3 Φ 5 ,(15.5.4.8) 15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 691 where A n , B n are arbitrary constants. Let us substitute (15.5.4.8) into (15.5.4.7) and collect the terms proportional to Φ 4 and Φ 5 to obtain (A 1 Ψ 1 + A 2 Ψ 2 + A 3 Ψ 3 + Ψ 4 )Φ 4 +(B 1 Ψ 1 + B 2 Ψ 2 + B 3 Ψ 3 + Ψ 5 )Φ 5 = 0. Equating the expressions in parentheses to zero, we have Ψ 4 =–A 1 Ψ 1 – A 2 Ψ 2 – A 3 Ψ 3 , Ψ 5 =–B 1 Ψ 1 – B 2 Ψ 2 – B 3 Ψ 3 . (15.5.4.9) Formulas (15.5.4.8) and (15.5.4.9) give solutions to equation (15.5.4.7). Other solutions are found likewise. Example 1. Consider the nonlinear hyperbolic equation ∂ 2 w ∂t 2 = a ∂ ∂x w ∂w ∂x + f(t)w + g(t), (15.5.4.10) where f(t)andg(t) are arbitrary functions. We look for generalized separable solutions of the form w(x, t)=ϕ(x)ψ(t)+χ(t). (15.5.4.11) Substituting (15.5.4.11) into (15.5.4.10) and collecting terms yield aψ 2 (ϕϕ x ) x + aψχϕ xx +(fψ – ψ tt )ϕ + fχ + g – χ tt = 0. This equation can be represented as a functional equation (15.5.4.4) in which Φ 1 =(ϕϕ x ) x , Φ 2 = ϕ xx , Φ 3 = ϕ, Φ 4 = 1, Ψ 1 = aψ 2 , Ψ 2 = aψχ, Ψ 3 = fψ – ψ tt , Ψ 4 = fχ + g – χ tt . (15.5.4.12) On substituting (15.5.4.12) into (15.5.4.5), we obtain the following overdetermined system of ordinary differ- ential equations for the functions ϕ = ϕ(x), ψ = ψ(t), and χ = χ(t): (ϕϕ x ) x = A 1 ϕ + A 2 , ϕ xx = A 3 ϕ + A 4 , fψ – ψ tt =–A 1 aψ 2 – A 3 aψχ, fχ+ g – χ tt =–A 2 aψ 2 – A 4 aψχ. (15.5.4.13) The first two equations in (15.5.4.13) are compatible only if A 1 = 6B 2 , A 2 = B 2 1 – 4B 0 B 2 , A 3 = 0, A 4 = 2B 2 , (15.5.4.14) where B 0 , B 1 ,andB 2 are arbitrary constants, and the solution is given by ϕ(x)=B 2 x 2 + B 1 x + B 0 . (15.5.4.15) On substituting the expressions (15.5.4.14) into the last two equations in (15.5.4.13), we obtain the following system of equations for ψ(t)andχ(t): ψ tt = 6aB 2 ψ 2 + f(t)ψ, χ tt =[2aB 2 ψ + f(t)]χ + a(B 2 1 – 4B 0 B 2 )ψ 2 + g(t). (15.5.4.16) Relations (15.5.4.11), (15.5.4.15) and system (15.5.4.16) determine a generalized separable solution of equation (15.5.4.10). The first equation in (15.5.4.16) can be solved independently; it is linear if B 2 = 0 and is integrable by quadrature for f(t) = const. The second equation in (15.5.4.16) is linear in χ (for ψ known). Equation (15.5.4.10) does not have other solutions with the form (15.5.4.11) if f and g are arbitrary functions and ϕ 0, ψ 0,andχ 0. 692 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Remark. It can be shown that equation (15.5.4.10) has a more general solution with the form w(x, y)=ϕ 1 (x)ψ 1 (t)+ϕ 2 (x)ψ 2 (t)+ψ 3 (t), ϕ 1 (x)=x 2 , ϕ 2 (x)=x, (15.5.4.17) where the functions ψ i = ψ i (t) are determined by the ordinary differential equations ψ 1 = 6aψ 2 1 + f(t)ψ 1 , ψ 2 =[6aψ 1 + f(t)]ψ 2 , ψ 3 =[2aψ 1 + f(t)]ψ 3 + aψ 2 2 + g(t). (15.5.4.18) (The prime denotes a derivative with respect to t.) The second equation in (15.5.4.18) has a particular solution ψ 2 = ψ 1 . Hence, its general solution can be represented as (see Polyanin and Zaitsev, 2003) ψ 2 = C 1 ψ 1 + C 2 ψ 1 dt ψ 2 1 . The solution obtained in Example 1 corresponds to the special case C 2 = 0. Example 2. Consider the nonlinear equation ∂ 2 w ∂x∂t + ∂w ∂x 2 – w ∂ 2 w ∂x 2 = a ∂ 3 w ∂x 3 , (15.5.4.19) which arises in hydrodynamics (see Polyanin and Zaitsev, 2004). We look for exact solutions of the form w = ϕ(t)θ(x)+ψ(t). (15.5.4.20) Substituting (15.5.4.20) into (15.5.4.19) yields ϕ t θ x – ϕψθ xx + ϕ 2 (θ x ) 2 – θθ xx – aϕθ xxx = 0. This functional differential equation can be reduced to the functional equation (15.5.4.4) by setting Φ 1 = ϕ t , Φ 2 = ϕψ, Φ 3 = ϕ 2 , Φ 4 = aϕ, Ψ 1 = θ x , Ψ 2 =–θ xx , Ψ 3 =(θ x ) 2 – θθ xx , Ψ 4 =–θ xxx . (15.5.4.21) On substituting these expressions into (15.5.4.5), we obtain the system of ordinary differential equations ϕ t = A 1 ϕ 2 + A 2 aϕ, ϕψ = A 3 ϕ 2 + A 4 aϕ, (θ x ) 2 – θθ xx =–A 1 θ x + A 3 θ xx , θ xxx = A 2 θ x – A 4 θ xx . (15.5.4.22) It can be shown that the last two equations in (15.5.4.22) are compatible only if the function θ and its derivative are linearly dependent, θ x = B 1 θ + B 2 . (15.5.4.23) The six constants B 1 , B 2 , A 1 , A 2 , A 3 ,andA 4 must satisfy the three conditions B 1 (A 1 + B 2 – A 3 B 1 )=0, B 2 (A 1 + B 2 – A 3 B 1 )=0, B 2 1 + A 4 B 1 – A 2 = 0. (15.5.4.24) Integrating (15.5.4.23) yields θ = B 3 exp(B 1 x)– B 2 B 1 if B 1 ≠ 0, B 2 x + B 3 if B 1 = 0, (15.5.4.25) where B 3 is an arbitrary constant. The first two equations in (15.5.4.22) lead to the following expressions for ϕ and ψ: ϕ = ⎧ ⎪ ⎨ ⎪ ⎩ A 2 a C exp(–A 2 at)–A 1 if A 2 ≠ 0, – 1 A 1 t + C if A 2 = 0, ψ = A 3 ϕ + A 4 a, (15.5.4.26) where C is an arbitrary constant. 15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 693 Formulas (15.5.4.25), (15.5.4.26) and relations (15.5.4.24) allow us to find the following solutions of equation (15.5.4.19) with the form (15.5.4.20): w = x + C 1 t + C 2 + C 3 if A 2 = B 1 = 0, B 2 =–A 1 ; w = C 1 e –λx + 1 λt + C 2 + aλ if A 2 = 0, B 1 =–A 4 , B 2 =–A 1 – A 3 A 4 ; w = C 1 e –λ(x+aβt) + a(λ + β)ifA 1 = A 3 = B 2 = 0, A 2 = B 2 1 + A 4 B 1 ; w = aβ + C 1 e –λx 1 + C 2 e –aλβt + a(λ – β)ifA 1 = A 3 B 1 – B 2 , A 2 = B 2 1 + A 4 B 1 , where C 1 , C 2 , C 3 , β,andλ are arbitrary constants (these can be expressed in terms of the A k and B k ). The analysis of the second solution (15.5.4.6) of the functional equation (15.5.4.4) in view of (15.5.4.21) leads to the following two more general solutions of the differential equation (15.5.4.19): w = x t + C 1 + ψ(t), w = ϕ(t)e –λx – ϕ t (t) λϕ(t) + aλ, where ϕ(t)andψ(t) are arbitrary functions, and C 1 and λ are arbitrary constants. 15.5.5. Titov–Galaktionov Method 15.5.5-1. Method description. Linear subspaces invariant under a nonlinear operator. Consider the nonlinear evolution equation ∂w ∂t = F [w], (15.5.5.1) where F [w] is a nonlinear differential operator with respect to x, F [w] ≡ F x, w, ∂w ∂x , , ∂ m w ∂x m .(15.5.5.2) Definition .Afinite-dimensional linear subspace n = ϕ 1 (x), , ϕ n (x) (15.5.5.3) formed by linear combinations of linearly independent functions ϕ 1 (x), , ϕ n (x) is called invariant under the operator F if F [ n ] ⊆ n . This means that there exist functions f 1 , , f n such that F n i=1 C i ϕ i (x) = n i=1 f i (C 1 , , C n )ϕ i (x)(15.5.5.4) for arbitrary constants C 1 , , C n . Let the linear subspace (15.5.5.3) be invariant under the operator F. Then equation (15.5.5.1) possesses generalized separable solutions of the form w(x, t)= n i=1 ψ i (t)ϕ i (x). (15.5.5.5) Here, the functions ψ 1 (t), , ψ n (t) are described by the autonomous system of ordinary differential equations ψ i = f i (ψ 1 , , ψ n ), i = 1, , n,(15.5.5.6) where the prime denotes a derivative with respect to t. The following example illustrates the scheme for constructing generalized separable solutions. 694 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Example 1. Consider the nonlinear second-order parabolic equation ∂w ∂t = a ∂ 2 w ∂x 2 + ∂w ∂x 2 + kw 2 + bw + c.(15.5.5.7) Obviously, the nonlinear differential operator F [w]=aw xx +(w x ) 2 + kw 2 + bw + c (15.5.5.8) for k > 0 has a two-dimensional invariant subspace 2 = 1,cos(x √ k ) . Indeed, for arbitrary C 1 and C 2 we have F C 1 + C 2 cos(x √ k ) = k(C 2 1 + C 2 2 )+bC 1 + c + C 2 (2kC 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 ), (15.5.5.9) where the functions ψ 1 (t)andψ 2 (t) are determined by the autonomous system of ordinary differential equations ψ 1 = k(ψ 2 1 + ψ 2 2 )+bψ 1 + c, ψ 2 = ψ 2 (2kψ 1 – ak + b). (15.5.5.10) Remark 1. Example 3 below shows how one can find all two-dimensional linear subspaces invariant under the nonlinear differential operator (15.5.5.8). Remark 2. For k > 0, the nonlinear differential operator (15.5.5.8) has a three-dimensional invariant subspace 3 = 1,sin(x √ k ), cos(x √ k ) ; see Example 3. Remark 3. For k < 0, the nonlinear differential operator (15.5.5.8) has a three-dimensional invariant subspace 3 = 1, sinh(x √ –k ), cosh(x √ –k ) ; see Example 3. Remark 4. A more general equation (15.5.5.7), 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 (15.5.5.9), where the functions ψ 1 (t)andψ 2 (t) are determined by the system of ordinary differential equations (15.5.5.10). 15.5.5-2. Some generalizations. Likewise, one can consider a more general equation of the form L 1 [w]=L 2 [U], U = F [w], (15.5.5.11) where L 1 [w]andL 2 [U] are linear differential operators with respect to t, L 1 [w] ≡ s 1 i=0 a i (t) ∂ i w ∂t i , L 2 [U] ≡ s 2 j=0 b j (t) ∂ j U ∂t j ,(15.5.5.12) and F [w] is a nonlinear differential operator with respect to x, F [w] ≡ F t, x, w, ∂w ∂x , , ∂ m w ∂x m ,(15.5.5.13) and may depend on t as a parameter. 15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 695 Let the linear subspace (15.5.5.3) be invariant under the operator F , i.e., for arbitrary constants C 1 , , C n the following relation holds: F n i=1 C i ϕ i (x) = n i=1 f i (t, C 1 , , C n )ϕ i (x). (15.5.5.14) Then equation (15.5.5.11) possesses generalized separable solutions of the form (15.5.5.5), where the functions ψ 1 (t), , ψ n (t) are described by the system of ordinary differential equations L 1 ψ i (t) = L 2 f i (t, ψ 1 , , ψ n ) , i = 1, , n.(15.5.5.15) Example 2. Consider the equation a 2 (t) ∂ 2 w ∂t 2 + a 1 (t) ∂w ∂t = ∂w ∂x ∂ 2 w ∂x 2 , (15.5.5.16) 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 (15.5.5.16) is a special case of equation (15.5.5.11), 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 (15.5.5.16) possesses generalized separable solutions of the form w(x, t)=ψ 1 (t)+ψ 2 (t)x 3/2 + ψ 3 (t)x 3 , where the functions ψ 1 (t), ψ 2 (t), and ψ 3 (t) are described by the system of ordinary differential equations a 2 (t)ψ 1 + a 1 (t)ψ 1 = 9 8 ψ 2 2 , a 2 (t)ψ 2 + a 1 (t)ψ 2 = 45 4 ψ 2 ψ 3 , a 2 (t)ψ 3 + a 1 (t)ψ 3 = 18ψ 2 3 . Remark. The operator F [w]=w x w xx also has a four-dimensional invariant subspace 4 = 1, x, x 2 , x 3 . Therefore, equation (15.5.5.16) has a generalized separable solution of the form w(x, t)=ψ 1 (t)+ψ 2 (t)x + ψ 3 (t)x 2 + ψ 4 (t)x 3 . 15.5.5-3. How to find linear subspaces invariant under a given nonlinear operator. The most difficult part in using the Titov–Galaktionov method for the construction of exact solutions to specific equations is to find linear subspaces invariant under a given nonlinear operator. In order to determine basis functions ϕ i = ϕ i (x), let us substitute the linear combination n i=1 C i ϕ i (x) into the differential operator (15.5.5.2). Thisisassumed to result in anexpression like F n i=1 C i ϕ i (x) = A 1 (C)Φ 1 (X)+A 2 (C)Φ 2 (X)+···+ A k (C)Φ k (X) + B 1 (C)ϕ 1 (x)+B 2 (C)ϕ 2 (x)+···+ B n (C)ϕ n (x), (15.5.5.17) where A j (C)andB i (C) depend on C 1 , , C n only, and the functionals Φ j (X) depend on x and are independent of C 1 , , C n : A j (C) ≡ A j (C 1 , , C n ), j = 1, , k, B i (C) ≡ B i (C 1 , , C n ), i = 1, , n, Φ j (X) ≡ Φ j x, ϕ 1 , ϕ 1 , ϕ 1 , , ϕ n , ϕ n , ϕ n . (15.5.5.18) . follows: A 1 = C 1,1 and A 2 = C 2,1 in the first solution, and A 1 =–1/C 1,2 and A 2 = C 1,1 /C 1,2 in the second solution. The functions on the right-hand sides of the formulas in (15.5.4.3). is linear in χ (for ψ known). Equation (15.5.4.10) does not have other solutions with the form (15.5.4.11) if f and g are arbitrary functions and ϕ 0, ψ 0 ,and 0. 692 NONLINEAR PARTIAL DIFFERENTIAL. C 3 , β ,and are arbitrary constants (these can be expressed in terms of the A k and B k ). The analysis of the second solution (15.5.4.6) of the functional equation (15.5.4.4) in view of (15.5.4.21) leads