696 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Here, for simplicity, the formulas are written out for the case of a second-order differential operator. For higher-order operators, the right-hand sides of relations (15.5.5.18) will contain higher-order derivatives of ϕ i . The functionals and functions Φ 1 (X), , Φ k (X), ϕ 1 (x), , ϕ n (x) together are assumed to be linearly independent, and the A j (C)are linearly independent functions of C 1 , , C n . The basis functions are determined by solving the (usually overdetermined) system of ordinary differential equations Φ j x, ϕ 1 , ϕ 1 , ϕ 1 , , ϕ n , ϕ n , ϕ n = p j,1 ϕ 1 + p j,2 ϕ 2 + ···+ p j,n ϕ n , j = 1, , k, (15.5.5.19) where p j,i are some constants independent of the parameters C 1 , , C n .Ifforsome collection of the constants p i,j , system (15.5.5.19) is solvable (in practice, it suffices to find a particular solution), then the functions ϕ i = ϕ i (x)define a linear subspace invariant under the nonlinear differential operator (15.5.5.2). In this case, the functions appearing on the right-hand side of (15.5.5.4) are given by f i (C 1 , , C n )=p 1,i A 1 (C 1 , , C n )+p 2,i A 2 (C 1 , , C n )+··· + p k,i A k (C 1 , , C n )+B i (C 1 , , C n ). Remark. The analysis of nonlinear differential operators is useful to begin with looking for two- dimensional invariant subspaces of the form 2 = {1, ϕ(x)}. Proposition 1. Let a nonlinear differential operator F [w] admit a two-dimensional invariant subspace of the form 2 = {1, ϕ(x)},whereϕ(x)=pϕ 1 (x)+qϕ 2 (x), p and q are arbitrary constants, and the functions 1, ϕ 1 (x), ϕ 2 (x) are linearly independent. Then the operator F [w] also admits a three-dimensional invariant subspace 2 = {1, ϕ 1 (x), ϕ 2 (x)}. Proposition 2. Let two nonlinear differential operators F 1 [w]andF 2 [w] admit one and the same invariant subspace n = {ϕ 1 (x), , ϕ n (x)}. Then the nonlinear operator pF 1 [w]+qF 2 [w], where p and q are arbitrary constants, also admits the same invariant subspace. Example 3. Consider the nonlinear differential operator (15.5.5.8). We look for its invariant subspaces of the form 2 = {1, ϕ(x)}.Wehave F [C 1 + C 2 ϕ(x)] = C 2 2 [(ϕ x ) 2 + kϕ 2 ]+C 2 aϕ xx + kC 2 1 + bC 1 + c +(bC 2 + 2kC 1 C 2 )ϕ. Here, Φ 1 (X)=(ϕ x ) 2 + kϕ 2 and Φ 2 (X)=aϕ xx . Hence, the basis function ϕ(x) is determined by the overdetermined system of ordinary differential equations (ϕ x ) 2 + kϕ 2 = p 1 + p 2 ϕ, ϕ xx = p 3 + p 4 ϕ, (15.5.5.20) where p 1 = p 1,1 , p 2 = p 1,2 , p 3 = p 2,1 /a,andp 4 = p 2,2 /a. Let us investigate system (15.5.5.20) for consistency. To this end, we differentiate the first equation with respect to x and then divide by ϕ x to obtain ϕ xx = –kϕ + p 2 /2. Using this relation to eliminate the second derivative from the second equation in (15.5.5.20), we get (p 4 + k)ϕ + p 3 – 1 2 p 2 = 0. For this equation to be satisfied, the following identities must hold: p 4 =–k, p 3 = 1 2 p 2 . (15.5.5.21) The simultaneous solution of system (15.5.5.20) under condition (15.5.5.21) is given by ϕ(x)=px 2 + qx if k = 0 (p 1 = q 2 , p 2 = 4p), ϕ(x)=p sin x √ k + q cos x √ k if k > 0 (p 1 = kp 2 + kq 2 , p 2 = 0), ϕ(x)=p sinh x √ –k + q cosh x √ –k if k < 0 (p 1 =–kp 2 + kq 2 , p 2 = 0), (15.5.5.22) where p and q are arbitrary constants. Since formulas (15.5.5.22) involve two arbitrary parameters p and q, it follows from Proposition 1 that the nonlinear differential operator (15.5.5.8) admits the following invariant subspaces: 3 = 1, x, x 2 if k = 0, 3 = 1,sin(x √ k ), cos(x √ k ) if k > 0, 3 = 1, sinh(x √ –k ), cosh(x √ –k ) if k < 0. 15.6. METHOD OF FUNCTIONAL SEPARATION OF VARIABLES 697 15.6. Method of Functional Separation of Variables 15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities 15.6.1-1. Structure of functional separable solutions. Suppose a nonlinear equation for w = w(x, y) is obtained from a linear mathematical physics equation for z = z(x, y) by a nonlinear change of variable w = F (z). Then, if the linear equation for z admits separable solutions, the transformed nonlinear equation for w will have exact solutions of the form w(x, y)=F(z), where z = n m=1 ϕ m (x)ψ m (y). (15.6.1.1) It is noteworthy that many nonlinear partial differential equations that are not reducible to linear equations have exact solutions of the form (15.6.1.1) as well. We will call such solutions functional separable solutions. In general, the functions ϕ m (x), ψ m (y), and F (z) in (15.6.1.1) are not known in advance and are to be identified. Main idea: The functional differential equation resulting from the substitution of (15.6.1.1) in the original partial differential equation should be reduced to the standard bilinear functional equation (15.5.1.3) or to a functional differential equation of the form (15.5.1.3)–(15.5.1.4), and then the results of Subsections 15.5.3–15.5.5 should be used. Remark. The function F (z) can be determined by a single ordinary differential equation or by an overde- termined system of equations; both possibilities must be taken into account. 15.6.1-2. Solution by reduction to equations with quadratic (or power) nonlinearities. In some cases, solutions of the form (15.6.1.1) can be searched for in two stages. First, one looks for a transformation that would reduce the original equation to an equation with a quadratic (or power) nonlinearity. Then the methods outlined in Subsections 15.5.3–15.5.5 are used for finding solutions of the resulting equation. Table 15.5 gives examples of nonlinear heat equations with power, exponential, and logarithmic nonlinearities reducible, by simple substitutions of the form w = F (z), to quadratically nonlinear equations. For these equations, it can be assumed that the form of the function F (z) in solution (15.6.1.1) is given a priori. 15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions 15.6.2-1. Special functional separable and generalized traveling-wave solutions. To simplify the analysis, some of the functions in (15.6.1.1) can be specified a priori and the other functions will be defined in the analysis. We call such solutions special functional separable solutions. A generalized separable solution (see Section 15.5) is a functional separable solution of the special form corresponding to F (z)=z. Below we consider two simplest functional separable solutions of special forms: w = F (z), z = ϕ 1 (x)y + ϕ 2 (x); w = F (z), z = ϕ(x)+ψ(y). (15.6.2.1) 698 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 15.5 Some nonlinear heat equations reducible to quadratically nonlinear equations by a transformation of the form w = F (z); the constant σ is expressed in terms of the coefficients of the transformed equation Original equation Transformation Transformed equation Form of solutions ∂w ∂t = a ∂ ∂x w n ∂w ∂x + bw + cw 1–n w = z 1/n ∂z ∂t = az ∂ 2 z ∂x 2 + a n ∂z ∂x 2 + bnz + cn z = ϕ(t)x 2 + ψ(t)x + χ(t) ∂w ∂t = a ∂ ∂x w n ∂w ∂x + bw n+1 + cw w = z 1/n ∂z ∂t = az ∂ 2 z ∂x 2 + a n ∂z ∂x 2 + bnz 2 + cnz z = ϕ(t)e σx + ψ(t)e –σx + χ(t) z = ϕ(t)sin(σx)+ψ(t)cos(σx)+χ(t) ∂w ∂t = a ∂ ∂x e λw ∂w ∂x + b + ce –λw w = 1 λ ln z ∂z ∂t = az ∂ 2 z ∂x 2 + bλz + cλ z = ϕ(t)x 2 + ψ(t)x + χ(t) ∂w ∂t = a ∂ ∂x e λw ∂w ∂x + be λw + c w = 1 λ ln z ∂z ∂t = az ∂ 2 z ∂x 2 + bz 2 + cλz z = ϕ(t)e σx + ψ(t)e –σx + χ(t) z = ϕ(t)sin(σx)+ψ(t)cos(σx)+χ(t) ∂w ∂t = a ∂ 2 w ∂x 2 + bw ln w + cw w = e z ∂z ∂t = a ∂ 2 z ∂x 2 + a ∂z ∂x 2 + bz + c z = ϕ(t)x 2 + ψ(t)x + χ(t) ∂w ∂t = a ∂ 2 w ∂x 2 + bw ln 2 w + cw w = e z ∂z ∂t = a ∂ 2 z ∂x 2 + a ∂z ∂x 2 + bz 2 + c z = ϕ(t)e σx + ψ(t)e –σx + χ(t) z = ϕ(t)sin(σx)+ψ(t)cos(σx)+χ(t) The first solution (15.6.2.1) will be called a generalized traveling-wave solution (x and y can be swapped). After substituting this solution into the original equation, one should eliminate y with the help of the expression for z. This will result in a functional differential equation with two arguments, x and z. Its solution may be obtained with the methods outlined in Subsections 15.5.3–15.5.5. Remark 1. In functional separation of variables, searching for solutions in the forms w = F ϕ(x)+ψ(y) [it is the second solution in (15.6.2.1)] and w = F ϕ(x)ψ(y) leads to equivalent results because the two forms are functionally equivalent. Indeed, we have F ϕ(x)ψ(y) = F 1 ϕ 1 (x)+ψ 1 (y) ,whereF 1 (z)=F (e z ), ϕ 1 (x)=lnϕ(x), and ψ 1 (y)=lnψ(y). Remark 2. In constructing functional separable solutions with the form w = F ϕ(x)+ψ(y) [it is the second solution in (15.6.2.1)], it is assumed that ϕ const and ψ const. Example 1. Consider the third-order nonlinear equation ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = a ∂ 2 w ∂y 2 n–1 ∂ 3 w ∂y 3 , which describes a boundary layer of a power-law fluid on a flat plate; w is the stream function, x and y are coordinates along and normal to the plate, and n is a rheological parameter (the value n = 1 corresponds to a Newtonian fluid). Searching for solutions in the form w = w(z), z = ϕ(x)y + ψ(x) leads to the equation ϕ x (w z ) 2 = aϕ 2n (w zz ) n–1 w zzz , which is independent of ψ. Separating the variables and integrating yields ϕ(x)=(ax + C) 1/(1–2n) , ψ(x) is arbitrary, and w = w(z) is determined by solving the ordinary differential equation (w z ) 2 =(1 – 2n)(w zz ) n–1 w zzz . 15.6.2-2. General scheme for constructing generalized traveling-wave solutions. For visualization, the general scheme for constructing generalized traveling-wave solutions for evolution equations is displayed in Fig. 15.3. 15.6. METHOD OF FUNCTIONAL SEPARATION OF VARIABLES 699 Write out the functional differential equation in two arguments Obtain (i) functional equation, (ii) determining system of ODEs Solve the determining system of ordinary differential equations Write out generalized solution of original equationtraveling-wave Search for generalized solutionstraveling-wave Apply splitting procedure Treat functional equation (i) Figure 15.3. Algorithm for constructing generalized traveling-wave solutions for evolution equations. Abbre- viation: ODE stands for ordinary differential equation. Example 2. Consider the nonstationary heat equation with a nonlinear source ∂w ∂t = ∂ 2 w ∂x 2 + F(w). (15.6.2.2) We look for functional separable solutions of the special form w = w(z), z = ϕ(t)x + ψ(t). (15.6.2.3) The functions w(z), ϕ(t), ψ(t), and F(w) are to be determined. On substituting (15.6.2.3) into (15.6.2.2) and on dividing by w z ,wehave ϕ t x + ψ t = ϕ 2 w zz w z + F(w) w z .(15.6.2.4) We express x from (15.6.2.3) in terms of z and substitute into (15.6.2.4) to obtain a functional differential equation with two variables, t and z: –ψ t + ψ ϕ ϕ t – ϕ t ϕ z + ϕ 2 w zz w z + F(w) w z = 0, which can be treated as the functional equation (15.5.4.4), where Φ 1 =–ψ t + ψ ϕ ϕ t , Φ 2 =– ϕ t ϕ , Φ 3 = ϕ 2 , Φ 4 = 1, Ψ 1 = 1, Ψ 2 = z, Ψ 3 = w zz w z , Ψ 4 = F(w) w z . 700 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Substituting these expressions into relations (15.5.4.5) yields the system of ordinary differential equations –ψ t + ψ ϕ ϕ t = A 1 ϕ 2 + A 2 ,– ϕ t ϕ = A 3 ϕ 2 + A 4 , w zz w z =–A 1 – A 3 z, F(w) w z =–A 2 – A 4 z, (15.6.2.5) where A 1 , , A 4 are arbitrary constants. Case 1.ForA 4 ≠ 0, the solution of system (15.6.2.5) is given by ϕ(t)= C 1 e 2A 4 t – A 3 A 4 –1/2 , ψ(t)=–ϕ(t) A 1 ϕ(t) dt + A 2 dt ϕ(t) + C 2 , w(z)=C 3 exp – 1 2 A 3 z 2 – A 1 z dz + C 4 , F(w)=–C 3 (A 4 z + A 2 )exp – 1 2 A 3 z 2 – A 1 z , (15.6.2.6) where C 1 , , C 4 are arbitrary constants. The dependence F = F(w)isdefined by the last two relations in parametric form (z is considered the parameter). If A 3 ≠ 0 in (15.6.2.6), the source function is expressed in terms of elementary functions and the inverse of the error function. In the special case A 3 = C 4 = 0, A 1 =–1,andC 3 = 1, the source function can be represented in explicit form as F(w)=–w(A 4 ln w + A 2 ). Case 2.ForA 4 = 0, the solution to the first two equations in (15.6.2.5) is given by ϕ(t)= 1 √ 2A 3 t + C 1 , ψ(t)= C 2 √ 2A 3 t + C 1 – A 1 A 3 – A 2 3A 3 (2A 3 t + C 1 ), and the solutions to the other equations are determined by the last two formulas in (15.6.2.6) where A 4 = 0. Remark. The algorithm presented in Fig. 15.3 can also be used for finding exact solutions of the more general form w = σ(t)F (z)+ϕ 1 (t)x + ψ 2 (t), where z = ϕ 1 (t)x + ψ 2 (t). For an example of this sort of solution, see Subsection 15.7.2 (Example 1). 15.6.3. Differentiation Method 15.6.3-1. Basic ideas of the method. Reduction to a standard equation. In general, the substitution of expression (15.6.1.1) into the nonlinear partial differential equation under study leads to a functional differential equation with three arguments—two arguments are usual, x and y, and the third is composite, z. In some cases, the resulting equation can be reduced by differentiation to a standard functional differential equation with two arguments (either x or y is eliminated). To solve the two-argument equation, one can use the methods outlined in Subsections 15.5.3–15.5.5. 15.6.3-2. Examples of constructing functional separable solutions. Below we consider specific examples illustrating the application of the differentiation method for constructing functional separable solutions of nonlinear equations. Example 1. Consider the nonlinear heat equation ∂w ∂t = ∂ ∂x F(w) ∂w ∂x .(15.6.3.1) We look for exact solutions with the form w = w(z), z = ϕ(x)+ψ(t). (15.6.3.2) 15.6. METHOD OF FUNCTIONAL SEPARATION OF VARIABLES 701 On substituting (15.6.3.2) into (15.6.3.1) and dividing by w z , we obtain the functional differential equation with three variables ψ t = ϕ xx F(w)+(ϕ x ) 2 H(z), (15.6.3.3) where H(z)=F(w) w zz w z + F z (w), w = w(z). (15.6.3.4) Differentiating (15.6.3.3) with respect to x yields ϕ xxx F(w)+ϕ x ϕ xx [F z (w)+2H(z)] + (ϕ x ) 3 H z = 0.(15.6.3.5) This functional differential equation with two variables can be treated as the functional equation (15.5.4.2). This three-term functional equation has two different solutions. Accordingly, we consider two cases. Case 1. The solutions of the functional differential equation (15.6.3.5) are determined from the system of ordinary differential equations F z + 2H = 2A 1 F, H z = A 2 F, ϕ xxx + 2A 1 ϕ x ϕ xx + A 2 (ϕ x ) 3 = 0, (15.6.3.6) where A 1 and A 2 are arbitrary constants. The first two equations (15.6.3.6) are linear and independent of the third equation. Their general solution is given by F = ⎧ ⎨ ⎩ e A 1 z (B 1 e kz + B 2 e –kz )ifA 2 1 > 2A 2 , e A 1 z (B 1 + B 2 z)ifA 2 1 = 2A 2 , e A 1 z [B 1 sin(kz)+B 2 cos(kz)] if A 2 1 < 2A 2 , H =A 1 F– 1 2 F z , k = |A 2 1 – 2A 2 |.(15.6.3.7) Substituting H of (15.6.3.7) into (15.6.3.4) yields an ordinary differential equation for w = w(z). On integrating this equation, we obtain w = C 1 e A 1 z |F(z)| –3/2 dz + C 2 ,(15.6.3.8) where C 1 and C 2 are arbitrary constants. The expression of F in (15.6.3.7) together with expression (15.6.3.8) defines the function F = F(w) in parametric form. Without full analysis, we will study the case A 2 = 0 (k = A 1 )andA 1 ≠ 0 in more detail. It follows from (15.6.3.7) and (15.6.3.8) that F(z)=B 1 e 2A 1 z + B 2 , H = A 1 B 2 , w(z)=C 3 (B 1 + B 2 e –2A 1 z ) –1/2 + C 2 (C 1 = A 1 B 2 C 3 ). (15.6.3.9) Eliminating z yields F(w)= B 2 C 2 3 C 2 3 – B 1 w 2 . (15.6.3.10) The last equation in (15.6.3.6) with A 2 = 0 has the first integral ϕ xx + A 1 (ϕ x ) 2 = const. The corresponding general solution is given by ϕ(x)=– 1 2A 1 ln D 2 D 1 1 sinh 2 A 1 √ D 2 x + D 3 for D 1 > 0 and D 2 > 0, ϕ(x)=– 1 2A 1 ln – D 2 D 1 1 cos 2 A 1 √ –D 2 x + D 3 for D 1 > 0 and D 2 < 0, ϕ(x)=– 1 2A 1 ln – D 2 D 1 1 cosh 2 A 1 √ D 2 x + D 3 for D 1 < 0 and D 2 > 0, (15.6.3.11) where D 1 , D 2 ,andD 3 are constants of integration. In all three cases, the following relations hold: (ϕ x ) 2 = D 1 e –2A 1 ϕ + D 2 , ϕ xx =–A 1 D 1 e –2A 1 ϕ . (15.6.3.12) We substitute (15.6.3.9) and (15.6.3.12) into the original functional differential equation (15.6.3.3). With reference to the expression of z in (15.6.3.2), we obtain the following equation for ψ = ψ(t): ψ t =–A 1 B 1 D 1 e 2A 1 ψ + A 1 B 2 D 2 . Its general solution is given by ψ(t)= 1 2A 1 ln B 2 D 2 D 4 exp(–2A 2 1 B 2 D 2 t)+B 1 D 1 , (15.6.3.13) where D 4 is an arbitrary constant. Formulas (15.6.3.2), (15.6.3.9) for w, (15.6.3.11), and (15.6.3.13) define three solutions of the nonlinear equation (15.6.3.1) with F(w) of the form (15.6.3.10) [recall that these solutions correspond to the special case A 2 = 0 in (15.6.3.7) and (15.6.3.8)]. 702 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Case 2. The solutions of the functional differential equation (15.6.3.5) are determined from the system of ordinary differential equations ϕ xxx = A 1 (ϕ x ) 3 , ϕ x ϕ xx = A 2 (ϕ x ) 3 , A 1 F + A 2 (F z + 2H)+H z = 0. (15.6.3.14) The first two equations in (15.6.3.14) are compatible in the two cases A 1 = A 2 = 0 =⇒ ϕ(x)=B 1 x + B 2 , A 1 = 2A 2 2 =⇒ ϕ(x)=– 1 A 2 ln |B 1 x + B 2 |. (15.6.3.15) The first solution in (15.6.3.15) eventually leads to the traveling-wave solution w = w(B 1 x + B 2 t) of equa- tion (15.6.3.1) and the second solution to the self-similar solution of the form w = w(x 2 /t). In both cases, the function F(w) in (15.6.3.1) is arbitrary. Example 2. Consider the nonlinear Klein–Gordon equation ∂ 2 w ∂t 2 – ∂ 2 w ∂x 2 = F(w). (15.6.3.16) We look for functional separable solutions in additive form: w = w(z), z = ϕ(x)+ψ(t). (15.6.3.17) Substituting (15.6.3.17) into (15.6.3.16) yields ψ tt – ϕ xx + (ψ t ) 2 –(ϕ x ) 2 g(z)=h(z), (15.6.3.18) where g(z)=w zz /w z , h(z)=F w(z) /w z . (15.6.3.19) On differentiating (15.6.3.18) first with respect to t and then with respect to x and on dividing by ψ t ϕ x ,we have 2(ψ tt – ϕ xx ) g z + (ψ t ) 2 –(ϕ x ) 2 g zz = h zz . Eliminating ψ tt – ϕ xx from this equation with the aid of (15.6.3.18), we obtain (ψ t ) 2 –(ϕ x ) 2 (g zz – 2gg z )=h zz – 2g z h. (15.6.3.20) This relation holds in the following cases: g zz – 2gg z = 0, h zz – 2g z h = 0 (case 1), (ψ t ) 2 = Aψ + B,(ϕ x ) 2 =–Aϕ + B – C, h zz – 2g z h =(Az + C)(g zz – 2gg z ) (case 2), (15.6.3.21) where A, B,andC are arbitrary constants. We consider both cases. Case 1.Thefirst two equations in (15.6.3.21) enable one to determine g(z)andh(z). Integrating the first equation once yields g z = g 2 + const. Further, the following cases are possible: g = k, (15.6.3.22a) g =–1/(z + C 1 ), (15.6.3.22b) g =–k tanh(kz + C 1 ), (15.6.3.22c) g =–k coth(kz + C 1 ), (15.6.3.22d) g = k tan(kz + C 1 ), (15.6.3.22e) where C 1 and k are arbitrary constants. The second equation in (15.6.3.21) has a particular solution h = g(z). Hence, its general solution is expressed by [e.g., see Polyanin and Zaitsev (2003)]: h = C 2 g(z)+C 3 g(z) dz g 2 (z) , (15.6.3.23) where C 2 and C 3 are arbitrary constants. The functions w(z)andF(w) are found from (15.6.3.19) as w(z)=B 1 G(z) dz + B 2 , F(w)=B 1 h(z)G(z), where G(z)=exp g(z)dz , (15.6.3.24) and B 1 and B 2 are arbitrary constants (F is defined parametrically). . NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Here, for simplicity, the formulas are written out for the case of a second-order differential operator. For higher-order operators, the right-hand sides of. (z); the constant σ is expressed in terms of the coefficients of the transformed equation Original equation Transformation Transformed equation Form of solutions ∂w ∂t = a ∂ ∂x w n ∂w ∂x +. an arbitrary constant. Formulas (15.6.3.2), (15.6.3.9) for w, (15.6.3.11), and (15.6.3.13) define three solutions of the nonlinear equation (15.6.3.1) with F(w) of the form (15.6.3.10) [recall