15.8. CLASSICAL METHOD OF STUDYING SYMMETRIES OF DIFFERENTIAL EQUATIONS 731 15.8.4-2. Group invariants. Local transformations of derivatives. A universal invariant (or, for short, an invariant) of a group (15.8.4.1) and a operator (15.8.4.3) is a function I(x, w) that satisfies the condition I(¯x, ¯w)=I(x, w). The expansion in a series in powers of the small parameter ε gives rise to the linear partial differential equation for I: XI = ξ i (x, w) ∂I ∂x i + ζ(x, w) ∂I ∂w = 0.(15.8.4.4) From the theory of first-order partial differential equations it follows that the group (15.8.4.1) and the operator (15.8.4.3) have n functionally independent universal invariants. On the other side, this means that any function F (x, w) that is invariant under the group (15.8.4.1) can be written as a function of n invariants. In the new variable (15.8.4.1), the derivatives are transformed as follows: ∂ ¯w ∂ ¯x i ∂w ∂x i + εζ i , ∂ 2 ¯w ∂ ¯x i ∂ ¯x j ∂ 2 w ∂x i ∂x j + εζ ij , ∂ 3 ¯w ∂ ¯x i ∂ ¯x j ∂ ¯x k ∂ 3 w ∂x i ∂x j ∂x k + εζ ijk , (15.8.4.5) The coordinates ζ i , ζ ij ,andζ ijk of the first three prolongations are expressed as ζ i = D i (ζ)–p s D i (ξ s ), ζ ij = D j (ζ i )–q is D j (ξ s ), ζ ijk = D k (ζ ij )–r ijs D k (ξ s ), (15.8.4.6) where summation is assumed over the repeated index s and the following short notation partial derivatives are used: p i = ∂w ∂x i , q ij = ∂ 2 w ∂x i ∂x j , r ijk = ∂ 3 w ∂x i ∂x j ∂x k , D i = ∂ ∂x i + p i ∂ ∂w + q ij ∂ ∂p j + r ijk ∂ ∂q jk + ··· , with D i being the total differential operator with respect to x i . 15.8.4-3. Invariant condition. Splitting procedure. Invariant solutions. We will consider partial differential equations of order m in n independent variables F x, w, ∂w ∂x i , ∂ 2 w ∂x i ∂x j , ∂ 3 w ∂x i ∂x j ∂x k , = 0,(15.8.4.7) where i, j, k = 1, , n. The group analysis of equation (15.8.4.7) consists of several stages. At the first stage, let us require that equation (15.8.4.7) be invariant (must preserve its form) under transfor- mations (15.8.4.1), so that F ¯x, ¯w, ∂ ¯w ∂ ¯x i , ∂ 2 ¯w ∂x i ∂ ¯x j , ∂ 3 ¯w ∂x i ∂ ¯x j ∂ ¯x k , = 0.(15.8.4.8) 732 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Let us expand this expression into a series in powers of ε about ε = 0 taking into account that the leading term (15.8.4.7) vanishes. Using formulas (15.8.4.1) and (15.8.4.5) and retaining terms to the first order of ε,weget X m F x, w, ∂w ∂x i , ∂ 2 w ∂x i ∂x j , ∂ 3 w ∂x i ∂x j ∂x k , F =0 = 0,(15.8.4.9) where X m F = ξ i ∂F ∂x i + ζ ∂F ∂w + ζ i ∂F ∂w x i + ζ ij ∂F ∂w x i x j + ζ ijk ∂F ∂w x i x j x k + ··· (15.8.4.10) The coordinates ζ i , ζ ij ,andζ ijk of the first three prolongations are defined by formulas (15.8.4.5)–(15.8.4.6). Summation is assumed over repeated indices. Relation (15.8.4.9) is called the invariance condition and the operator X m is called the mth prolongation of the group generator; the partial derivatives of F with respect to all m derivatives of w appear last in (15.8.4.10). At the second stage, one of the highest mth-order derivatives is eliminated from (15.8.4.9) using equation (15.8.4.7). The resulting relation is then represented as a poly- nomial in “independent variables,” the various combinations of the remaining derivatives, which are the products of different powers of w x , w y , w xx , w xy , All the coefficients of this polynomial—they depend on x, w, ξ i ,andζ only and are independent of the derivatives of w —are further equated to zero. As a result, the invariance condition is split into an overdetermined linear determining system. The third stage involves solving the determining system and finding admissible coordi- nates ξ i and ζ of the group generator (15.8.4.3). For mth-order equations in two independent variables, the invariant solutions are defined in a similar way as for second-order equations. In this case, the procedure of constructing invariant solutions (for known coordinates of the group generator) is identical to that described in Subsection 15.8.3. 15.9. Nonclassical Method of Symmetry Reductions ∗ 15.9.1. Description of the Method. Invariant Surface Condition Consider a second-order equation in two independent variables of the form F x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2 = 0.(15.9.1.1) The results of the classical group analysis (see Section 15.8) can be substantially ex- tended if, instead of finding invariants of an admissible infinitesimal operator X by means of solving the characteristic system of equations dx ξ(x, y, w) = dy η(x, y, w) = dw ζ(x, y, w) , one imposes the corresponding invariant surface condition (Bluman and Cole, 1969) ξ(x, y, w) ∂w ∂x + η(x, y, w) ∂w ∂y = ζ(x, y, w). (15.9.1.2) * Prior to reading this section, the reader may find it useful to get acquainted with Section 15.8. 15.9. NONCLASSICAL METHOD OF SYMMETRY REDUCTIONS 733 Equation (15.9.1.1) and condition (15.9.1.2) are supplemented by the invariance condition X 2 F x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2 F =0 = 0,(15.9.1.3) which coincides with equation (15.8.2.3). All three equations (15.9.1.1)–(15.9.1.3) are used for the construction of exact solutions of the original equation (15.9.1.1). It should be observed that in this case, the determin- ing equations obtained for the unknown functions ξ(x, y, w), η(x, y, w), and ζ(x, y, w)by the splitting procedure are nonlinear. The symmetries determined by the invariant sur- face (15.9.1.2) are called nonclassical. Figure 15.5 is intended to clarify the general scheme for constructing of exact solutions of a second-order evolution equation by the nonclassical method on the basis of the invariant surface condition (15.9.1.2). Remark. Apart from the algorithm shown in Fig. 15.5, its modification can also be used. Instead of solving the characteristic system of ordinary differential equations, the derivative w t is eliminated from (15.9.1.1)– (15.9.1.2) after finding the coordinates ξ, η,andζ. Then the resulting equation is solved, which can be treated as an ordinary differential equation for x with parameter t. 15.9.2. Examples: The Newell–Whitehead Equation and a Nonlinear Wave Equation Example 1. Consider the Newell–Whitehead equation w t = w xx + w – w 3 ,(15.9.2.1) which corresponds to the left-hand side F =–w t + w xx + w – w 3 of equation (15.9.1.1) with y = t. Without loss of generality, we set η = 1 in the invariant surface condition (15.9.1.2) with y = t, thus assuming that η ≠ 0.Wehave ∂w ∂t + ξ(x, t, w) ∂w ∂x = ζ(x, t, w). (15.9.2.2) The invariance condition is obtained by a procedure similar to the classical algorithm (see Subsec- tion 15.8.2). Namely, we apply the operator X 2 = ξ∂ x + η∂ t + ζ∂ w + ζ 1 ∂ w x + ζ 2 ∂ w t + ζ 11 ∂ w xx + ζ 12 ∂ w xt + ζ 22 ∂ w tt (15.9.2.3) to equation (15.9.2.1). Taking into account that ∂ x = ∂ t = ∂ w x = ∂ w xt = ∂ w tt = 0, since the equation is explicitly independent of x, t, w x , w xt , w tt , we get the invariance condition in the form ζ 2 = ζ(3w 2 – 1)+ζ 11 . Substituting here the expressions (15.8.1.9) and (15.8.1.14) for the coordinates ζ 2 and ζ 11 of the first and second prolongations, with y = t and η = 1, we obtain ζ t – ξ t w x + ζ w w t – ξ w w x w t = ζ(–3w 2 + 1)+ζ xx +(2ζ wx – ξ xx )w x +(ζ ww – 2ξ wx )w 2 x – ξ ww w 3 x +(ζ w – 2ξ x – 3ξ w w x )w xx .(15.9.2.4) Let us express the derivatives w t and w xx with the help of (15.9.2.1)–(15.9.2.2) via the other quantities: w t = ζ – ξw x , w xx = ζ – ξw x + w(w 2 – 1). (15.9.2.5) Inserting these expressions into the invariance condition (15.9.2.4), we arrive at cubic polynomial in the remaining “independent” derivative w x . Equating all functional coefficients of the various powers of this polynomial to zero, we get the determining system w 3 x : ξ ww = 0, w 2 x : ζ ww – 2(ξ wx – ξξ w )=0, w x : 2ζ wx – 2ξ w ζ – 3w(w 2 – 1)ξ w – ξ xx + 2ξξ x + ξ t = 0, 1: ζ t – ζ xx + 2ξ x ζ +(2ξ x – ζ w )w(w 2 – 1)+(3w 2 – 1)ζ = 0, (15.9.2.6) 734 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Calculate the coordinates of the prolonged operator Derive the determining system of PDEs Solve the characteristic system Impose the invariant surface condition (3) (2) (1) Figure 15.5. Algorithm for the construction of exact solutions by a nonclassical method for second-order evolution equations. Here, ODE stands for ordinary differential equation and PDE for partial differential equation. which consists of only four equations. The analysis of the system (15.9.2.6) allows us to conclude that ξ = ξ(x, t), ζ =–ξ x w, where the function ξ(x, t) satisfies the system ξ t – 3ξ xx + 2ξξ x = 0, ξ xt – ξ xxx + 2ξ 2 x + 2ξ x = 0. (15.9.2.7) The associated invariant surface condition has the form (15.9.2.2): w t + ξw x =–ξ x w.(15.9.2.8) We look for a stationary particular solution to equation (15.9.2.7) in the form ξ = ξ(x). Let us differentiate the first equation of (15.9.2.7) with respect to x and then eliminate the third derivative, using the second equation 15.9. NONCLASSICAL METHOD OF SYMMETRY REDUCTIONS 735 of (15.9.2.7), from the resulting expression. This will give a second-order equation. Eliminating from it the second derivative, using the first equation of (15.9.2.7), after simple rearrangements we obtain (6ξ x – 2ξ 2 + 9)ξ x = 0.(15.9.2.9) Equating the expression in parentheses in (15.9.2.9) to zero, we get a separable first-order equation. Its general solution can be written as ξ =– 3 √ 2 A exp √ 2 x + B A exp √ 2 x – B =– 3 √ 2 A exp √ 2 2 x + B exp – √ 2 2 x A exp √ 2 2 x – B exp – √ 2 2 x , (15.9.2.10) where A and B are arbitrary constants. The characteristic system corresponding to (15.9.2.8), dt 1 = dx ξ =– dw ξ x w , (15.9.2.11) admits the first integrals t + 2 3 ln A exp √ 2 2 x + B exp – √ 2 2 x = C 1 , ξw = C 2 , (15.9.2.12) where C 1 and C 2 are arbitrary constants. For convenience, we introduce the new constant C 1 =exp 3 2 C 1 andlookforasolutionintheformC 2 =– 3 √ 2 C 1 F (C 1 ). Inserting (15.9.2.12) here and taking into account (15.9.2.10), we obtain the solution structure w(x, t)= A exp 1 2 ( √ 2x + 3t) – B exp 1 2 (– √ 2x + 3t) F (z), z = A exp 1 2 ( √ 2x + 3t) + B exp 1 2 (– √ 2x + 3t) . (15.9.2.13) Substituting (15.9.2.13) in the original equation, we find the equation for F = F (z): F zz = 2F 3 . (15.9.2.14) The general solution of equation (15.9.2.14) is expressed in terms of elliptic functions. This equation admits the following particular solutions: F = 1 z + C ,whereC is an arbitrary constant. Remark 1. To the degenerate case ξ x = 0 in (15.9.2.9) there corresponds a traveling-wave solution. Remark 2. It is apparent from this example that using the invariant surface condition (15.9.1.2) gives much more freedom in determining the coordinates ξ, η, ζ as compared with the classical scheme presented in Section 15.8. This stems from the fact that in the classical scheme, the invariance condition is split with respect to two derivatives, w x and w xx , which are considered to be independent (see Example 2 in Subsection 15.8.2). In the nonclassical scheme, the derivatives w x and w xx are related by the second equation of (15.9.2.5) and the invariance conditions are split in only one derivative, w x . This is why in the classical scheme the determining system consists of a larger number of equations, which impose additional constraints on the unknown quantities, as compared with the nonclassical scheme. In particular, the classical scheme fails to fi nd exact solutions to equation Newell–Whitehead (15.9.2.1) in the form (15.9.2.13). Example 2. Consider the nonlinear wave equation ∂ 2 w ∂t 2 = w ∂ 2 w ∂x 2 , (15.9.2.15) which corresponds to the left-hand side F = w tt – ww xx of equation (15.9.1.1) with y = t. Let us add the invariant surface condition (15.9.2.2). Applying the prolonged operator (15.9.2.3) to equation (15.9.2.15) gives the invariance condition. Taking into account that ∂ x = ∂ t = ∂ w x = ∂ w t = ∂ w xt = 0 (since the equation is independent of x, t, w x , w t , w xt explicitly) and η = 1, we get the invariance condition ζ 22 = ζw xx + wζ 11 . Substituting here the expressions (15.8.1.14) for the coordinates ζ 11 and ζ 22 of the second prolongation with y = t and η = 1,wehave ζ tt – ξ tt w x + 2ζ wt w t – 2ξ wt w x w t + ζ ww w 2 t – ξ ww w x w 2 t – 2(ξ t + ξ w w t )w xt +(ζ w – ξ w w x )w tt = ζw xx + w ζ xx +(2ζ wx – ξ xx )w x +(ζ ww – 2ξ wx )w 2 x – ξ ww w 3 x +(ζ w – 2ξ x – 3ξ w w x )w xx . (15.9.2.16) 736 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS From the invariant surface condition (15.9.2.2) and equation (15.9.2.15) it follows that w t = ζ – ξw x , w tt = ww xx , w xt = ζ x – ξ x w x – ξw xx . (15.9.2.17) The last formula has been obtained by differentiating the first one with respect to x. Substituting w t , w tt , w xt from (15.9.2.17) into (15.9.2.16) yields a polynomial in two “independent” derivatives, w x and w xx . Equating the functional coefficients of this polynomial to zero, we arrive at the determining system: w x w xx :(ξ 2 – w)ξ w = 0, w xx : 2ξξ t + 2wξ x + 2ξξ w ζ – ζ = 0, w 3 x :(ξ 2 – w)ξ ww = 0, w 2 x :(ξ 2 – w)ζ ww + 2ξξ wt + 2ξξ ww ζ – 2ξξ x ξ w + 2wξ wx = 0, w x : ξ tt + 2ξζ wt + 2ξ wt ζ + 2ξζζ ww + ξ ww ζ 2 – 2ξ t ξ x – 2ξ x ξ w ζ – 2ξξ w ζ x + 2wζ wx – wξ xx = 0, 1: ζ tt + 2ζζ wt + ζ 2 ζ ww – 2ξ t ζ x – 2ξ w ζζ x – wζ xx = 0. From the first equation it follows that 1) ξ = ξ(x, t); (15.9.2.18) 2) ξ = √ w. (15.9.2.19) Consider both cases. 1 ◦ .Forξ = ξ(x, t), the third equation of the determining system (15.9.2.18) is satisfied identically. From the second equation it follows that ζ = 2wξ x + 2ξξ t . (15.9.2.20) The fourth equation is also satisfied identically in view of (15.9.2.18) and (15.9.2.20). Substituting (15.9.2.18) and (15.9.2.20) into the fifth and sixth equations of the determining system yields two solutions: ξ = αt + β, ζ = 2α(αt + β)(first solution); ξ = αx + β, ζ = 2αw (second solution); (15.9.2.21) where α and β are arbitrary constants. First solution. The characteristic system of ordinary differential equations associated with the first solution (15.9.2.21), with α = 2 and β = 0,hastheform dt 1 = dx 2t = dw 8t . Find the first integrals: C 1 = x – t 2 , C 2 = w – 4t 2 . Following the scheme shown in Fig. 15.5, we look for a solution in the form w – 4t 2 = Φ(x – t 2 ). Substituting w = Φ(z)+4t 2 , z = x – t 2 (15.9.2.22) into (15.9.2.15) yields the following autonomous ordinary differential equation for Φ = Φ(z): ΦΦ zz + 2Φ z = 8. It is easy to integrate. Reducing its order gives a separable equation. As a result, one can obtain an exact solution to equation (15.9.2.15) of the form (15.9.2.22). Second solution. The characteristic system of ordinary differential equations associated with the second solution of (15.9.2.21), with α = 1 and β = 0,hastheform dt 1 = dx x = dw 2w . Find the first integrals: C 1 =ln|x| – t, C 2 = w/x 2 . Following the scheme shown in Fig. 15.5, we look for a solution in the form w/x 2 = Φ(ln |x| – t). Substituting w = x 2 Φ(z), z =ln|x| – t into (15.9.2.15) yields the autonomous ordinary differential equation (Φ – 1)Φ zz + 3ΦΦ z + 2Φ 2 = 0, which admits order reduction via the substitution U(Φ)=Φ z . 2 ◦ . The second case of (15.9.2.19) gives rise to the trivial solution ζ = 0 (as follows from the fourth equation of the determining system), which generates the obvious solution w = const. 15.10. DIFFERENTIAL CONSTRAINTS METHOD 737 15.10. Differential Constraints Method 15.10.1. Description of the Method 15.10.1-1. Preliminary remarks. A simple example. Basic idea: Try to find an exact solution to a complex equation by analyzing it in conjunction with a simpler, auxiliary equation, called a differential constraint. In Subsections 15.4.1 and 15.4.3, we have considered examples of additive separable solutions of nonlinear equations in the form w(x, y)=ϕ(x)+ψ(y). (15.10.1.1) At the initial stage, the functions ϕ(x)andψ(y) are assumed arbitrary and are to be determined in the subsequent analysis. Differentiating the expression (15.10.1.1) with respect to y, we obtain ∂w ∂y = f(y)(f = ψ y ). (15.10.1.2) Conversely, relation (15.10.1.2) implies a representation of the solution in the form (15.10.1.1). Further, differentiating (15.10.1.2) with respect to x gives ∂ 2 w ∂x∂y = 0.(15.10.1.3) Conversely, from (15.10.1.3) we obtain a representation of the solution in the form (15.10.1.1). Thus, the problem of finding exact solutions of the form (15.10.1.1) for a specific partial differential equation may be replaced by an equivalent problem of finding exact solutions of the given equation supplemented with the condition (15.10.1.2) or (15.10.1.3). Such supplementary conditions in the form of one or several differential equations will be called differential constraints. Prior to giving a general description of the differential constraints method, we demon- strate its features by a simple example. Example. Consider the third-order nonlinear equation ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = a ∂ 3 w ∂y 3 , (15.10.1.4) which occurs in the theory of the hydrodynamic boundary layer. Let us seek a solution of equation (15.10.1.4) satisfying the linear first-order differential constraint ∂w ∂x = ϕ(y). (15.10.1.5) Here, the function ϕ(y) cannot be arbitrary, in general, but must satisfy the condition of compatibility of equations (15.10.1.4) and (15.10.1.5). The compatibility condition is a differential equation for ϕ(y)andisa consequence of equations (15.10.1.4), (15.10.1.5), and those obtained by their differentiation. Successively differentiating (15.10.1.5) with respect to different variables, we calculate the derivatives w xx = 0, w xy = ϕ y , w xxy = 0, w xyy = ϕ yy , w xyyy = ϕ yyy . (15.10.1.6) Differentiating (15.10.1.4) with respect to x yields w 2 xy + w y w xxy – w xx w yy – w x w xyy = aw xyyy . (15.10.1.7) . 15.5. Algorithm for the construction of exact solutions by a nonclassical method for second-order evolution equations. Here, ODE stands for ordinary differential equation and PDE for partial differential equation. which. solution in the form (15.10.1.1). Thus, the problem of finding exact solutions of the form (15.10.1.1) for a specific partial differential equation may be replaced by an equivalent problem of finding. condition of compatibility of equations (15.10.1.4) and (15.10.1.5). The compatibility condition is a differential equation for ϕ(y)andisa consequence of equations (15.10.1.4), (15.10.1.5), and those