15.2. TRANSFORMATIONS OF EQUATIONS OF MAT HE M AT I C AL PHYSICS 661 With (15.2.3.5)–(15.2.3.6), we find the second derivatives ∂ 2 w ∂x 2 = J ∂ 2 u ∂η 2 , ∂ 2 w ∂x∂y =–J ∂ 2 u ∂ξ∂η , ∂ 2 w ∂y 2 = J ∂ 2 u ∂ξ 2 , where J = ∂ 2 w ∂x 2 ∂ 2 w ∂y 2 – ∂ 2 w ∂x∂y 2 , 1 J = ∂ 2 u ∂ξ 2 ∂ 2 u ∂η 2 – ∂ 2 u ∂ξ∂η 2 . The Legendre transformation (15.2.3.5), with J ≠ 0, allows us to rewrite a general second-order equation with two independent variables F x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2 = 0 (15.2.3.7) in the form F ∂u ∂ξ , ∂u ∂η , ξ ∂u ∂ξ + η ∂u ∂η – u, ξ, η, J ∂ 2 u ∂η 2 ,–J ∂ 2 u ∂ξ∂η , J ∂ 2 u ∂ξ 2 = 0.(15.2.3.8) Sometimes equation (15.2.3.8) may be simpler than (15.2.3.7). Let u = u(ξ,η) be a solution of equation (15.2.3.8). Then the formulas (15.2.3.5) define the corresponding solution of equation (15.2.3.7) in parametric form. Remark. The Legendre transformation may result in the loss of solutions for which J = 0. Example 1. The equation of steady-state transonic gas flow a ∂w ∂x ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = 0 is reduced by the Legendre transformation (15.2.3.5) to the linear equation with variable coefficients aξ ∂ 2 u ∂η 2 + ∂ 2 u ∂ξ 2 = 0. Example 2. The Legendre transformation (15.2.3.5) reduces the nonlinear equation f ∂w ∂x , ∂w ∂y ∂ 2 w ∂x 2 + g ∂w ∂x , ∂w ∂y ∂ 2 w ∂x∂y + h ∂w ∂x , ∂w ∂y ∂ 2 w ∂y 2 = 0 to the following linear equation with variable coefficients: f(ξ,η) ∂ 2 u ∂η 2 – g(ξ, η) ∂ 2 u ∂ξ∂η + h(ξ, η) ∂ 2 u ∂ξ 2 = 0. 15.2.3-3. Euler transformation. The Euler transformation belongs to the class of contact transformations and is defined by the relations x = ∂u ∂ξ , y = η, w = xξ – u.(15.2.3.9) Differentiating the last relation in (15.2.3.9) with respect to x and y and taking into account the other two relations, we find that ∂w ∂x = ξ, ∂w ∂y =– ∂u ∂η .(15.2.3.10) 662 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Differentiating these expressions in x and y,wefind the second derivatives: w xx = 1 u ξξ , w xy =– u ξη u ξξ , w yy = u 2 ξη – u ξξ u ηη u ξξ .(15.2.3.11) The subscripts indicate the corresponding partial derivatives. The Euler transformation (15.2.3.9) is employed in finding solutions and linearization of certain nonlinear partial differential equations. The Euler transformation (15.2.3.9) allows us to reduce a general second-order equation with two independent variables F x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2 = 0 (15.2.3.12) to the equation F u ξ , η,ξu ξ – u, ξ,–u η , 1 u ξξ ,– u ξη u ξξ , u 2 ξη – u ξξ u ηη u ξξ = 0.(15.2.3.13) In some cases, equation (15.2.3.13) may become simpler than equation (15.2.3.12). Let u = u(ξ, η) be a solution of equation (15.2.3.13). Then formulas (15.2.3.9) define the corresponding solution of equation (15.2.3.12) in parametric form. Remark. The Euler transformation may result in the loss of solutions for which w xx = 0. Example 3. The nonlinear equation ∂w ∂y ∂ 2 w ∂x 2 + a = 0 is reduced by the Euler transformation (15.2.3.9)–(15.2.3.11) to the linear heat equation ∂u ∂η = a ∂ 2 u ∂ξ 2 . Example 4. The nonlinear equation ∂ 2 w ∂x∂y = a ∂w ∂y ∂ 2 w ∂x 2 (15.2.3.14) can be linearized with the help of the Euler transformation (15.2.3.9)–(15.2.3.11) to obtain ∂ 2 u ∂ξ∂η = a ∂u ∂η . Integrating this equation yields the general solution u = f (ξ)+g(η)e aξ , (15.2.3.15) where f(ξ)andg(η) are arbitrary functions. Using (15.2.3.9) and (15.2.3.15), we obtain the general solution of the original equation (15.2.3.14) in parametric form: w = xξ – f(ξ)–g(y)e aξ , x = f ξ (ξ)+ag(y)e aξ . Remark. In the degenerate case a = 0, the solution w = ϕ(y)x + ψ(y)islost,whereϕ(y)andψ(y)are arbitrary functions; see also the previous remark. 15.2. TRANSFORMATIONS OF EQUATIONS OF MAT HE M AT I C AL PHYSICS 663 15.2.3-4. Legendre transformation with many variables. For a function of many variables w = w(x 1 , , x n ), the Legendre transformation and its inverse are defined as Legendre transformation Inverse Legendre transformation x 1 = X 1 , , x k–1 = X k–1 , X 1 = x 1 , , X k–1 = x k–1 , x k = ∂W ∂X k , , x n = ∂W ∂X n , X k = ∂w ∂x k , , X n = ∂w ∂x n , w(x)= n i=k X i ∂W ∂X i – W (X), W (X)= n i=k x i ∂w ∂x i – w(x), where x = {x 1 , , x n }, X = {X 1 , , X n }, and the derivatives are related by ∂w ∂x 1 =– ∂W ∂X 1 , , ∂w ∂x k–1 =– ∂W ∂X k–1 . 15.2.4. B ¨ acklund Transformations. Differential Substitutions 15.2.4-1. B ¨ acklund transformations for second-order equations. Let w = w(x, y) be a solution of the equation F 1 x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2 = 0,(15.2.4.1) and let u = u(x, y) be a solution of another equation F 2 x, y, u, ∂u ∂x , ∂u ∂y , ∂ 2 u ∂x 2 , ∂ 2 u ∂x∂y , ∂ 2 u ∂y 2 = 0.(15.2.4.2) Equations (15.2.4.1) and (15.2.4.2) are said to be related by the B ¨ acklund transformation Φ 1 x, y, w, ∂w ∂x , ∂w ∂y , u, ∂u ∂x , ∂u ∂y = 0, Φ 2 x, y, w, ∂w ∂x , ∂w ∂y , u, ∂u ∂x , ∂u ∂y = 0 (15.2.4.3) if the compatibility of the pair (15.2.4.1), (15.2.4.3) implies equation (15.2.4.2), and the compatibility of the pair (15.2.4.2), (15.2.4.3) implies (15.2.4.1). If, for some specific solution u = u(x, y) of equation (15.2.4.2), one succeeds in solving equations (15.2.4.3) for w = w(x, y), then this function w = w(x, y) will be a solution of equation (15.2.4.1). B ¨ acklund transformations may preserve the form of equations* (such transformations are used for obtaining new solutions) or establish relations between solutions of different equations (such transformations are used for obtaining solutions of one equation from solutions of another equation). * In such cases, these are referred to as auto-B ¨ acklund transformations. 664 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Example 1. The Burgers equation ∂w ∂t = w ∂w ∂x + ∂ 2 w ∂x 2 (15.2.4.4) is related to the linear heat equation ∂u ∂t = ∂ 2 u ∂x 2 (15.2.4.5) by the B ¨ acklund transformation ∂u ∂x – 1 2 uw = 0, ∂u ∂t – 1 2 ∂(uw) ∂x = 0. (15.2.4.6) Eliminating w from (15.2.4.6), we obtain equation (15.2.4.5). Conversely, let u(x, t) be a nonzero solution of the heat equation (15.2.4.5). Dividing (15.2.4.5) by u, differentiating the resulting equation with respect to x, and taking into account that (u t /u) x =(u x /u) t ,we obtain u x u t = u xx u x .(15.2.4.7) From the first equation in (15.2.4.6) we have u x u = w 2 =⇒ u xx u – u x u 2 = w x 2 =⇒ u xx u = w x 2 + 1 4 w 2 .(15.2.4.8) Replacing the expressions in parentheses in (15.2.4.7) with the right-hand sides of the first and the last relation (15.2.4.8), we obtain the Burgers equation (15.2.4.4). Example 2. Let us demonstrate that Liouville’s equation ∂ 2 w ∂x∂y = e λw (15.2.4.9) is connected with the linear wave equation ∂ 2 u ∂x∂y = 0 (15.2.4.10) by the B ¨ acklund transformation ∂u ∂x – ∂w ∂x = 2k λ exp 1 2 λ(w + u) , ∂u ∂y + ∂w ∂y =– 1 k exp 1 2 λ(w – u) , (15.2.4.11) where k ≠ 0 is an arbitrary constant. Indeed, let us differentiate the first relation of (15.2.4.11) with respect to y and the second equation with respect to x. Then, taking into account that u yx = u xy and w yx = w xy and eliminating the combinations of the first derivatives using (15.2.4.11), we obtain ∂ 2 u ∂x∂y – ∂ 2 w ∂x∂y = k exp 1 2 λ(w + u) ∂u ∂y + ∂w ∂y = – exp(λw), ∂ 2 u ∂x∂y + ∂ 2 w ∂x∂y = λ 2k exp 1 2 λ(w – u) ∂u ∂x – ∂w ∂x = exp(λw). (15.2.4.12) Adding relations (15.2.4.12) together, we get the linear equation (15.2.4.10). Subtracting the latter equation from the former gives the nonlinear equation (15.2.4.9). Example 3. The nonlinear heat equation with a exponential source w xx + w yy = ae βw is connected with the Laplace equation u xx + u yy = 0 15.2. TRANSFORMATIONS OF EQUATIONS OF MAT HE M AT I C AL PHYSICS 665 by the B ¨ acklund transformation u x + 1 2 βw y = 1 2 aβ 1/2 exp 1 2 βw sin u, u y – 1 2 βw x = 1 2 aβ 1/2 exp 1 2 βw cos u. This fact can be proved in a similar way as in Example 2. Remark 1. It is significant that unlike the contact transformations, the B ¨ acklund transformations are determined by the specific equations (a B ¨ acklund transformation that relates given equations does not always exist). Remark 2. For two nth-order evolution equations of the forms ∂w ∂t = F 1 x, w, ∂w ∂x , , ∂ n w ∂x n , ∂u ∂t = F 2 x, u, ∂u ∂x , , ∂ n u ∂x n , aB ¨ acklund transformation is sometimes sought in the form Φ x, w, ∂w ∂x , , ∂ m w ∂x m , u, ∂u ∂x , , ∂ k u ∂x k = 0 containing derivatives in only one variable x (the second variable, t, is present implicitly through the functions w, u). This transformation can be regarded as an ordinary differential equation in one of the dependent variables. 15.2.4-2. Nonlocal transformations based on conservation laws. Consider a differential equation written as a conservation law, ∂ ∂x F w, ∂w ∂x , ∂w ∂y , + ∂ ∂y G w, ∂w ∂x , ∂w ∂y , = 0.(15.2.4.13) The transformation dz = F(w,w x , w y , ) dy – G(w, w x , w y , ) dx, dη = dy (15.2.4.14) dz = ∂z ∂x dx + ∂z ∂y dy =⇒ ∂z ∂x =–G, ∂z ∂y = F determines the passage from the variables x and y to the new independent variables z and η according to the rule ∂ ∂x =–G ∂ ∂z , ∂ ∂y = ∂ ∂η + F ∂ ∂z . Here, F and G are the same as in (15.2.4.13). The transformation (15.2.4.14) preserves the order of the equation under consideration. Remark. Often one may encounter transformations (15.2.4.14) that are supplemented with a transforma- tion of the unknown function in the form u = ϕ(w). Example 4. Consider the nonlinear heat equation ∂w ∂t = ∂ ∂x f(w) ∂w ∂x , (15.2.4.15) which represents a special case of equation (15.2.4.13) for y = t, F = f (w)w x ,andG =–w. 666 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS In this case, transformation (15.2.4.14) has the form dz = wdx+[f(w)w x ] dt, dη = dt (15.2.4.16) and determines a transformation from the variables x and y to the new independent variables z and η according to the rule ∂ ∂x = w ∂ ∂z , ∂ ∂t = ∂ ∂η +[f(w)w x ] ∂ ∂z . Applying transformation (15.2.4.16) to equation (15.2.4.15), we obtain ∂w ∂η = w 2 ∂ ∂z f(w) ∂w ∂z . (15.2.4.17) The substitution w = 1/u reduces (15.2.4.17) to an equation of the form (15.2.4.15), ∂u ∂η = ∂ ∂z 1 u 2 f 1 u ∂u ∂z . In the special case of f(w)=aw –2 , the nonlinear equation (15.2.4.15) is reduced to the linear equation u η = au zz by the transformation (15.2.4.16). 15.2.5. Differential Substitutions In mathematical physics, apart from the B ¨ acklund transformations, one sometimes resorts to the so-called differential substitutions. For second-order differential equations, differential substitutions have the form w = Ψ x, y, u, ∂u ∂x , ∂u ∂y . A differential substitution increases the order of an equation (if it is inserted into an equation for w) and allows us to obtain solutions of one equation from those of another. The relationship between the solutions of the two equations is generally not invertible and is, in a sense, unilateral. A differential substitution may decrease the order of an equation (if it is inserted into an equation for u). A differential substitution may be obtained as a consequence of a B ¨ acklund transformation (although this is not always the case). A differential substitution may decrease the order of an equation (when the equation for u is regarded as the original one). In general, differential substitutions are defined by formulas (15.2.3.1), where the func- tion X, Y ,andW can be defined arbitrarily. Example 1. Consider once again the Burgers equation (15.2.4.4). The first relation in (15.2.4.6) can be rewritten as the differential substitution (the Hopf–Cole transformation) w = 2u x u .(15.2.5.1) Substituting (15.2.5.1) into (15.2.4.4), we obtain the equation 2u tx u – 2u t u x u 2 = 2u xxx u – 2u x u xx u 2 , which can be converted to ∂ ∂x 1 u ∂u ∂t – ∂ 2 u ∂x 2 = 0.(15.2.5.2) Thus, using formula (15.2.5.1), one can transform each solution of the linear heat equation (15.2.4.5) into a solution of the Burgers equation (15.2.4.4). The converse is not generally true. Indeed, it follows from (15.2.5.2) that a solution of equation (15.2.4.4) generates a solution of the more general equation ∂u ∂t – ∂ 2 u ∂x 2 = f(t)u, where f(t) is a function of t. 15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 667 Example 2. The equation of a steady-state laminar hydrodynamic boundary layer at a flat plate has the form (see Schlichting, 1981) ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = a ∂ 3 w ∂y 3 ,(15.2.5.3) where w is the stream function, x and y are the coordinates along and across the flow, and a is the kinematic viscosity of the fluid. The von Mises transformation (a differential substitution) ξ = x, η = w, u(ξ, η)= ∂w ∂y ,wherew = w(x, y), (15.2.5.4) decreases the order of equation (15.2.5.3) and brings it to the simpler nonlinear heat equation ∂u ∂ξ = a ∂ ∂η u ∂u ∂η .(15.2.5.5) When deriving equation (15.2.5.5), the following formulas for the computation of the derivatives have been used: ∂ ∂y = u ∂ ∂η , ∂ ∂x = ∂ ∂ξ + ∂w ∂x ∂ ∂η , ∂w ∂y = u, ∂ 2 w ∂y 2 = u ∂u ∂η , ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = u ∂u ∂ξ , ∂ 3 w ∂y 3 = u ∂ ∂η u ∂u ∂η . 15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. Similarity Method 15.3.1. Preliminary Remarks There are a number of methods for the construction of exact solutions to equations of mathematical physics that are based on the reduction of the original equations to equations in fewer dependent and/or independent variables. The main idea is to find such variables and, by passing to them, to obtain simpler equations. In particular, in this way, finding exact solutions of some partial differential equations in two independent variables may be reduced to finding solutions of appropriate ordinary differential equations (or systems of ordinary differential equations). Naturally, the ordinary differential equations thus obtained do not give all solutions of the original partial differential equation, but provide only a class of solutions with some specific properties. The simplest classes of exact solutions described by ordinary differential equations involve traveling-wave solutions and self-similar solutions. The existence of such solutions is usually due to the invariance of the equations in question under translations and scaling transformations. Traveling-wave solutions and self-similar solutions often occur in various applications. Below we consider some characteristic features of such solutions. It is assumed that the unknown w depends on two variables, x and t,wheret plays the role of time and x is a spatial coordinate. 15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations 15.3.2-1. General form of traveling-wave solutions. Traveling-wave solutions,bydefinition, are of the form w(x, t)=W (z), z = kx – λt,(15.3.2.1) . (15.2.4.3) for w = w(x, y), then this function w = w(x, y) will be a solution of equation (15.2.4.1). B ¨ acklund transformations may preserve the form of equations* (such transformations are used for. corresponding partial derivatives. The Euler transformation (15.2.3.9) is employed in finding solutions and linearization of certain nonlinear partial differential equations. The Euler transformation. solution of equation (15.2.3.13). Then formulas (15.2.3.9) define the corresponding solution of equation (15.2.3.12) in parametric form. Remark. The Euler transformation may result in the loss of solutions