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668 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS where λ/k plays the role of the wave propagation velocity (the sign of λ can be arbitrary, the value λ = 0 corresponds to a stationary solution, and the value k = 0 corresponds to a space-homogeneous solution). Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time* instants are obtained from one another by appropriate shifts (translations) along the x-axis. Consequently, a Cartesian coordinate system moving with a constant speed can be introduced in which the profile of the desired quantity is stationary. For k > 0 and λ > 0, the wave (15.3.2.1) travels along the x-axis to the right (in the direction of increasing x). A traveling-wave solution is found by directly substituting the representation (15.3.2.1) into the original equation and taking into account the relations w x = kW , w t =–λW ,etc. (the prime denotes a derivative with respect to z). Traveling-wave solutions occur for equations that do not explicitly involve independent variables, F w, ∂w ∂x , ∂w ∂t , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂t , ∂ 2 w ∂t 2 , = 0.(15.3.2.2) Substituting (15.3.2.1) into (15.3.2.2), we obtain an autonomous ordinary differential equa- tion for the function W (z): F (W ,kW ,–λW , k 2 W ,–kλW , λ 2 W , )=0, where k and λ are arbitrary constants. Example 1. The nonlinear heat equation ∂w ∂t = ∂ ∂x f(w) ∂w ∂x (15.3.2.3) admits a traveling-wave solution. Substituting (15.3.2.1) into (15.3.2.3), we arrive at the ordinary differential equation k 2 [f(W )W ] + λW = 0. Integrating this equation twice yields its solution in implicit form: k 2 f(W )dW λW + C 1 =–z + C 2 , where C 1 and C 2 are arbitrary constants. Example 2. Consider the homogeneous Monge–Amp ` ere equation ∂ 2 w ∂x∂t 2 – ∂ 2 w ∂x 2 ∂ 2 w ∂t 2 = 0.(15.3.2.4) Inserting (15.3.2.1) into this equation, we obtain an identity. Therefore, equation (15.3.2.4) admits solutions of the form w = W (kx – λt), where W (z) is an arbitrary function and k and λ are arbitrary constants. 15.3.2-2. Invariance of solutions and equations under translation transformations. Traveling-wave solutions (15.3.2.1) are invariant under the translation transformations x = ¯x + Cλ, t = ¯ t + Ck,(15.3.2.5) where C is an arbitrary constant. * We also use the term traveling-wave solution in the cases where the variable t plays the role of a spatial coordinate. 15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 669 It should be observed that equations of the form (15.3.2.2) are invariant (i.e., preserve their form) under transformation (15.3.2.5); furthermore, these equations are also invariant under general translations in both independent variables: x = ¯x + C 1 , t = ¯ t + C 2 ,(15.3.2.6) where C 1 and C 2 are arbitrary constants. The property of the invariance of specific equations under translation transformations (15.3.2.5) or (15.3.2.6) is inseparably linked with the existence of traveling-wave solutions to such equations (the former implies the latter). Remark 1. Traveling-wave solutions, which stem from the invariance of equations under translations, are simplest invariant solutions. Remark 2. The condition of invariance of equations under translations is not a necessary condition for the existence of traveling-wave solutions. It can be verified directly that the second-order equation F w, w x , w t , xw x + tw t , w xx , w xt , w tt = 0 does not admit transformations of the form (15.3.2.5) and (15.3.2.6) but has an exact traveling-wave solution (15.3.2.1) described by the ordinary differential equation F (W , kW ,–λW , zW , k 2 W ,–kλW , λ 2 W = 0. 15.3.2-3. Functional equation describing traveling-wave solutions. Let us demonstrate that traveling-wave solutions can bedefined as solutions of the functional equation w(x, t)=w(x + Cλ, t + Ck), (15.3.2.7) where k and λ are some constants and C is an arbitrary constant. Equation (15.3.2.7) states that the unknown function does not change under increasing both arguments by proportional quantities, with C being the coefficient of proportionality. For C = 0, equation (15.3.2.7) turns into an identity. Let us expand (15.3.2.7) into a series in powers of C about C = 0, then divide the resulting expression by C, and proceed to the limit as C → 0 to obtain the linear first-order partial differential equation λ ∂w ∂x + k ∂w ∂t = 0. The general solution to this equation is constructed by the method of characteristics (see Paragraph 13.1.1-1) and has the form (15.3.2.1), which was to be proved. 15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations 15.3.3-1. General form of self-similar solutions. Similarity method. By definition, a self-similar solution is a solution of the form w(x, t)=t α U(ζ), ζ = xt β .(15.3.3.1) The profiles of these solutions at different time instants are obtained from one another by a similarity transformation (like scaling). 670 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Self-similar solutions exist if the scaling of the independent and dependent variables, t = C ¯ t, x = C k ¯x, w = C m ¯w,whereC ≠ 0 is an arbitrary constant, (15.3.3.2) for some k and m (|k| + |m| ≠ 0), is equivalent to the identical transformation. This means that the original equation F (x, t, w, w x , w t , w xx , w xt , w tt , )=0,(15.3.3.3) when subjected to transformation (15.3.3.2), turns into the same equation in the new vari- ables, F (¯x, ¯ t, ¯w, ¯w ¯x , ¯w ¯ t , ¯w ¯x¯x , ¯w ¯x ¯ t , ¯w ¯ t ¯ t , )=0.(15.3.3.4) Here, the function F is the same as in the original equation (15.3.3.3); it is assumed that equation (15.3.3.3) is independent of the parameter C. Let us find the connection between the parameters α, β in solution (15.3.3.1) and the parameters k, m in the scaling transformation (15.3.3.2). Suppose w = Φ(x, t)(15.3.3.5) is a solution of equation (15.3.3.3). Then the function ¯w = Φ(¯x, ¯ t)(15.3.3.6) is a solution of equation (15.3.3.4). In view of the explicit form of solution (15.3.3.1), if follows from (15.3.3.6) that ¯w = ¯ t α U(¯x ¯ t β ). (15.3.3.7) Using (15.3.3.2) to return to the new variables in (15.3.3.7), we get w = C m–α t α U C –k–β xt β .(15.3.3.8) By construction, this function satisfies equation (15.3.3.3) and hence is its solution. Let us require that solution (15.3.3.8) coincide with (15.3.3.1), so that the condition for the uniqueness of the solution holds for any C ≠ 0. To this end, we must set α = m, β =–k.(15.3.3.9) In practice, the above existence criterion is checked: if a pair of k and m in (15.3.3.2) has been found, then a self-similar solution is defined by formulas (15.3.3.1) with parame- ters (15.3.3.9). The method for constructing self-similar solutions on the basis of scaling transformations (15.3.3.2) is called the similarity method. It is significant that these transformations involve the arbitrary constant C as a parameter. To make easier to understand, Fig. 15.1 depicts the basic stages for constructing self- similar solutions. 15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 671 Here is a free parameter and , are some numbers C km Look for a self-similar solution Substitute into the original equation Figure 15.1. A simple scheme that is often used in practice for constructing self-similar solutions. 15.3.3-2. Examples of self-similar solutions to mathematical physics equations. Example 1. Consider the heat equation with a nonlinear power-law source term ∂w ∂t = a ∂ 2 w ∂x 2 + bw n . (15.3.3.10) The scaling transformation (15.3.3.2) converts equation (15.3.3.10) into C m–1 ∂ ¯w ∂ ¯ t = aC m–2k ∂ 2 ¯w ∂ ¯x 2 + bC mn ¯w n . Equating the powers of C yields the following system of linear algebraic equations for the constants k and m: m – 1 = m – 2k = mn. This system admits a unique solution: k = 1 2 , m = 1 1–n . Using this solution together with relations (15.3.3.1) and (15.3.3.9), we obtain self-similar variables in the form w = t 1/(1–n) U(ζ), ζ = xt –1/2 . Inserting these into (15.3.3.10), we arrive at the following ordinary differential equation for the function U(ζ): aU ζζ + 1 2 ζU ζ + 1 n – 1 U + bU n = 0. 672 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Example 2. Consider the nonlinear equation ∂ 2 w ∂t 2 = a ∂ ∂x w n ∂w ∂x , (15.3.3.11) which occurs in problems of wave and gas dynamics. Inserting (15.3.3.2) into (15.3.3.11) yields C m–2 ∂ 2 ¯w ∂ ¯ t 2 = aC mn+m–2k ∂ ∂ ¯x ¯w n ∂ ¯w ∂ ¯x . Equating the powers of C results in a single linear equation, m – 2 = mn + m – 2k. Hence, we obtain k = 1 2 mn + 1,wherem is arbitrary. Further, using (15.3.3.1) and (15.3.3.9), we find self-similar variables: w = t m U(ζ), ζ = xt – 1 2 mn–1 (m is arbitrary). Substituting these into (15.3.3.11), one obtains an ordinary differential equation for the function U(ζ). Table 15.1 gives examples of self-similar solutions to some other nonlinear equations of mathematical physics. TABLE 15.1 Some nonlinear equations of mathematical physics that admit self-similar solutions Equation Equation name Form of solutions Determining equation ∂w ∂t = ∂ ∂x f(w) ∂w ∂x Unsteady heat equation w = w(z), z = xt –1/2 [f(w)w ] + 1 2 zw = 0 ∂w ∂t = a ∂ ∂x w n ∂w ∂x +bw k Heat equation with source w = t p u(z), z =xt q , p = 1 1–k , q = k–n–1 2(1–k) a(u n u ) –qzu +bu k –pu = 0 ∂w ∂t = a ∂ 2 w ∂x 2 +bw ∂w ∂x Burgers equation w = t –1/2 u(z), z =xt –1/2 au +buu + 1 2 zu + 1 2 u = 0 ∂w ∂t = a ∂ 2 w ∂x 2 +b ∂w ∂x 2 Potential Burgers equation w = w(z), z = xt –1/2 aw +b(w ) 2 + 1 2 zw = 0 ∂w ∂t = a ∂w ∂x k ∂ 2 w ∂x 2 Filtration equation w = t p u(z), z =xt q , p =– (k+2)q+1 k , q is any a(u ) k u = qzu +pu ∂w ∂t = f ∂w ∂x ∂ 2 w ∂x 2 Filtration equation w = t 1/2 u(z), z =xt –1/2 2f(u )u +zu –u =0 ∂ 2 w ∂t 2 = ∂ ∂x f(w) ∂w ∂x Wave equation w = w(z), z = x/t (z 2 w ) =[f(w)w ] ∂ 2 w ∂t 2 = a ∂ ∂x w n ∂w ∂x Wave equation w = t 2k u(z), z =xt –nk–1 , k is any 2k(2k–1) (nk+1) 2 u+ nk–4k+2 nk+1 zu +z 2 u = a (nk+1) 2 (u n u ) ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = aw n Heat equation with source w = x 2 1–n u(z), z =y/x (1 +z 2 )u – 2(1+n) 1–n zu + 2(1+n) (1–n) 2 u–au n = 0 ∂ 2 w ∂x 2 +a ∂w ∂y ∂ 2 w ∂y 2 = 0 Equation of steady transonic gas flow w = x –3k–2 u(z), z =x k y, k is any a k+1 u u + k 2 k+1 z 2 u –5kzu +3(3k +2)u = 0 ∂w ∂t = a ∂ 3 w ∂x 3 +bw ∂w ∂x Korteweg–de Vries equation w = t –2/3 u(z), z =xt –1/3 au +buu + 1 3 zu + 2 3 u = 0 ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = a ∂ 3 w ∂y 3 Boundary-layer equation w = x λ+1 u(z), z =x λ y, λ is any (2λ+1)(u ) 2 –(λ+1)uu = au The above method for constructing self-similar solutions is also applicable to systems of partial differential equations. Let us illustrate this by a specificexample. 15.3. TRAVELING-WAV E ,SELF-SIMILAR, AND OTHER SIMPLE SOLUTIONS.SIMILARITY METHOD 673 Example 3. Consider the system of equations of a steady-state laminar boundary hydrodynamic boundary layer at a flat plate (see Schlichting, 1981) u ∂u ∂x + v ∂u ∂y = a ∂ 2 u ∂y 2 , ∂u ∂x + ∂v ∂y = 0. (15.3.3.12) Let us scale the independent and dependent variables in (15.3.3.12) according to x = C ¯x, y = C k ¯y, u = C m ¯u, v = C n ¯v. (15.3.3.13) Multiplying these relations by appropriate constant factors, we have ¯u ∂ ¯u ∂ ¯x + C n–m–k+1 ¯v ∂ ¯u ∂ ¯y = C –m–2k+1 a ∂ 2 ¯u ∂ ¯y 2 , ∂ ¯u ∂ ¯x + C n–m–k+1 ∂ ¯v ∂ ¯y = 0. (15.3.3.14) Let us require that the form of the equations of the transformed system (15.3.3.14) coincide with that of the original system (15.3.3.12). This condition results in two linear algebraic equations, n – m – k + 1 = 0 and –2k – m + 1 = 0. On solving them for m and n, we obtain m = 1 – 2k, n =–k, (15.3.3.15) where the exponent k can be chosen arbitrarily. To find a self-similar solution, let us use the procedure outlined in Fig. 15.1. The following renaming should be done: x → y, t → x, w → u (for u)andx → y, t → x, w → v, m → n (for v). This results in u(x, y)=x 1–2k U(ζ), v(x, y)=x –k V (ζ), ζ = yx –k , (15.3.3.16) where k is an arbitrary constant. Inserting (15.3.3.16) into the original system (15.3.3.12), we arrive at a system of ordinary differential equations for U = U(ζ)andV = V (ζ): U (1 – 2k)U – kζU ζ + VU ζ = aU ζζ , (1 – 2k)U – kζU ζ + V ζ = 0. 15.3.3-3. More general approach based on solving a functional equation. The algorithm for theconstruction of a self-similar solution, presented inParagraph 15.3.3-1, relies on representing this solution in the form (15.3.3.1) explicitly. However, there is a more general approach that allows the derivation of relation (15.3.3.1) directly from the condition of the invariance of equation (15.3.3.3) under transformations (15.3.3.2). Indeed, let us assume that transformations (15.3.3.2) convert equation (15.3.3.3) into the same equation (15.3.3.4). Let (15.3.3.5) be a solution of equation (15.3.3.3). Then (15.3.3.6) will be a solution of equation (15.3.3.4). Switching back to the original variables (15.3.3.2) in (15.3.3.6),we obtain w = C m Φ C –k x, C –1 t .(15.3.3.17) By construction, this function satisfies equation (15.3.3.3) and hence is its solution. Let us require that solution (15.3.3.17) coincide with (15.3.3.5), so that the uniqueness condition for the solution is met for any C ≠ 0. This results in the functional equation Φ(x, t)=C m Φ C –k x, C –1 t .(15.3.3.18) For C = 1, equation (15.3.3.18) is satisfied identically. Let us expand (15.3.3.18) in a power series in C about C = 1, then divide the resulting expression by (C – 1), and proceed to the limit as C → 1. This results in a linear first-order partial differential equation for Φ: kx ∂Φ ∂x + t ∂Φ ∂t – mΦ = 0.(15.3.3.19) 674 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS The associated characteristic system of ordinary differential equations (see Paragraph 13.1.1-1) has the form dx kx = dt t = dΦ mΦ . Its first integrals are xt –k = A 1 , t –m Φ = A 2 , where A 1 and A 2 are arbitrary constants. The general solution of the partial differential equation (15.3.3.19) is sought in the form A 2 = U(A 1 ), where U(A) is an arbitrary function (see Paragraph 13.1.1-1). As a result, one obtains a solution of the functional equation (15.3.3.18) in the form Φ(x, t)=t m U(ζ), ζ = xt –k .(15.3.3.20) Substituting (15.3.3.20) into (15.3.3.5) yields the self-similar solution (15.3.3.1) with parameters (15.3.3.9). 15.3.3-4. Some remarks. Remark 1. Self-similar solutions (15.3.3.1) with α = 0 arise in problems with simple initial and boundary conditions of the form w = w 1 at t = 0 (x > 0), w = w 2 at x = 0 (t > 0), where w 1 and w 2 are some constants. Remark 2. Self-similar solutions, which stem from the invariance of equations under scaling transforma- tions, are considered among the simplest invariant solutions. The condition for the existence of a transformation (15.3.3.2) preserving the form of the given equation is sufficient for the existence of a self-similar solution. However, this condition is not necessary: there are equations that do not admit transformations of the form (15.3.3.2) but have self-similar solutions. For example, the equation a ∂ 2 w ∂x 2 + b ∂ 2 w ∂t 2 =(bx 2 + at 2 )f(w) has a self-similar solution w = w(z), z = xt =⇒ w – f(w)=0, but does not admit transformations of the form (15.3.3.2). In this equation, a and b can be arbitrary functions of the arguments x, t, w, w x , w t , w xx , Remark 3. Traveling-wave solutions are closely related to self-similar solutions. Indeed, setting t =lnτ, x =lny in (15.3.2.1), we obtain a self-similar representation of a traveling wave: w = W k ln(yτ –λ/k ) = U(yτ –λ/k ), where U(z)=W (k ln z). 15.3.4. Equations Invariant Under Combinations of Translation and Scaling Transformations, and Their Solutions 15.3.4-1. Exponential self-similar (limiting self-similar) solutions. Exponential self-similar solutions are solutions of the form w(x, t)=e αt V (ξ), ξ = xe βt .(15.3.4.1) . Therefore, equation (15.3.2.4) admits solutions of the form w = W (kx – λt), where W (z) is an arbitrary function and k and λ are arbitrary constants. 15.3.2-2. Invariance of solutions and equations. invariance of equations under scaling transforma- tions, are considered among the simplest invariant solutions. The condition for the existence of a transformation (15.3.3.2) preserving the form of the. Invariance of Equations Under Scaling Transformations 15.3.3-1. General form of self-similar solutions. Similarity method. By definition, a self-similar solution is a solution of the form w(x,
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