Handbook of mathematics for engineers and scienteists part 112 potx

15.10. DIFFERENTIAL CONSTRAINTS METHOD 745 Example. From the class of nonlinear heat equations with a source ∂w ∂t = ∂ ∂x  f 1 (w) ∂w ∂x  + f 2 (w) (15.10.3.1) one singles out equations that admit invariant manifolds of the form ∂ 2 w ∂x 2 = g 1 (w)  ∂w ∂x  2 + g 2 (w). (15.10.3.2) The functions f 2 (w), f 1 (w), g 2 (w), and g 1 (w) are to be determined in the further analysis. Eliminating the second derivative from (15.10.3.1) and (15.10.3.2), we obtain ∂w ∂t = ϕ(w)  ∂w ∂x  2 + ψ(w), (15.10.3.3) where ϕ(w)=f 1 (w)g 1 (w)+f  1 (w), ψ(w)=f 1 (w)g 2 (w)+f 2 (w). (15.10.3.4) The condition of invariance of the manifold (15.10.3.2) under equation (15.10.3.1) is obtained by differen- tiating (15.10.3.2) with respect to t: w xxt = 2g 1 w x w xt + g  1 w 2 x w t + g  2 w t . The derivatives w xxt , w xt ,andw t should be eliminated from this relation with the help of equations (15.10.3.2) and (15.10.3.3) and those obtained by their differentiation. As a result, we get (2ϕg 2 1 + 3ϕ  g 1 + ϕg  1 + ϕ  )w 4 x +(4ϕg 1 g 2 + 5ϕ  g 2 + ϕg  2 – g 1 ψ  – ψg  1 + ψ  )w 2 x + 2ϕg 2 2 + ψ  g 2 – ψg  2 = 0. Equating the coefficients of like powers of w x to zero, one obtains three equations, which, for convenience, may be written in the form (ϕ  + ϕg 1 )  + 2g 1 (ϕ  + ϕg 1 )=0, 4g 2 (ϕ  + ϕg 1 )+(ϕg 2 – ψg 1 )  + ψ  = 0, ϕ =– 1 2 (ψ/g 2 )  . (15.10.3.5) The first equation can be satisfied by taking ϕ  + ϕg 1 = 0. The corresponding particular solution of sys- tem (15.10.3.5) has the form ϕ =– 1 2 μ  , ψ = μg 2 , g 1 =– μ  μ  , g 2 =  2C 1 + C 2 √ |μ|  1 μ  , (15.10.3.6) where μ = μ(w) is an arbitrary function. Taking into account (15.10.3.4), we find the functional coefficients of the original equation (15.10.3.1) and the invariant set (15.10.3.2): f 1 =  C 3 – 1 2 w  μ  , f 2 =(μ – f 1 )g 2 , g 1 =– μ  μ  , g 2 =  2C 1 + C 2 √ |μ|  1 μ  . (15.10.3.7) Equation (15.10.3.2), together with (15.10.3.7), admits the first integral w 2 x =  4C 1 μ + 4C 2  |μ| + 2σ  t (t)  1 (μ  ) 2 , (15.10.3.8) where σ(t) is an arbitrary function. Let us eliminate w 2 x from (15.10.3.3) by means of (15.10.3.8) and substitute the functions ϕ and ψ from (15.10.3.6) to obtain the equation μ  w t =–C 2  |μ| – σ  t (t). (15.10.3.9) Let us dwell on the special case C 2 = C 3 = 0. Integrating equation (15.10.3.9) and taking into account that μ t = μ  w t yield μ =–σ(t)+θ(x), (15.10.3.10) where θ(x) is an arbitrary function. Substituting (15.10.3.10) into (15.10.3.8) and taking into account the relation μ x = μ  w x , we obtain θ 2 x – 4C 1 θ = 2σ t – 4C 1 σ. Equating both sides of this equation to zero and integrating the resulting ordinary differential equations, we find the functions on the right-hand side of (15.10.3.10): σ(t)=A exp(2C 1 t), θ(x)=C 1 (x + B) 2 ,(15.10.3.11) where A and B are arbitrary constants. Thus, an exact solution of equation (15.10.3.1) with the functions f 1 and f 2 from (15.10.3.7) can be represented in implicit form for C 2 = C 3 = 0 as follows: μ(w)=–A exp(2C 1 t)+C 1 (x + B) 2 . In the solution and the determining relations (15.10.3.7), the function μ(w) can be chosen arbitrarily. 746 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 15.10 Second-order differential constraints corresponding to some classes of exact solutions representable in explicit form No. Type of solution Structure of solution Differential constraints 1 Additive separable solution w = ϕ(x)+ψ(y) w xy = 0 2 Multiplicative separable solution w = ϕ(x)ψ(y) ww xy – w x w y = 0 3 Generalized separable solution w = ϕ(x)y 2 + ψ(x)y + χ(x) w yy – f(x)=0 4 Generalized separable solution w = ϕ(x)ψ(y)+χ(x) w yy – f(y)w y = 0 w xy – g(x)w y = 0 5 Functional separable solution w = f (z), z = ϕ(x)y + ψ(x) w yy – g(w)w 2 y = 0 6 Functional separable solution w = f (z), z = ϕ(x)+ψ(y) ww xy – g(w)w x w y = 0 15.10.4. Connection Between the Differential Constraints Method and Other Methods The differential constraints method is one of the most general methods for the construction of exact solutions to nonlinear partial differential equations. Many other methods can be treated as its particular cases.* 15.10.4-1. Generalized/functional separation of variables vs. differential constraints. Table 15.10 lists examples of second-order differential constraints that are essentially equiv- alent to most common forms of separable solutions. For functionalseparable solutions (rows 5 and 6), the function g can be expressed through f . Searching for a generalized separable solution of the form w(x, y)=ϕ 1 (x)ψ 1 (y)+···+ ϕ n (x)ψ n (y), with2n unknown functions, is equivalent to prescribing a differential constraint of order 2n; in general, the number of unknown functions ϕ i (x), ψ i (y) corresponds to the order of the differential equation representing the differential constraint. For the types of solutions listed in Table 15.10, it is preferable to use the methods of generalized and functional separation of variables, since these methods require less steps where it is necessary to solve intermediate differential equations. Furthermore, the method of differential constraints is ill-suited for the construction of exact solutions of third- and higher-order equations since they lead to cumbersome computations and rather complex equations (often, the original equations are simpler). Remark. The direct specification of a solution structure, on which the methods of generalized and functional separation of variables are based, may be treated as the use of a zeroth-order differential constraint. 15.10.4-2. Generalized similarity reductions and differential constraints. Consider a generalized similarity reduction based on a prescribed form of the desired solution, w(x, t)=F  x, t, u(z)  , z = z(x, t), (15.10.4.1) * The basic difficulty in applying the differential constraints method is due to the great generality of its statements and the necessity of selecting differential constraints suitable for specific classes of equations. This is why for the construction of exact solutions of nonlinear equations, it is often preferable to use simpler (but less general) methods. 15.10. DIFFERENTIAL CONSTRAINTS METHOD 747 where F (x, t, u)andz(x, t) should be selected so as to obtain ultimately a single ordinary differential equation for u(z); see Remark 3 in Subsection 15.7.1-2. Let us show that employing the solution structure (15.10.4.1) is equivalent to searching for a solution with the help of a first-order quasilinear differential constraint ξ(x, t) ∂w ∂t + η(x, t) ∂w ∂x = ζ(x, t, w). (15.10.4.2) Indeed, first integrals of the characteristic system of ordinary differential equations dt ξ(x, t) = dx η(x, t) = dw ζ(x, t, w) have the form z(x, t)=C 1 , ϕ(x, t, w)=C 2 ,(15.10.4.3) where C 1 and C 2 are arbitrary constants. Therefore, the general solution of equation (15.10.4.2) can be written as follows: ϕ(x, t, w)=u  z(x, t)  ,(15.10.4.4) where u(z) is an arbitrary function. On solving (15.10.4.4) for w, we obtain a representation of the solution in the form (15.10.4.1). 15.10.4-3. Nonclassical method of symmetry reductions and differential constraints. The nonclassical method of symmetry reductions can be restated in terms of the differential constraints method. This can be demonstrated by the following example with a general second-order equation F  x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2  = 0.(15.10.4.5) Let us supplement equation (15.10.4.5) with two differential constraints ξ ∂w ∂x + η ∂w ∂y = ζ,(15.10.4.6) ξ ∂F ∂x +η ∂F ∂y +ζ ∂F ∂w +ζ 1 ∂F ∂w x +ζ 2 ∂F ∂w y +ζ 11 ∂F ∂w xx +ζ 12 ∂F ∂w xy +ζ 22 ∂F ∂w yy = 0,(15.10.4.7) where ξ = ξ(x, y, w), η = η(x, y, w), and ζ = ζ(x, y, w) are unknown functions, and the coordinates of the first and the second prolongations ζ i and ζ ij are defined by formu- las (15.8.1.9) and (15.8.1.14). The differential constraint (15.10.4.7) coincides with the invariance condition for equation (15.10.4.5); see (15.8.2.3)–(15.8.2.4). The method for the construction of exact solutions to equation (15.10.4.5) based on using the first-order partial differential equation (15.10.4.6) and the invariance condi- tion (15.10.4.7) corresponds to the nonclassical method of symmetry reductions (see Sub- section 15.9.1). 748 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.11. Painlev ´ e Test for Nonlinear Equations of Mathematical Physics 15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method Description Basic idea: Solutions of partial differential equations are sought in the form of series expansions containing a movable pole singularity. The position of the pole is defined by an arbitrary function. 15.11.1-1. Simple scheme for studying nonlinear partial differential equations. For a clearer exposition, we will be considering equations of mathematical physics in two independent variables, x and t, and one dependent variable, w, explicitly independent of x and t. A solution sought in a small neighborhood of a manifold x – x 0 (t)=0 is the form of the following series expansion (Jimbo, Kruskal, and Miwa, 1982): w(x, t)= 1 ξ p ∞  m=0 w m (t)ξ m , ξ = x – x 0 (t). (15.11.1.1) Here, the exponent p is a positive integer, so that the movable singularity is of the pole type. The function x 0 (t) is assumed arbitrary, and the w m are assumed to depend on derivatives of x 0 (t). The representation (15.11.1.1) is substituted into the given equation. The exponent p and the leading term u 0 (t)arefirst determined from the balance of powers in the expansion. Terms with like powers of ξ are further collected. In the resulting polynomial, the coef- ficients of the different powers of ξ are all equated to zero to obtain a system of ordinary differential equations for the functions w m (t). The solution obtained is general if the expansion (15.11.1.1) involves arbitrary functions, with the number of them equal to the order of the equation in question. 15.11.1-2. General scheme for analysis of nonlinear partial differential equations. A solution of a partial differential equation is sought in a neighborhood of a singular manifold ξ(x, t)=0 in the form of a generalized series expansion symmetric in the independent variables (Weiss, Tabor, and Carnevalle, 1983): w(x, t)= 1 ξ p ∞  m=0 w m (x, t)ξ m , ξ = ξ(x, t), (15.11.1.2) where ξ t ξ x ≠ 0. Here and henceforth, the subscripts x and t denote partial derivatives; the function ξ(x, t) is assumed arbitrary and the w m are assumed to be dependent on derivatives of ξ(x, t). Expansion (15.11.1.1) is a special case of expansion (15.11.1.2), when the equation of the singular manifold has been resolved for x. The requirement that there are no movable singularities implies that p is a positive integer. The solution will be general if the number of arbitrary functions appearing in the coefficients w m (x, t) and the expansion variable ξ(x, t) coincides with the order of the equation. 15.11. PAINLEV ´ E TEST FOR NONLINEAR EQUATIONS OF MAT HE M AT I C A L PHYSICS 749 Substituting (15.11.1.2) into the equation, collect terms with like powers of ξ,and equating the coefficients to zero, one arrives at the following recurrence relations for the expansion coefficients: k m w m = Φ m (w 0 , w 1 , , w m–1 , ξ t , ξ x , ). (15.11.1.3) Here, k m are polynomials of degree n with integer argument m of the form k m =(m + 1)(m – m 1 )(m – m 2 ) (m – m n–1 ), (15.11.1.4) where n is the order of the equation concerned. If the roots of the polynomial, m 1 , m 2 , , m n–1 , called Fuchs indices (resonances), are all nonnegative integers and the consistency conditions Φ m=m j = 0 (j = 1, 2, , n – 1)(15.11.1.5) hold, the equation is said to pass the Painlev ´ etest. Equations that pass the Painlev ´ etest are often classified as integrable, which is supported by the fact that such equations are reducible to linear equations in many known cases. 15.11.1-3. Basic steps of the Painlev ´ e test for nonlinear equations. For nonlinear equations of mathematical physics, the Painlev ´ e test is convenient to carry out in several steps. At the first and second steps, one determines the leading term in the expansion (15.11.1.1) and the Fuchs indices; this allows to verify the necessary conditions for the Painlev ´ e test without making full computations. For the sake of clarity, the basic steps in performing the Painlev ´ e test for nonlinear equations, using the expansion (15.11.1.1), are shown in Figure 15.7. Remark. An equation fails to pass the Painlev ´ e test if any of the following conditions holds: p < 0, p is noninteger, p is complex, m j < 0, m j is noninteger, or m j is complex (at least for one j,wherej = 1, , n–1). 15.11.1-4. Some remarks. Truncated expansions. The numerous researchers have established that many known integrable nonlinear equations of mathematical physics pass the Painlev ´ e test. New equations possessing this property have also been found. As a simple check whether a specific equation passes the Painlev ´ e test, one may use the simple scheme based on the expansion (15.11.1.1). The associated important technical simplifications as compared with the expansion (15.11.1.2) are due to the fact that (w m ) x = 0 and ξ x = 1. The general expansion (15.11.1.2), involving more cumbersome but yet more informa- tive computations, can prove useful after the Painlev ´ e property has been established at the simple check. It may help reveal many important properties of equations and their solutions. In some cases, a truncated expansion, w = w 0 ξ p + w 1 ξ p–1 + ···+ w p ,(15.11.1.6) can be useful for constructing exact solutions and finding a B ¨ acklund transformation lin- earizing the original equation. This expansion corresponds to zero values of the expansion coefficients w m with m > p in (15.11.1.2); see Examples 1 and 2 in Subsection 15.11.2. The Painlev ´ e test for a nonlinearpartial differential equation can be performed forspecial classes of its exact solutions, usually traveling-wave solutions and self-similar solutions, which are determined by ordinary differential equations. If the ordinary differential equation obtained fails the Painlev ´ e test, then the original partial differential equation also fails the test. If the ordinary differential equation passes the Painlev ´ e test, then the original partial differential equation normally also passes the test. 750 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS First step: look for the leading term of expansion (15.11.1.2) Look for Fuchs indices (resonances) Substitute expansion (15.11.1.2) into the equation Check the consistency conditions (15.11.1.5) Figure 15.7. Basic steps of the Painlev ´ e test for nonlinear equations of mathematical physics. It is assumed that ξ = x – x 0 (t) for the simple scheme and ξ = ξ(x, t) for the general scheme. 15.11.2. Examples of Performing the Painlev ´ e Test and Truncated Expansions for Studying Nonlinear Equations This Subsection treats some common nonlinear equations ofmathematical physics. For their analysis, the simple scheme of the Painlev ´ e test will be used first; this scheme is based on the expansion (15.11.1.1) [see also the scheme in Fig. 15.7 with ξ = x–x 0 (t)]. The truncated expansion (15.11.1.6) will then be used for constructing B ¨ acklund transformations. 15.11. PAINLEV ´ E TEST FOR NONLINEAR EQUATIONS OF MAT HE M AT I C A L PHYSICS 751 Example 1. Consider the Burgers equation ∂w ∂t + w ∂w ∂x = ν ∂ 2 w ∂x 2 . (15.11.2.1) First step. Substitute the leading term of expansion (15.11.1.1) in equation (15.11.2.1) and multiply the resulting relation by ξ p+2 (the product ξ p+2 w xx is equal to unity by order of magnitude). This results in w  0 ξ 2 + pw 0 x  0 ξ – pw 2 0 ξ 1–p = νp(p + 1)w 0 , where ξ = x – x 0 , x 0 = x 0 (t), w 0 = w 0 (t), and the prime denotes a derivative with respect to t.Wefind from the balance of the leading terms, which corresponds to dropping two leftmost terms, that p = 1, w 0 =–2ν (m = 0). (15.11.2.2) Since p is a positive integer, the first necessary condition of the Painlev ´ e test is satisfied. Second step. In order to find the Fuchs indices (resonances), we substitute the binomial w =–2νξ –1 + w m ξ m–1 into the leading terms ww x and νw xx of equation (15.11.2.1). Collecting coefficients of like powers of w m , we get ν(m + 1)(m – 2)w m ξ m–3 + ···= 0. Equating (m + 1)(m – 2) to zero yields the Fuchs index m 1 = 2. Since it is a positive integer, the second necessary condition of the Painlev ´ e test is satisfied. Third step. We substitute the expansion (15.11.1.1) (according to the second step, we have to consider terms up to number m = 2 inclusive), w =–2νξ –1 + w 1 + w 2 ξ + ··· , into the Burgers equation (15.11.2.1), collect terms of like powers of ξ = x – x 0 (t), and then equate the coefficients of the different powers to zero to obtain a system of equations for the w m : ξ –2 : 2ν(w 1 – x  0 )=0, ξ –1 : 0×w 2 = 0. (15.11.2.3) From the second relation in (15.11.2.3) it follows that the function w 2 = w 2 (t) can be chosen arbitrarily. Therefore the Burgers equation (15.11.2.1) passes the Painlev ´ e test and its solution has two arbitrary functions, x 0 = x 0 (t)andw 2 = w 2 (t), as required. It follows from the first relation in (15.11.2.3) that w 1 = x  0 (t). The solution to equation (15.11.2.1) can be written as w(x, t)=– 2ν x – x 0 (t) + x  0 (t)+w 2 (t)[x – x 0 (t)] 2 + ··· , where x 0 (t)andw 2 (t) are arbitrary functions. Cole–Hopf transformation. For further analysis of the Burgers equation (15.11.2.1), we use a truncated expansion of the general form (15.11.1.6) with p = 1: w = w 0 ξ + w 1 , (15.11.2.4) where w 0 = w 0 (x, t), w 1 = w 1 (x, t), and ξ = ξ(x, t). Substitute (15.11.2.4) in (15.11.2.1) and collect the terms of equal powers in ξ to obtain ξ –3 w 0 ξ x  w 0 + 2νξ x  + ξ –2  w 0 ξ t – w 0 (w 0 ) x + w 0 w 1 ξ x – 2ν(w 0 ) x ξ x – νw 0 ξ xx  + ξ –1  –(w 0 ) t – w 0 (w 1 ) x – w 1 (w 0 ) x + ν(w 0 ) xx  –(w 1 ) t – w 1 (w 1 ) x + ν(w 1 ) xx = 0, where the subscripts x and t denote partial derivatives. Equating the coefficients of like powers of ξ to zero, we get the system of equations w 0 + 2νξ x = 0, w 0 ξ t – w 0 (w 0 ) x + w 0 w 1 ξ x – 2ν(w 0 ) x ξ x – νw 0 ξ xx = 0, (w 0 ) t + w 0 (w 1 ) x + w 1 (w 0 ) x – ν(w 0 ) xx = 0, (w 1 ) t + w 1 (w 1 ) x – ν(w 1 ) xx = 0, (15.11.2.5) . generality of its statements and the necessity of selecting differential constraints suitable for specific classes of equations. This is why for the construction of exact solutions of nonlinear. common forms of separable solutions. For functionalseparable solutions (rows 5 and 6), the function g can be expressed through f . Searching for a generalized separable solution of the form w(x,. w), η = η(x, y, w), and ζ = ζ(x, y, w) are unknown functions, and the coordinates of the first and the second prolongations ζ i and ζ ij are defined by formu- las (15.8.1.9) and (15.8.1.14). The