766 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.13. Conservation Laws and Integrals of Motion 15.13.1. Basic Definitions and Examples 15.13.1-1. General form of conservation laws. Consider a partial differential equation with two independent variables F x, t, w, ∂w ∂x , ∂w ∂t , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂t , ∂ 2 w ∂x 2 , = 0.(15.13.1.1) A conservation law for this equation has the form ∂T ∂t + ∂X ∂x = 0,(15.13.1.2) where T = T x, t, w, ∂w ∂x , ∂w ∂t , , X = X x, t, w, ∂w ∂x , ∂w ∂t , .(15.13.1.3) The left-hand side of the conservation law (15.13.1.2) must vanish for all (sufficiently smooth) solutions of equation (15.13.1.1). In simplest cases, the substitution of rela- tions (15.13.1.3) into the conservation law (15.13.1.2) followed by differentiation and ele- mentary transformations leads to a relation that coincides with (15.13.1.1) up to a functional factor. The quantities T and X in (15.13.1.2) are called a density and a flow, respectively. The density and the flow in the conservation law (15.13.1.2) are not uniquely determined; they can be changed according to the rules T =⇒ T + ϕ(x), X =⇒ X + ψ(t), where ϕ(x) and ψ(t) are arbitrary functions, or, in general, by the rules T =⇒ aT + ∂Φ ∂x , X =⇒ aX – ∂Φ ∂t , where Φ = Φ(x, t) is an arbitrary function of two variables and a ≠ 0 is any number. For nonstationary equations with n spatial variables x 1 , , x n , conservation laws have the form ∂T ∂t + n k=1 ∂X k ∂x k = 0. Partial differential equations can have several (sometimes infinitely many) conservation laws or none at all. 15.13.1-2. Integrals of motion. If the total variation of the quantity X on the interval a ≤ x ≤ b is equal to zero, which means that the relation X(a)=X(b) holds, then the following “integral of motion” takes place: b a Tdx= const (for all t). (15.13.1.4) For many specific equations, relations of the form (15.13.1.4) have a clear physical meaning and are used for approximate analytical solution of the corresponding problems, as well as for the verification of results obtained by numerical methods. 15.13. CONSERVATION LAWS AND INTEGRALS OF MOTION 767 15.13.1-3. Conservation laws for some nonlinear equations of mathematical physics. Example 1. The Korteweg–de Vries equation ∂w ∂t + ∂ 3 w ∂x 3 – 6w ∂w ∂x = 0 admits infinitely many conservation laws of the form (15.13.1.2). The first three are determined by T 1 = w, X 1 = w xx – 3w 2 ; T 2 = w 2 , X 2 = 2ww xx – w 2 x – 4w 3 ; T 3 = w 2 x + 2w 3 , X 3 = 2w x w xxx – w 2 xx + 6w 2 w xx – 12ww 2 x – 9w 4 , where the subscripts denote partial derivatives with respect to x. Example 2. The sine-Gordon equation ∂ 2 w ∂x∂t –sinw = 0 also has infinitely many conservation laws. The first three are described by the formulas T 1 = w 2 x , X 1 = 2 cos w; T 2 = w 4 x – 4w 2 xx , X 2 = 4w 2 x cos w; T 3 = 3w 6 x – 12w 2 x w 2 xx + 16w 3 x w xxx + 24w 2 xxx , X 3 =(2w 4 x – 24w 2 xx )cosw. Example 3. The nonhomogeneous Monge–Amp ` ere equation ∂ 2 w ∂x∂y 2 – ∂ 2 w ∂x 2 ∂ 2 w ∂y 2 = F (x, y), where F (x, y) is an arbitrary function, admits the conservation law ∂ ∂x ∂w ∂x ∂ 2 w ∂y 2 + ∂ ∂y – ∂w ∂x ∂ 2 w ∂x∂y + y a F (x, z) dz = 0. 15.13.2. Equations Admitting Variational Formulation. Noetherian Symmetries 15.13.2-1. Lagrangian. Euler–Lagrange equation. Noetherian symmetries. Here we consider second-order equations in two independent variables F x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂x 2 = 0 (15.13.2.1) that admit the variational formulation of minimizing a functional of the form Z[w]= S L(x, y, w, w x , w y ) dx dy.(15.13.2.2) The function L = L(x, y, w, w x , w y ) is called a Lagrangian. It is well known that a minimum of the functional (15.13.2.2) corresponds to the Euler– Lagrange equation ∂L ∂w – D x ∂L ∂w x – D y ∂L ∂w y = 0,(15.13.2.3) 768 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS where D x and D y are the total differential operators in x and y: D x = ∂ ∂x + w x ∂ ∂w + w xx ∂ ∂w x + w xy ∂ ∂w y , D y = ∂ ∂y + w y ∂ ∂w + w xy ∂ ∂w x + w yy ∂ ∂w y . The original equation (15.13.2.1) must be a consequence of equation (15.13.2.3). A symmetry that preserves the differential form Ω = L(x, y, w, w x , w y ) dx dy is called a Noetherian symmetry of the Lagrangian L. In order to obtain Noetherian symmetries, one should find point transformations ¯x = f 1 (x, y, w, ε), ¯y = f 2 (x, y, w, ε), ¯w = g(x, y,w, ε)(15.13.2.4) such that preserve the differential form, ¯ Ω = Ω, i.e., ¯ Ld¯xd¯y = Ldxdy.(15.13.2.5) Calculating the differentials d¯x, d¯y and taking into account (15.13.2.4), we obtain d¯x = D x f 1 dx, d¯y = D y f 2 dy, and therefore, relation (15.13.2.5) can be rewritten as (L – ¯ LD x f 1 D y f 2 ) dx dy = 0, which is equivalent to L – ¯ LD x f 1 D y f 2 = 0.(15.13.2.6) Let us associate the point transformation (15.13.2.4) with the prolongation operator X = ξ∂ x + η∂ y + ζ∂ w + ζ 1 ∂ w x + ζ 2 ∂ w y ,(15.13.2.7) where the coordinates of the first prolongation, ζ 1 and ζ 2 ,aredefined by formulas (15.8.1.9). Then, by the usual procedure, from (15.13.2.6) one obtains the invariance condition in the form X(L)+L(D x ξ + D y η)=0.(15.13.2.8) Noetherian symmetries are determined by (15.13.2.8). Each Noetherian symmetry operator X generates a conservation law, D x ξL +(ζ – ξw x – ηw y ) ∂L ∂w x + D y ηL +(ζ – ξw x – ηw y ) ∂L ∂w y = 0.(15.13.2.9) 15.13. CONSERVATION LAWS AND INTEGRALS OF MOTION 769 15.13.2-2. Examples of constructing conservation laws using Noetherian symmetries. Example 1. Consider the stationary heat equation with nonlinear source ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 – f(w)=0.(15.13.2.10) This equation admits the variational formulation of minimizing the functional (15.13.2.2) with the La- grangian L = w 2 x + w 2 y + 2F (w), F (w)= f(w) dw.(15.13.2.11) This can be verified by substituting (15.13.2.11) into the Euler–Lagrange equation (15.13.2.3). In (15.13.2.11), it is assumed that F (w) ≥ 0. Substituting (15.13.2.11) into the invariance condition (15.13.2.8) and taking into account (15.13.2.7) and (15.8.1.9), after some rearrangements we obtain a polynomial in the derivatives w x and w y : –ξ w w 3 x – η w w 2 x w y – ξ w w x w 2 y – η w w 3 y +(2ζ w – ξ x + η y )w 2 x – 2(η x + ξ y )w x w y +(2ζ w – η y + ξ x )w 2 y + 2(ζ x + Fξ w )w x + 2(ζ y + Fη w )w y + 2fζ + 2F(ξ x + η y )=0.(15.13.2.12) Equating the functional coefficients of w 3 x , w 2 x w y , w x ,andw y to zero, we find that ξ w = 0, η w = 0, ζ x = 0,and ζ y = 0. Consequently, ξ = ξ(x, y), η = η(x, y), ζ = ζ(w). (15.13.2.13) In (15.13.2.12), equating the coefficients of the remaining powers of the derivatives to zero, we obtain w 2 x : 2ζ w – ξ x + η y = 0, w 2 y : 2ζ w – η y + ξ x = 0, w x w y : η x + ξ y = 0, 1: fζ + F(ξ x + η y )=0. (15.13.2.14) For arbitrary f = f (w), it follows from the last equation in (15.13.2.14) that ζ = 0, ξ x + η y = 0.(15.13.2.15) From the first equation in (15.13.2.14) and the second equation in (15.13.2.15), with ζ = 0, it follows that ξ x = 0, η y = 0 or ξ = ξ(y), η = η(x). Substituting these expressions into the third equation of (15.13.2.14), we get ξ = C 1 y + C 2 , η =–C 1 x + C 3 , where C 1 , C 2 ,andC 3 are arbitrary constants. Therefore, for arbitrary f(w), Noetherian symmetries of the Lagrangian (15.13.2.11) are defined by the three operators X 1 = ∂ x (C 2 = 1, C 1 = C 3 = 0); X 2 = ∂ y (C 3 = 1, C 1 = C 2 = 0); X 3 = y∂ x – x∂ y (C 1 = 1, C 2 = C 3 = 0). (15.13.2.16) In accordance with (15.13.2.9), these operators determine three conservation laws: D x (–w 2 x + w 2 y + 2F + D y –2w x w y = 0 (ξ = 1, η = ζ = 0); D x –2w x w y + D y w 2 x – w 2 y + 2F = 0 (η = 1, ξ = ζ = 0); D x –yw 2 x + yw 2 y + 2xw x w y + 2yF + D y –xw 2 x + xw 2 y – 2yw x w y – 2xF = 0 (ξ = y, η =–x, ζ = 0), (15.13.2.17) with the function F = F(w)defined by (15.13.2.11). Remark 1. The operators (15.13.2.16) could be found by a symmetry analysis of the original differential equation (15.13.2.10), as in Example 1 in Subsection 15.8.2. Remark 2. In the variational formulation of equation (15.13.2.11), it is assumed that F (w) ≥ 0.However, the conservation laws (15.13.2.17) are valid for any F (w). This situation is also typical of other equations; to obtain conservation laws, it usually suffices that the equation in question is representable as the Euler–Lagrange equation (15.13.2.3) (the variational formulation may be unnecessary). 770 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Example 2. The equation of minimal surfaces (1 + w 2 y )w xx – 2w x w y w xy +(1 + w 2 x )w yy = 0 corresponds to the functional (15.13.2.2) with Lagrangian L = 1 + w 2 x + w 2 y . The admissible point operators X 1 = ∂ x , X 2 = ∂ y , X 3 = x∂ x + y∂ y + w∂ w , X 4 = y∂ x – x∂ y , X 5 = ∂ w are found fromthe invariance condition (15.13.2.8), as was the casein Example 1(the procedure is also described in Subsection 15.13.2). These operators determine Noetherian symmetries and correspond to conservation laws: X 1 : D x L – w x ∂L ∂w x + D y –w x ∂L ∂w y = 0, X 2 : D x –w y ∂L ∂w x + D y L – w y ∂L ∂w y = 0, X 3 : D x Lx +(w – xw x – yw y ) ∂L ∂w x + D y Ly +(w – xw x – yw y ) ∂L ∂w y = 0, X 4 : D x Ly +(yw x – xw y ) ∂L ∂w x + D x –Ly +(yw x – xw y ) ∂L ∂w y = 0, X 5 : D x w x 1 + w 2 x + w 2 y + D y w y 1 + w 2 x + w 2 y = 0. 15.14. Nonlinear Systems of Partial Differential Equations 15.14.1. Overdetermined Systems of Two Equations 15.14.1-1. Overdetermined systems of first-order equations in one unknown. Consider an overdetermined system of two quasilinear first-order equations z x = F (x, y, z), z y = G(x, y,z), (15.14.1.1) in one unknown function z = z(x, y). In thegeneral case, the system is unsolvable. To derive anecessary consistency condition for system (15.14.1.1), let us differentiate the first equation with respect to y and the second with respect to x. Eliminate the first derivatives from the resulting relations using the initial equations of the system to obtain z xy = F y + F z z y = F y + GF z , z yx = G x + G z z x = G x + FG z . Equating the expressions of the second derivatives z xy and z yx to each other, we obtain a necessary condition for consistency of system (15.14.1.1): F y + GF z = G x + FG z .(15.14.1.2) Consider two possible situations. First case. Suppose the substitution of the right-hand sides of equations (15.14.1.1) into the necessary condition (15.14.1.2) results in an equation with x, y,andz. Treating it as an algebraic (transcendental) equation for z,wefind z = z(x, y). The direct substitution of the expression z = z(x, y) into both equations (15.14.1.1) gives an answer to the question whether it is a solution of the system in question or not. 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 771 Example 1. Consider the quadratically nonlinear system z x = yz, z y = z 2 + axz. (15.14.1.3) In this case, the consistency condition (15.14.1.2) can be written, after simple rearrangements, in the form z(a – 1 + yz)=0. This equations is not satisfied identically and gives rise to two possible expressions for z: z = 0 or z = 1 – a y . Substituting them into (15.14.1.3), we see that the former is a solution of the system and the latter is not. Second case. Suppose the substitution of the right-hand sides of equations (15.14.1.1) into the necessary condition (15.14.1.2) turns it into an identity. In this case, the system has a family of solutions that depends at least on one arbitrary constant. If the partial derivatives F y , F z , G x , G z are continuous and condition (15.14.1.2) is satisfied identically, then system (15.14.1.1) has a unique solution z = z(x, y) that takes a given value, z = z 0 ,atx = x 0 and y = y 0 . A simple way to find solutions in this case is as follows. The first equation of system (15.14.1.1) is treated as an ordinary differential equation in x with parameter y.Onefinds its solution, where the role of the arbitrary constant is played by an arbitrary function ϕ(y). Substituting this solution into the second equation of system (15.14.1.1), one determines the function ϕ(y). Example 2. Consider the nonlinear system z x = ae y–z , z y = be y–z + 1. (15.14.1.4) In this case, the consistency condition (15.14.1.2) is satisfied identically. To solve the first equation, let us make the change of variable w = e z to obtain the linear equation w x = ae y . Its general solution has the form w = ae y x + ϕ(y), where ϕ(y) is an arbitrary function. Going back to the original variable, we find the general solution of the first equation of system (15.14.1.4): z =ln[ae y x + ϕ(y)]. (15.14.1.5) Substituting this solution into the second equation of system (15.14.1.4), we obtain a linear first-order equation for ϕ = ϕ(y): ϕ y = ϕ + be y . Its general solution is ϕ =(by + C)e y ,whereC is an arbitrary constant. Substituting this solution into (15.14.1.5), we obtain the following solution of system (15.14.1.4): z =ln[ae y x +(by + C)e y ]=y +ln(ax + by + C). 15.14.1-2. Other overdetermined systems of equations in one unknown. Section 15.10 describes a consistency analysis for overdetermined systems of partial differ- ential equations in the context of the differential constraints method; one of the equations there is called a differential constraint. Detailed consideration is given to systems consisting of one second-order equation and one first-order equation (see Subsection 15.10.2), and also to some systems consisting of two second-order equations (see Subsection 15.10.3). Examples of solving such systems are also given. Table 15.11 presents some overdetermined systems of equations with consistency con- ditions; the derivation of the consistency conditions for these systems can be found in Section 15.10. 772 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 15.11 Some overdetermined systems of equations No. First equation of system Second equation of system Consistency conditions 1 ∂w ∂t = ∂ ∂x f(w) ∂w ∂x + g(w) ∂w ∂t = ϕ(w) (15.10.2.8) 2 ∂ 2 w ∂x∂t = f(w) ∂w ∂x = ϕ(t)g(w) (15.10.2.23) 3 ∂w ∂t = ∂ ∂x f 1 (w) ∂w ∂x + f 2 (w) ∂ 2 w ∂x 2 = g 1 (w) ∂w ∂x 2 + g 2 (w) (15.10.3.5) 4 ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = a ∂ 3 w ∂y 3 ∂w ∂x = ϕ(y) (15.10.1.8) 15.14.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems 15.14.2-1. Pfaffian equations. A Pfaffian equation is an equation of the form P (x, y, z) dx + Q(x, y, z) dy + R(x, y, z) dz = 0.(15.14.2.1) Equation (15.14.2.1) always has a solution x = x 0 , y = y 0 , z = z 0 ,wherex 0 , y 0 ,andz 0 are arbitrary constants. Such simple solutions are not considered below. We will distinguish between the following two cases: 1. Find a two-dimensional solution to the Pfaffian equation, when the three variables x, y, z are connected by a single relation (a certain condition must hold for such a solution to exist). 2. Find a one-dimensional solution to the Pfaffian equation, when the three variables x, y, z are connected by two relations. 15.14.2-2. Condition for integrability of the Pfaffian equation by a single relation. Let a solution of the Pfaffian equation be representable in the form z = z(x, y), where z is the unknown function and x, y are independent variables. From equation (15.14.2.1) we find the expression for the differential: dz =– P R dx – Q R dy.(15.14.2.2) On the other hand, since z = z(x, y), we have dz = ∂z ∂x dx + ∂z ∂y dy.(15.14.2.3) Equating the right-hand sides of (15.14.2.2) and (15.14.2.3) to each other and taking into account the independence of the differentials dx and dy, we obtain an overdetermined system of equations of the form (15.14.1.1): z x =–P/R, z y =–Q/R.(15.14.2.4) . NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.13. Conservation Laws and Integrals of Motion 15.13.1. Basic Definitions and Examples 15.13.1-1. General form of conservation laws. Consider a partial. place: b a Tdx= const (for all t). (15.13.1.4) For many specific equations, relations of the form (15.13.1.4) have a clear physical meaning and are used for approximate analytical solution of the corresponding. problems, as well as for the verification of results obtained by numerical methods. 15.13. CONSERVATION LAWS AND INTEGRALS OF MOTION 767 15.13.1-3. Conservation laws for some nonlinear equations of mathematical