570 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS Here, η and ξ are arbitrary numbers and τ > 0. We assume that problem (13.1.4.10), (13.1.4.12) has a unique bounded solution. The stable generalized solution of the Cauchy problem (13.1.4.8), (13.1.4.9) is given by w(x, y – 0)=W x, x, y, ξ – (x, y) , w(x, y + 0)=W x, x, y, ξ + (x, y) , (13.1.4.13) where ξ – (x, y)andξ + (x, y) denote, respectively, the greatest lower bound and the least upper bound of the set of values {ξ = ξ n } for which the function I(x, y, ξ)= ξ 0 ϕ(η)–W (0, x, y, η) dη (13.1.4.14) takes the minimum value for fixed x and y (x >0). If function (13.1.4.14) takes the minimum value for a single ξ = ξ 1 ,thenξ – = ξ + and relation (13.1.4.14) describes the classical smooth solution. 13.2. Nonlinear Equations 13.2.1. Solution Methods 13.2.1-1. Complete, general, and singular integrals. A nonlinear first-order partial differential equation with two independent variables has the general form F (x, y, w, p, q)=0,wherep = ∂w ∂x , q = ∂w ∂y .(13.2.1.1) Such equations are encountered in analytical mechanics, calculus of variations, optimal control, differential games, dynamic programming, geometric optics, differential geometry, and other fields. In this subsection, we consider only smooth solutions w = w(x, y) of equation (13.2.1.1), which are continuously differentiable with respect to both arguments (Subsection 13.2.3 deals with nonsmooth solutions). 1 ◦ . Let a particular solution of equation (13.2.1.1), w = Ξ(x, y, C 1 , C 2 ), (13.2.1.2) depending on two parameters C 1 and C 2 , be known. The two-parameter family of so- lutions (13.2.1.2) is called a complete integral of equation (13.2.1.1) if the rank of the matrix M = Ξ 1 Ξ x1 Ξ y1 Ξ 2 Ξ x2 Ξ y2 (13.2.1.3) is equal to two in the domain being considered (for example, this is valid if Ξ x1 Ξ y2 – Ξ x2 Ξ y1 ≠ 0). In equation (13.2.1.3), Ξ n denotes the partial derivative of Ξ with respect to C n (n = 1, 2), Ξ xn is the second partial derivative with respect to x and C n ,andΞ yn is the second partial derivative with respect to y and C n . In some cases, a complete integral can be found using the method of undetermined coefficients by presetting an appropriate structure of the particular solution sought. (The complete integral is determined by the differential equation nonuniquely.) 13.2. NONLINEAR EQUATIONS 571 Example 1. Consider the equation ∂w ∂x = a ∂w ∂y n + b. We seek a particular solution as the sum w = C 1 y + C 2 + C 3 x. Substituting this expression into the equation yields the relation C 3 = aC n 1 + b for the coefficients C 1 and C 3 . With this relation, we find a complete integral in the form w = C 1 y + aC n 1 + b x + C 2 . A complete integral of equation (13.2.1.1) is often written in implicit form:* Ξ(x, y, w, C 1 , C 2 )=0.(13.2.1.4) 2 ◦ .Thegeneral integral of equation (13.2.1.1) can be represented in parametric form by using the complete integral (13.2.1.2) [or (13.2.1.4)] and the two equations C 2 = f (C 1 ), ∂Ξ ∂C 1 + ∂Ξ ∂C 2 f (C 1 )=0, (13.2.1.5) where f is an arbitrary function and the prime stands for the derivative. In a sense, the general integral plays the role of the general solution depending on an arbitrary function (the questions whether it describes all solutions calls for further analysis). Example 2. For the equation considered in the first example, the general integral can be written in parametric form by using the relations w = C 1 y + aC n 1 + b x + C 2 , C 2 = f(C 1 ), y + anC n–1 1 x + f (C 1 )=0. Eliminating C 2 from these relations and renaming C 1 by C, one can represent the general integral in a more graphic manner in the form w = Cy + aC n + b x + f(C), y =–anC n–1 x + f (C). 3 ◦ . Singular integrals of equation (13.2.1.1) can be found without invoking a complete inte- gral by eliminating p and q from the following system of three algebraic (or transcendental) equations: F = 0, F p = 0, F q = 0, where the first equation coincides with equation (13.2.1.1). 13.2.1-2. Method of separation of variables. Equations of special form. The method of separation of variables implies searching for a complete integral as the sum or product of functions of various arguments. Such solutions are called additive separable and multiplicative separable, respectively. Presented below are structures of complete integrals for some classes of nonlinear equations admitting separation of variables. 1 ◦ . If the equation does not depend explicitly on y and w, i.e., F (x, w x , w y )=0, then one can seek a complete integral in the form of the sum of two functions with different arguments w = C 1 y + C 2 + u(x). The new unknown function u is determined by solving the following ordinary differential equation: F (x, u x , C 1 )=0. Expressing u x from this equation in terms of x, one arrives at a separable differential equation for u = u(x). * In equations (13.2.1.2) and (13.2.1.4), the symbol Ξ denotes different functions. 572 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 2 ◦ . Consider an equation with separated variables F 1 (x, w x )=F 2 (y, w y ). Then one can seek a complete integral as the sum of two functions with different arguments, w = u(x)+v(y)+C 1 , which are determined by the following two ordinary differential equations: F 1 (x, u x )=C 2 , F 2 (y, v y )=C 2 . 3 ◦ . Let the equation have the form (generalizes the equation of Item 2 ◦ ) F 1 (x, w x )+F 2 (y, w y )=aw. Then one can seek a complete integral as the sum of two functions with different arguments, w = u(x)+v(y)+C 1 , which are determined by the following two ordinary differential equations: F 1 (x, u x )–au = aC 1 + C 2 , F 2 (y, v y )–av =–C 2 , where C 1 is an arbitrary constant. 4 ◦ . Suppose the equation can be rewritten in the form F ϕ(x, w x ), y, w y = 0. Then one can seek a complete integral as the sum of two functions with different arguments, w(x, y)=u(x)+v(y)+C 1 , which are determined by the following two ordinary differential equations: ϕ(x, u x )=C 2 , F (C 2 , y, v y )=0, where C 2 is an arbitrary constant. 5 ◦ . Let the equation have the form F 1 (x, w x /w)=w k F 2 (y, w y /w). Then one can seek a complete integral in the form of the product of two functions with different arguments, w = u(x)v(y), which are determined by the following two ordinary differential equations: F 1 (x, u x /u)=C 1 u k , F 2 (y, v y /v)=C 1 v –k , where C 1 is an arbitrary constant. 6 ◦ . Table 13.2 lists complete integrals of the above and some other nonlinear equations of general form involving arbitrary functions with several arguments. Section T7.3 presents complete integrals for many more nonlinear first-order partial differential equations with two independent variables than in Table 13.2. 13.2. NONLINEAR EQUATIONS 573 TABLE 13.2 Complete integrals for some special types of nonlinear first-order partial differential equations; C 1 and C 2 are arbitrary constants No. Equations and comments Complete integrals Auxiliary equations 1 F (w x , w y )=0, does not depend on x, y,andw implicitly w = C 1 + C 2 x + C 3 y F (C 2 , C 3 )=0 2 F (x,w x , w y )=0, does not depend on y and w implicitly w = C 1 y + C 2 + u(x) F (x, u x , C 1 )=0 3 F (w, w x , w y )=0, does not depend on x and y implicitly w = u(z), z = C 1 x + C 2 y F (u, C 1 u z , C 2 u z )=0 4 F 1 (x, w x )=F 2 (y, w y ), separated equation w = u(x)+v(y)+C 1 F 1 (x, u x )=C 2 , F 2 (y, v y )=C 2 5 F 1 (x, w x )+F 2 (y, w y )=aw, generalizes equation 4 w = u(x)+v(y) F 1 (x, u x )–au = C 1 , F 2 (y, v y )–av =–C 1 6 F 1 (x, w x )=e aw F 2 (y, w y ), generalizes equation 4 w = u(x)+v(y) F 1 (x, u x )=C 1 e au , F 2 (y, v y )=C 1 e –av 7 F 1 (x, w x /w)=w k F 2 (y, w y /w), can be reduced to equation 6 by the change of variable w = e z w = u(x)v(y) F 1 (x, u x /u)=C 1 u k , F 2 (y, v y /v)=C 1 v –k 8 w = xw x + yw y + F (w x , w y ), Clairaut equation w = C 1 x + C 2 y + F (C 1 , C 2 ) — 9 F (x, w x , w y , w – yw y )=0, generalizes equation 2 w = C 1 y + u(x) F x, u x , C 1 , u = 0 10 F (w, w x , w y , xw x + yw y )=0, generalizes equations 3 and 8 w = u(z), z = C 1 x + C 2 y F (u, C 1 u z , C 2 u z , zu z )=0 11 F ϕ(x, w x ), y, w y = 0, generalizes equation 4 w = u(x)+v(y)+C 1 ϕ(x, u x )=C 2 , F (C 2 , y, v y )=0 13.2.1-3. Lagrange–Charpit method. Suppose that a first integral, Φ(x, y, w, p, q)=C 1 ,(13.2.1.6) of the characteristic system of ordinary differential equations dx F p = dy F q = dw pF p + qF q =– dp F x + pF w =– dq F y + qF w (13.2.1.7) is known. Here, p = ∂w ∂x , q = ∂w ∂y , F x = ∂F ∂x , F y = ∂F ∂y , F w = ∂F ∂w , F p = ∂F ∂p , F q = ∂F ∂q . We assume that solution (13.2.1.6) and equation (13.2.1.1) can be solved for the deriva- tives p and q, i.e., p = ϕ 1 (x, y, w, C 1 ), q = ϕ 2 (x, y, w, C 1 ). (13.2.1.8) 574 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS The first equation of this system can be treated as an ordinary differential equation with independent variable x and parameter y.Onfinding the solution of this equation depending on an arbitrary function ψ(y), one substitutes this solution into the second equation to arrive at an ordinary differential equation for ψ. On determining ψ(y) and on substituting it into the general solution of the first equation of (13.2.1.8), one finds a complete integral of equation (13.2.1.1). In a similar manner, one can start solving system (13.2.1.8) with the second equation, treating it as an ordinary differential equation with independent variable y and parameter x. Example 3. Consider the equation ywp 2 – q = 0,wherep = ∂w ∂x , q = ∂w ∂y . In this case, the characteristic system (13.2.1.7) has the form dx 2ywp =– dy 1 = dw 2ywp 2 – q =– dp yp 3 =– dq wp 2 + yp 2 q . By making use of the original equation, we simplify the denominator of the third ratio to obtain an integrable combination: dw/(ywp 2 )=–dp/(yp 3 ). This yields the first integral p = C 1 /w. Solving the original equation for q, we obtain the system p = C 1 w , q = C 2 1 y w . The general solution of the first equation has the form w 2 = 2C 1 x + ψ(y), where ψ(y) is an arbitrary function. With this solution, it follows from the second equation of the system that ψ (y)=2C 2 1 y. Thus, ψ(y)=C 2 1 y 2 +C 2 . Finally, we arrive at a complete integral of the form w 2 = 2C 1 x + C 2 1 y 2 + C 2 . Note that the general solution of the completely integrable Pfaff equation (see Subsec- tion 15.14.2) dw = ϕ 1 (x, y, w, C 1 ) dx + ϕ 2 (x, y, w, C 1 ) dy (13.2.1.9) is a complete integral of equation (13.2.1.1). Here, the functions ϕ 1 and ϕ 2 are the same as in system (13.2.1.8). Remark. The relation F (x, y, w, p, q)=C is an obvious first integral of the characteristic system (13.2.1.7). Hence, the function Φ determining the integral (13.2.1.6) must differ from F . However, the use of rela- tion (13.2.1.1) makes it possible to reduce the order of system (13.2.1.7) by one. 13.2.1-4. Construction of a complete integral with the aid of two first integrals. Suppose two independent first integrals, Φ(x, y, w, p, q)=C 1 , Ψ(x, y, w, p, q)=C 2 ,(13.2.1.10) of the characteristic system of ordinary differential equations (13.2.1.7) are known. Assume that the functions F , Φ,andΨ determining equation (13.2.1.1) and the integrals (13.2.1.10) satisfy the two conditions (a) J ≡ ∂(F , Φ, Ψ) ∂(w, p, q) 0, (b) [Φ, Ψ] ≡ Φ p Φ x + pΦ w Ψ p Ψ x + pΨ w + Φ q Φ y + qΦ w Ψ q Ψ y + qΨ w ≡ 0, (13.2.1.11) where J is the Jacobian of F , Φ,andΨ with respect to w, p,andq,and[Φ, Ψ]isthe Jacobi–Mayer bracket. In this case, relations (13.2.1.1) and (13.2.1.10) form a parametric representation of the complete integral of equation (13.2.1.1) (p and q are considered to be parameters). Eliminating p and q from equations (13.2.1.1) and (13.2.1.10) followed by solving the obtained relation for w yields a complete integral in an explicit form w = w(x, y, C 1 , C 2 ). 13.2. NONLINEAR EQUATIONS 575 Example 4. Consider the equation pq – aw = 0,wherep = ∂w ∂x , q = ∂w ∂y . The characteristic system (13.2.1.7) has the form dx q = dy p = dw 2pq = dp ap = dq aq . Equating the first ratio with the fifth one and the second ratio with the fourth one, we obtain the first integrals q – ax = C 1 , p – ay = C 2 . Thus, F = pq – aw, Φ = q – ax,andΨ = p – ax. These functions satisfy conditions (13.2.1.11). Solving the equation and the first integrals for w yields a complete integral of the form w = 1 a (ax + C 1 )(ay + C 2 ). 13.2.1-5. Case where the equation does not depend on w explicitly. Suppose the original equation does not contain the unknown first explicitly, i.e., it has the form F (x, y, p, q)=0.(13.2.1.12) 1 ◦ . Given a one-parameter family of solutions w = Ξ(x, y, C 1 ) such that Ξ 1 const, a complete integral is given by w = Ξ(x, y, C 1 )+C 2 . 2 ◦ .Thefirst integral may be sought in the form Φ(x, y, p, q)=C 1 similar to that of equation (13.2.1.12). In this case, the characteristic system (13.2.1.7) is represented as dx F p = dy F q =– dp F x =– dq F y . The corresponding Pfaff equation (13.2.1.9) becomes dw = ϕ 1 (x, y, C 1 ) dx + ϕ 2 (x, y, C 1 ) dy. One may integrate this equation in quadrature, thus arriving at the following expression for the complete integral: w = x x 0 ϕ 1 (t, y, C 1 ) dt + y y 0 ϕ 2 (x 0 , s, C 1 ) ds + C 2 ,(13.2.1.13) where x 0 and y 0 are arbitrary numbers. 3 ◦ . Suppose that equation (13.2.1.12) can be solved for p or q, for example, p =–H(x, y, q). Then, by differentiating this relation with respect to y, we obtain a quasilinear equation for the derivative q in the form ∂q ∂x + ∂ ∂y H(x, y, q)=0, q = ∂w ∂y . This equation is simpler than the original one; qualitative features of it and solution methods can be found in Section 13.1.1. 576 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 13.2.1-6. Hamilton–Jacobi equation. Equation (13.2.1.1) solved for one of the derivatives, e.g., p + H(x, y, w, q)=0,wherep = ∂w ∂x , q = ∂w ∂y ,(13.2.1.14) is commonly referred to as the Hamilton–Jacobi equation* and the function H as the Hamiltonian. Equations of the form (13.2.1.14) are frequently encountered in various fields of mechanics, control theory, and differential games, where the variable x usually plays the role of time and the variable y the role of the spatial coordinate. To the Hamilton–Jacobi equation (13.2.1.14) there corresponds the function F (x, y, w, p, q)=p + H(x, y, w, q)in equation (13.2.1.1). The characteristic system (13.2.1.7) for equation (13.2.1.14) can be reduced, by taking into account the relation p =–H, to a simpler system consisting of three differential equations, y x = H q , w x = qH q – H, q x =–qH w – H y ,(13.2.1.15) which are independent of p; the left-hand sides of these equations are derivatives with respect to x. 13.2.2. Cauchy Problem. Existence and Uniqueness Theorem 13.2.2-1. Statement of the problem. Solution procedure. Consider the Cauchy problem for equation (13.2.1.1) subject to the initial conditions x = h 1 (ξ), y = h 2 (ξ), w = h 3 (ξ), (13.2.2.1) where ξ is a parameter (α ≤ ξ ≤ β)andtheh k (ξ) are given functions. The solution of this problem is carried out in several steps: 1 ◦ . First, one determines additional initial conditions for the derivatives, p = p 0 (ξ), q = q 0 (ξ). (13.2.2.2) To this end, one must solve the algebraic (or transcendental) system of equations F h 1 (ξ), h 2 (ξ), h 3 (ξ), p 0 , q 0 = 0,(13.2.2.3) p 0 h 1 (ξ)+q 0 h 2 (ξ)–h 3 (ξ)=0 (13.2.2.4) for p 0 and q 0 . Equation (13.2.2.3) results from substituting the initial data (13.2.2.1) into the original equation (13.2.1.1). Equation (13.2.2.4) is a consequence of the dependence of w on x and y and the relation dw = pdx+ qdy,wheredx, dy,anddw are calculated in accordance with the initial data (13.2.2.1). 2 ◦ . One solves the autonomous system dx F p = dy F q = dw pF p + qF q =– dp F x + pF w =– dq F y + qF w = dτ ,(13.2.2.5) which is obtained from (13.2.1.7) by introducing the additional variable τ (playing the role of time). * The Hamilton–Jacobi equation often means equation (13.2.1.14) that does not depend on w explicitly, i.e., the equation p + H(x, y, q)=0. . Method of separation of variables. Equations of special form. The method of separation of variables implies searching for a complete integral as the sum or product of functions of various arguments Jacobian of F , Φ ,and with respect to w, p,andq ,and[ Φ, Ψ]isthe Jacobi–Mayer bracket. In this case, relations (13.2.1.1) and (13.2.1.10) form a parametric representation of the complete integral of. lower bound and the least upper bound of the set of values {ξ = ξ n } for which the function I(x, y, ξ)= ξ 0 ϕ(η)–W (0, x, y, η) dη (13.1.4.14) takes the minimum value for fixed x and y (x