T9.3. ELLIPTIC EQUATIONS 1319 1 ◦ . Solutions: w(x, y)= 1 β ln 2(A 2 + B 2 ) aβ(Ax + By + C) 2 if aβ > 0, w(x, y)= 1 β ln 2(A 2 + B 2 ) aβ sinh 2 (Ax + By + C) if aβ > 0, w(x, y)= 1 β ln –2(A 2 + B 2 ) aβ cosh 2 (Ax + By + C) if aβ < 0, w(x, y)= 1 β ln 2(A 2 + B 2 ) aβ cos 2 (Ax + By + C) if aβ > 0, w(x, y)= 1 β ln 8C aβ – 2 β ln (x + A) 2 +(y + B) 2 – C , where A, B,andC are arbitrary constants. The first four solutions are of traveling-wave type and the last one is a radial symmetric solution with center at the point (–A,–B). 2 ◦ . Functional separable solutions: w(x, y)=– 2 β ln C 1 e ky 2aβ 2k cos(kx + C 2 ) , w(x, y)= 1 β ln 2k 2 (B 2 – A 2 ) aβ[A cosh(kx + C 1 )+B sin(ky + C 2 )] 2 , w(x, y)= 1 β ln 2k 2 (A 2 + B 2 ) aβ[A sinh(kx + C 1 )+B cos(ky + C 2 )] 2 , where A, B, C 1 , C 2 ,andk are arbitrary constants (x and y can be swapped to give another three solutions). 3 ◦ . General solution: w(x, y)=– 2 β ln 1 – 2aβΦ(z) Φ(z) 4|Φ z (z)| , where Φ = Φ(z) is an arbitrary analytic (holomorphic) function of the complex variable z = x+iy with nonzero derivative, and the bar over a symbol denotes the complex conjugate. 4. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = ae βw + be 2βw . 1 ◦ . Traveling-wave solutions: w(x, y)=– 1 β ln – aβ C 2 1 +C 2 2 +C 3 exp(C 1 x+C 2 y)+ a 2 β 2 –bβ(C 2 1 +C 2 2 ) 4C 3 (C 2 1 +C 2 2 ) 2 exp(–C 1 x–C 2 y) , w(x, y)=– 1 β ln aβ C 2 1 +C 2 2 + a 2 β 2 +bβ(C 2 1 +C 2 2 ) C 2 1 +C 2 2 sin(C 1 x+C 2 y +C 3 ) , where C 1 , C 2 ,andC 3 are arbitrary constants. 2 ◦ . For other exact solutions of this equation, see equation T9.3.1.7 with f (w)=ae βw + be 2βw . 1320 NONLINEAR MATH EM ATICA L PHYSICS EQUATIONS 5. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = αw ln(βw). 1 ◦ . Solutions: w(x, y)= 1 β exp 1 4 α(x + A) 2 + 1 4 α(y + B) 2 + 1 , w(x, y)= 1 β exp A(x + B) 2 Aα – 4A 2 (x + B)(y + C)+( 1 4 α – A)(y + C) 2 + 1 2 , where A, B,andC are arbitrary constants. 2 ◦ . There are exact solutions of the following forms: w(x, y)=F (z), z = Ax + By, w(x, y)=G(r), r = (x + C 1 ) 2 +(y + C 2 ) 2 , w(x, y)=f(x)g(y). 6. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = α sin(βw). 1 ◦ . Functional separable solution for α = β = 1: w(x, y)=4 arctan cot A cosh F cosh G , F = cos A √ 1 + B 2 (x – By), G = sin A √ 1 + B 2 (y + Bx), where A and B are arbitrary constants. 2 ◦ . Functional separable solution (generalizes the solution of Item 1 ◦ ): w(x, y)= 4 β arctan f(x)g(y) , where the functions f = f (x)andg = g(y) are determined by the first-order autonomous ordinary differential equations (f x ) 2 = Af 4 + Bf 2 + C,(g y ) 2 = Cg 4 +(αβ – B)g 2 + A, and A, B,andC are arbitrary constants. 3 ◦ . For other exact solutions of this equation, see equation T9.3.1.7 with f(w)=α sin(βw). 7. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = f(w). This is a stationary heat equation with a nonlinear source. 1 ◦ . Suppose w = w(x, y) is a solution of the equation in question. Then the functions w 1 = w( x + C 1 , y + C 2 ), w 2 = w(x cos β – y sin β, x sin β + y cos β), where C 1 , C 2 ,andβ are arbitrary constants, are also solutions of the equation (the plus or minus signs in w 1 are chosen arbitrarily). T9.3. ELLIPTIC EQUATIONS 1321 2 ◦ . Traveling-wave solution in implicit form: C + 2 A 2 + B 2 F (w) –1/2 dw = Ax + By + D, F (w)= f(w) dw, where A, B, C,andD are arbitrary constants. 3 ◦ . Solution with central symmetry about the point (–C 1 ,–C 2 ): w = w(ζ), ζ = (x + C 1 ) 2 +(y + C 2 ) 2 , where C 1 and C 2 are arbitrary constants and the function w = w(ζ) is determined by the ordinary differential equation w ζζ + ζ –1 w ζ = f (w). T9.3.2. Equations of the Form ∂ ∂x f(x) ∂w ∂x + ∂ ∂y g(y) ∂w ∂y = f(w) 1. ∂ ∂x ax n ∂w ∂x + ∂ ∂y by m ∂w ∂y = f(w). Functional separable solution for n ≠ 2 and m ≠ 2: w = w(r), r = b(2 – m) 2 x 2–n + a(2 – n) 2 y 2–m 1/2 . Here, the function w(r) is determined by the ordinary differential equation w rr + Ar –1 w r = Bf(w), where A = 4 – nm (2 – n)(2 – m) , B = 4 ab(2 – n) 2 (2 – m) 2 . 2. a ∂ 2 w ∂x 2 + ∂ ∂y be μy ∂w ∂y = f(w), ab >0. Functional separable solution for μ ≠ 0: w = w(ξ), ξ = bμ 2 (x + C 1 ) 2 + 4ae –μy 1/2 , where C 1 is an arbitrary constant and the function w(ξ)isdefined implicitly by C 2 + 2 abμ 2 F (w) –1/2 dw = C 3 ξ, F (w)= f(w) dw, with C 2 and C 3 being arbitrary constants. 3. ∂ ∂x ae βx ∂w ∂x + ∂ ∂y be μy ∂w ∂y = f(w), ab >0. Functional separable solution for βμ ≠ 0: w = w(ξ), ξ = bμ 2 e –βx + aβ 2 e –μy 1/2 , where the function w(ξ) is determined by the ordinary differential equation w ξξ – ξ –1 w ξ = Af(w), A = 4/(abβ 2 μ 2 ). 1322 NONLINEAR MATH EM ATICA L PHYSICS EQUATIONS 4. ∂ ∂x f(x) ∂w ∂x + ∂ ∂y g(y) ∂w ∂y = kw ln w. Multiplicative separable solution: w(x, y)=ϕ(x)ψ(y), where the functions ϕ(x)andψ(y) are determined by the ordinary differential equations [f(x)ϕ x ] x = kϕ ln ϕ + Cϕ,[g(y)ψ y ] y = kψ ln ψ – Cψ, and C is an arbitrary constant. T9.3.3. Equations of the Form ∂ ∂x f(w) ∂w ∂x + ∂ ∂y g(w) ∂w ∂y = h(w) 1. ∂ 2 w ∂x 2 + ∂ ∂y (αw + β) ∂w ∂y =0. Stationary Khokhlov–Zabolotskaya equation. It arises in acoustics and nonlinear mechan- ics. 1 ◦ . Solutions: w(x, y)=Ay – 1 2 A 2 αx 2 + C 1 x + C 2 , w(x, y)=(Ax + B)y – α 12A 2 (Ax + B) 4 + C 1 x + C 2 , w(x, y)=– 1 α y + A x + B 2 + C 1 x + B + C 2 (x + B) 2 – β α , w(x, y)=– 1 α β + λ 2 A(y + λx)+B , w(x, y)=(Ax + B) C 1 y + C 2 – β α , where A, B, C 1 , C 2 ,andλ are arbitrary constants. 2 ◦ . Generalized separable solution quadratic in y (generalizes the third solution of Item 1 ◦ ): w(x, y)=– 1 α(x + A) 2 y 2 + B 1 (x + A) 2 + B 2 (x + A) 3 y + C 1 x + A + C 2 (x + A) 2 – β α – αB 2 1 4(x + A) 2 – 1 2 αB 1 B 2 (x + A) 3 – 1 54 αB 2 2 (x + A) 8 , where A, B 1 , B 2 , C 1 ,andC 2 are arbitrary constants. 3 ◦ . See also equation T9.3.3.3 with f(w)=1 and g(w)=αw + β. T9.3. ELLIPTIC EQUATIONS 1323 2. ∂ 2 w ∂x 2 + ∂ ∂y ae βw ∂w ∂y =0, a >0. 1 ◦ . Additive separable solutions: w(x, y)= 1 β ln(Ay + B)+Cx + D, w(x, y)= 1 β ln(–aA 2 y 2 + By + C)– 2 β ln(–aAx + D), w(x, y)= 1 β ln(Ay 2 + By + C)+ 1 β ln p 2 aA cosh 2 (px + q) , w(x, y)= 1 β ln(Ay 2 + By + C)+ 1 β ln p 2 –aA cos 2 (px + q) , w(x, y)= 1 β ln(Ay 2 + By + C)+ 1 β ln p 2 –aA sinh 2 (px + q) , where A, B, C, D, p,andq are arbitrary constants. 2 ◦ . There are exact solutions of the following forms: w(x, y)=F (r), r = k 1 x + k 2 y; w(x, y)=G(z), z = y/x; w(x, y)=H(ξ)–2(k + 1)β –1 ln |x|, ξ = y|x| k ; w(x, y)=U(η)–2β –1 ln |x|, η = y + k ln |x|; w(x, y)=V (ζ)–2β –1 x, ζ = ye x , where k, k 1 ,andk 2 are arbitrary constants. 3 ◦ . For other solutions, see equation T9.3.3.3 with f (w)=1 and g(w)=ae βw . 3. ∂ ∂x f(w) ∂w ∂x + ∂ ∂y g(w) ∂w ∂y =0. This is a stationary anisotropic heat (diffusion) equation. 1 ◦ . Traveling-wave solution in implicit form: A 2 f(w)+B 2 g(w) dw = C 1 (Ax + By)+C 2 , where A, B, C 1 ,andC 2 are arbitrary constants. 2 ◦ . Self-similar solution: w = w(ζ), ζ = x + A y + B , where the function w(ζ) is determined by the ordinary differential equation [f(w)w ζ ] ζ +[ζ 2 g(w)w ζ ] ζ = 0.(1) Integrating (1) and taking w to be the independent variable, one obtains the Riccati equation Cζ w = g(w)ζ 2 + f(w), where C is an arbitrary constant. 1324 NONLINEAR MATH EM ATICA L PHYSICS EQUATIONS 3 ◦ . The original equation can be represented as the system of the equations f(w) ∂w ∂x = ∂v ∂y ,–g(w) ∂w ∂y = ∂v ∂x .(2) The hodograph transformation x = x(w, v), y = y(w, v), where w, v are treated as the independent variables and x, y as the dependent ones, brings (2) to the linear system f(w) ∂y ∂v = ∂x ∂w ,–g(w) ∂x ∂v = ∂y ∂w .(3) Eliminating y yields the following linear equation for x = x(w, v): ∂ ∂w 1 f(w) ∂x ∂w + g(w) ∂ 2 x ∂v 2 = 0. Likewise, we can obtain another linear equation for y = y(w, v) from system (3). T9.4. Other Second-Order Equations T9.4.1. Equations of Transonic Gas Flow 1. a ∂w ∂x ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 =0. This is an equation of steady transonic gas flow. 1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the function w 1 = C –3 1 C 2 2 w(C 1 x + C 3 , C 2 y + C 4 )+C 5 y + C 6 , where C 1 , , C 6 are arbitrary constants, is also a solution of the equation. 2 ◦ . Solutions: w(x, y)=C 1 xy + C 2 x + C 3 y + C 4 , w(x, y)=– (x + C 1 ) 3 3a(y + C 2 ) 2 + C 3 y + C 4 , w(x, y)= a 2 C 3 1 39 (y + A) 13 + 2 3 aC 2 1 (y + A) 8 (x + B)+3C 1 (y + A) 3 (x + B) 2 – (x + B) 3 3a(y + A) 2 , w(x, y)=–aC 1 y 2 + C 2 y + C 3 4 3C 1 (C 1 x + C 4 ) 3/2 , w(x, y)=–aA 3 y 2 – B 2 aA 2 x + C 1 y + C 2 4 3 (Ax + By + C 3 ) 3/2 , w(x, y)= 1 3 (Ay + B)(2C 1 x + C 2 ) 3/2 – aC 3 1 12A 2 (Ay + B) 4 + C 3 y + C 4 , w(x, y)=– 9aA 2 y + C 1 + 4A x + C 2 y + C 1 3/2 – (x + C 2 ) 3 3a(y + C 1 ) 2 + C 3 y + C 4 , w(x, y)=– 3 7 aA 2 (y + C 1 ) 7 + 4A(x + C 2 ) 3/2 (y + C 1 ) 5/2 – (x + C 2 ) 3 3a(y + C 1 ) 2 + C 3 y + C 4 , where A, B, C 1 , , C 4 are arbitrary constants (the first solution is degenerate). T9.4. OTHER SECOND-ORDER EQUATIONS 1325 3 ◦ . There are solutions of the following forms: w(x, y)=y –3k–2 U(z), z = xy k (self-similar solution, k is any number); w(x, y)=ϕ 1 (y)+ϕ 2 (y)x 3/2 + ϕ 3 (y)x 3 (generalized separable solution); w(x, y)=ψ 1 (y)+ψ 2 (y)x + ψ 3 (y)x 2 + ψ 4 (y)x 3 (generalized separable solution); w(x, y)=ψ 1 (y)ϕ(x)+ψ 2 (y) (generalized separable solution). 2. ∂ 2 w ∂y 2 + a y ∂w ∂y + b ∂w ∂x ∂ 2 w ∂x 2 =0. 1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the function w 1 = C –3 1 C 2 2 w(C 1 x + C 3 , C 2 y)+C 4 y 1–a + C 5 , where C 1 , , C 5 are arbitrary constants, is also a solution of the equation. 2 ◦ . Additive separable solution: w(x, y)=– bC 1 4(a + 1) y 2 + C 2 y 1–a + C 3 2 3C 1 (C 1 x + C 4 ) 3/2 , where C 1 , , C 4 are arbitrary constants. 3 ◦ . Generalized separable solutions: w(x, y)=– 9A 2 b 16(n + 1)(2n + 1 + a) y 2n+2 + Ay n (x + C) 3/2 + a – 3 9b (x + C) 3 y 2 , where A and C are arbitrary constants, and n = n 1,2 are roots of the quadratic equation n 2 +(a – 1)n + 5 4 (a – 3)=0. 4 ◦ . Generalized separable solution: w(x, y)=(Ay 1–a + B)(2C 1 x + C 2 ) 3/2 + 9bC 3 1 θ(y), θ(y)=– B 2 2(a + 1) y 2 – AB 3 – a y 3–a – A 2 2(2 – a)(3 – a) y 4–2a + C 3 y 1–a + C 4 , where A, B, C 1 , C 2 , C 3 ,andC 4 are arbitrary constants. 5 ◦ . There are solutions of the following forms: w(x, y)=y –3k–2 U(z), z = xy k (self-similar solution, k is any number); w(x, y)=ϕ 1 (y)+ϕ 2 (y)x 3/2 + ϕ 3 (y)x 3 (generalized separable solution); w(x, y)=ψ 1 (y)+ψ 2 (y)x + ψ 3 (y)x 2 + ψ 4 (y)x 3 (generalized separable solution); w(x, y)=ψ 1 (y)ϕ(x)+ψ 2 (y) (generalized separable solution). . equations (f x ) 2 = Af 4 + Bf 2 + C,(g y ) 2 = Cg 4 +(αβ – B)g 2 + A, and A, B,andC are arbitrary constants. 3 ◦ . For other exact solutions of this equation, see equation T9.3.1.7 with f(w)=α sin(βw). 7. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 =. (w). T9.3.2. Equations of the Form ∂ ∂x f(x) ∂w ∂x + ∂ ∂y g(y) ∂w ∂y = f(w) 1. ∂ ∂x ax n ∂w ∂x + ∂ ∂y by m ∂w ∂y = f(w). Functional separable solution for n ≠ 2 and m ≠ 2: w = w(r),. ϕ(x )and (y) are determined by the ordinary differential equations [f(x)ϕ x ] x = kϕ ln ϕ + Cϕ,[g(y)ψ y ] y = kψ ln ψ – Cψ, and C is an arbitrary constant. T9.3.3. Equations of the Form ∂ ∂x f(w) ∂w ∂x + ∂ ∂y g(w) ∂w ∂y =