T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1361 The first equation in (1) is a separable equation; its solution can be written out in implicit form. The second equation in (1) can be solved using the change of variable ψ = e ζ (it is reduced to a linear equation for ζ). Equation (2) admits exact solutions of the form θ =exp σ 2 (t)x 2 + σ 0 (t) , where the functions σ n (t) are described by the equations σ 2 = f(ϕ)σ 2 + 4aσ 2 2 , σ 0 = f(ϕ)σ 0 + 2a(n + 1)σ 2 . This system can be successively integrated, since the first equation is a Bernoulli equation and the second one is linear in the unknown. If f = const, equation (2) also has a traveling-wave solution θ = θ(kx – λt). T10.3.2-3. Arbitrary functions depend on the product of powers of the unknowns. 8. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + uf(x, u k w m ), ∂w ∂t = b x n ∂ ∂x x n ∂w ∂x + wg(x, u k w m ). Multiplicative separable solution: u = e –mλt y(x), w = e kλt z(x), where λ is an arbitrary constant and the functions y = y(x)andz = z(x) are determined by the system of ordinary differential equations ax –n (x n y x ) x + mλy + yf(x, y k z m )=0, bx –n (x n z x ) x – kλz + zg(x, y k z m )=0. 9. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + u 1+kn f u n w m , ∂w ∂t = b x n ∂ ∂x x n ∂w ∂x + w 1–km g u n w m . Self-similar solution: u =(C 1 t + C 2 ) – 1 kn y(ξ), w =(C 1 t + C 2 ) 1 km z(ξ), ξ = x √ C 1 t + C 2 , where C 1 and C 2 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations aξ –n (ξ n y ξ ) ξ + 1 2 C 1 ξy ξ + C 1 kn y + y 1+kn f y n z m = 0, bξ –n (ξ n z ξ ) ξ + 1 2 C 1 ξz ξ – C 1 km z + z 1–km g y n z m = 0. 1362 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 10. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + cu ln u + uf (x, u k w m ), ∂w ∂t = b x n ∂ ∂x x n ∂w ∂x + cw ln w + wg(x, u k w m ). Multiplicative separable solution: u =exp(Ame ct )y(x), w =exp(–Ake ct )z(x), where A is an arbitrary constant, and the functions y = y(x)andz = z(x) are determined by the system of ordinary differential equations ax –n (x n y x ) x + cy ln y + yf(x, y k z m )=0, bx –n (x n z x ) x + cz ln z + zg(x, y k z m )=0. T10.3.2-4. Arbitrary functions depend on u 2 w 2 . 11. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + uf(u 2 + w 2 ) – wg(u 2 + w 2 ), ∂w ∂t = a x n ∂ ∂x x n ∂w ∂x + wf(u 2 + w 2 ) + ug(u 2 + w 2 ). Time-periodic solution: u = r(x)cos θ(x)+C 1 t + C 2 , w = r(x)sin θ(x)+C 1 t + C 2 , where C 1 and C 2 are arbitrary constants, and the functions r = r(x)andθ = θ(x)are determined by the system of ordinary differential equations ar xx – ar(θ x ) 2 + an x r x + rf(r 2 )=0, arθ xx + 2ar x θ x + an x rθ x + rg(r 2 )–C 1 r = 0. 12. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + uf(u 2 – w 2 ) + wg(u 2 – w 2 ), ∂w ∂t = a x n ∂ ∂x x n ∂w ∂x + wf(u 2 – w 2 ) + ug(u 2 – w 2 ). Solution: u = r(x)cosh θ(x)+C 1 t + C 2 , w = r(x)sinh θ(x)+C 1 t + C 2 , where C 1 and C 2 are arbitrary constants, and the functions r = r(x)andθ = θ(x)are determined by the system of ordinary differential equations ar xx + ar(θ x ) 2 + an x r x + rf(r 2 )=0, arθ xx + 2ar x θ x + an x rθ x + rg(r 2 )–C 1 r = 0. T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1363 T10.3.2-5. Arbitrary functions have different arguments. 13. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + uf(u 2 + w 2 ) – wg w u , ∂w ∂t = a x n ∂ ∂x x n ∂w ∂x + wf(u 2 + w 2 ) + ug w u . Solution: u = r(x, t)cosϕ(t), w = r(x, t)sinϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation ϕ t = g(tan ϕ), (1) and the function r = r(x, t) is determined by the differential equation ∂r ∂t = a x n ∂ ∂x x n ∂r ∂x + rf(r 2 ). (2) The general solution of equation (1) is expressed in implicit form as dϕ g(tan ϕ) = t + C. Equation (2) admits a time-independent exact solution r = r(x). 14. ∂u ∂t = a x n ∂ ∂x x n ∂u ∂x + uf(u 2 – w 2 ) + wg w u , ∂w ∂t = a x n ∂ ∂x x n ∂w ∂x + wf(u 2 – w 2 ) + ug w u . Solution: u = r(x, t)coshϕ(t), w = r(x, t)sinhϕ(t), where the function ϕ = ϕ(t) is determined by the autonomous ordinary differential equation ϕ t = g(tanh ϕ), (1) and the function r = r(x, t) is determined by the differential equation ∂r ∂t = a x n ∂ ∂x x n ∂r ∂x + rf(r 2 ). (2) The general solution of equation (1) is expressed in implicit form as dϕ g(tanh ϕ) = t + C. Equation (2) admits a time-independent exact solution r = r(x). 1364 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS T10.3.3. Systems of the Form Δu = F (u, w), Δw = G(u, w) T10.3.3-1. Arbitrary functions depend on a linear combination of the unknowns. 1. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf (au – bw) + g(au – bw), ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = wf(au – bw) + h(au – bw). 1 ◦ . Solution: u = ϕ(x)+bθ(x, y), w = ψ(x)+aθ(x, y), where ϕ = ϕ(x)andψ = ψ(x) are determined by the autonomous system of ordinary differential equations ϕ xx = ϕf(aϕ – bψ)+g(aϕ – bψ), ψ xx = ψf(aϕ – bψ)+h(aϕ – bψ), and the function θ = θ(x, y) satisfi es the linear Schr ¨ odinger equation of the special form ∂ 2 θ ∂x 2 + ∂ 2 θ ∂y 2 = F (x)θ, F (x)=f(au – bw). Its solutions are determined by separation of variables. 2 ◦ . Let us multiply the first equation by a and the second one by –b and add the results together to obtain ∂ 2 ζ ∂x 2 + ∂ 2 ζ ∂y 2 = ζf(ζ)+ag(ζ)–bh(ζ), ζ = au – bw.(1) This equation will be considered in conjunction with the first equation of the original system ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf (ζ)+g(ζ). (2) Equation (1) can be treated separately. An extensive list of exact solutions to equations of this form for various kinetic functions F (ζ)=ζf(ζ)+ag(ζ)–bh(ζ) can be found in the book by Polyanin and Zaitsev (2004). Note two important solutions to equation (1): (i) In the general case, equation (1) admits an exact, traveling-wave solution ζ = ζ(z), where z = k 1 x + k 2 y with arbitrary constants k 1 and k 2 . (ii) If the condition ζf(ζ)+ag(ζ)–bh(ζ)=c 1 ζ + c 0 holds, equation (1) is a linear Helmholtz equation. Given a solution ζ = ζ(x, y) to equation (1), the function u = u(x, y) can be determined by solving the linear equation (2) and the function w = w(x, y) is found as w =(bu – ζ)/c. 2. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = e λu f(λu – σw), ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = e σw g(λu – σw). 1 ◦ . Solution: u = U(ξ)– 2 λ ln |x + C 1 |, w = W (ξ)– 2 σ ln |x + C 1 |, ξ = y + C 2 x + C 1 , T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1365 where C 1 and C 2 are arbitrary constants, and the functions U = U (ξ)andW = W (ξ)are determined by the system of ordinary differential equations (1 + ξ 2 )U ξξ + 2ξU ξ + 2 λ = e λU f(λU – σW), (1 + ξ 2 )W ξξ + 2ξW ξ + 2 σ = e σW g(λU – σW). 2 ◦ . Solution: u = θ(x, y), w = λ σ θ(x, y)– k σ , where k is a root of the algebraic (transcendental) equation λf(k)=σe –k g(k), and the function θ = θ(x, y) is described by the solvable equation ∂ 2 θ ∂x 2 + ∂ 2 θ ∂y 2 = f(k)e λθ . This equation is encountered in combustion theory; for its exact solutions, see Polyanin and Zaitsev (2004). T10.3.3-2. Arbitrary functions depend on the ratio of the unknowns. 3. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf u w , ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = wg u w . 1 ◦ . A space-periodic solution in multiplicative form (another solution is obtained by inter- changing x and y): u =[C 1 sin(kx)+C 2 cos(kx)]ϕ(y), w =[C 1 sin(kx)+C 2 cos(kx)]ψ(y), where C 1 , C 2 ,andk are arbitrary constants and the functions ϕ = ϕ(y)andψ = ψ(y)are determined by the autonomous system of ordinary differential equations ϕ yy = k 2 ϕ + ϕf(ϕ/ψ), ψ yy = k 2 ψ + ψg(ϕ/ψ). 2 ◦ . Solution in multiplicative form: u =[C 1 exp(kx)+C 2 exp(–kx)]U (y), w =[C 1 exp(kx)+C 2 exp(–kx)]W (y), where C 1 , C 2 ,andk are arbitrary constants and the functions U = U(y)andW = W (y)are determined by the autonomous system of ordinary differential equations U yy =–k 2 U + Uf(U/W), W yy =–k 2 W + Wg(U/W). 1366 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 3 ◦ . Degenerate solution in multiplicative form: u =(C 1 x + C 2 )U(y), w =(C 1 x + C 2 )W (y), where C 1 and C 2 are arbitrary constants and the functions U = U(y)andW = W (y)are determined by the autonomous system of ordinary differential equations U yy = Uf(U/W), W yy = Wg(U/W). Remark. The functions f and g in Items 1 ◦ –3 ◦ can be dependent on y. 4 ◦ . Solution in multiplicative form: u = e a 1 x+b 1 y ξ(z), w = e a 1 x+b 1 y η(z), z = a 2 x + b 2 y, where a 1 , a 2 , b 1 ,andb 2 are arbitrary constants, and the functions ξ = ξ(z)andη = η(z)are determined by the autonomous system of ordinary differential equations (a 2 2 + b 2 2 )ξ zz + 2(a 1 a 2 + b 1 b 2 )ξ z +(a 2 1 + b 2 1 )ξ = ξf(ξ/η), (a 2 2 + b 2 2 )η zz + 2(a 1 a 2 + b 1 b 2 )η z +(a 2 1 + b 2 1 )η = ηg(ξ/η). 5 ◦ . Solution: u = kθ(x, y), w = θ(x, y), where k is a root of the algebraic (transcendental) equation f (k)=g(k), and the function θ = θ(x, y) is described by the linear Helmholtz equation ∂ 2 θ ∂x 2 + ∂ 2 θ ∂y 2 = f(k)θ. For its exact solutions, see Subsection T8.3.3. 4. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf u w + u w h u w , ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = wg u w + h u w . Solution: u = kw, w = θ(x, y)– h(k) f(k) , where k is a root of the algebraic (transcendental) equation f(k)=g(k), and the function θ = θ(x, y) satisfi es the linear Helmholtz equation ∂ 2 θ ∂x 2 + ∂ 2 θ ∂y 2 = f (k)w. T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1367 5. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = u n f u w , ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = w n g u w . For f (z)=kz –m and g(z)=–kz n–m , the system describes an nth-order chemical reaction (of order n –m in the component u and of order m in the component w); to n = 2 and m = 1 there corresponds a second-order reaction, which often occurs in applications. 1 ◦ . Solution: u = r 2 1–n U(θ), w = r 2 1–n W (θ), r = (x + C 1 ) 2 +(y + C 2 ) 2 , θ = y + C 2 x + C 1 , where C 1 and C 2 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the autonomous system of ordinary differential equations U θθ + 4 (1 – n) 2 U = U n f U W , W θθ + 4 (1 – n) 2 W = W n g U W . 2 ◦ . Solution: u = kζ(x, y), w = ζ(x, y), where k is a root of the algebraic (transcendental) equation k n–1 f(k)=g(k), and the function ζ = ζ(x, y) satisfies the equation with a power-law nonlinearity ∂ 2 ζ ∂x 2 + ∂ 2 ζ ∂y 2 = g(k)ζ n . T10.3.3-3. Other systems. 6. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf (u n w m ), ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = wg(u n w m ). Solution in multiplicative form: u = e m(a 1 x+b 1 y) ξ(z), w = e –n(a 1 x+b 1 y) η(z), z = a 2 x + b 2 y, where a 1 , a 2 , b 1 ,andb 2 are arbitrary constants, and the functions ξ = ξ(z)andη = η(z)are determined by the autonomous system of ordinary differential equations (a 2 2 + b 2 2 )ξ zz + 2m(a 1 a 2 + b 1 b 2 )ξ z + m 2 (a 2 1 + b 2 1 )ξ = ξf(ξ n η m ), (a 2 2 + b 2 2 )η zz – 2n(a 1 a 2 + b 1 b 2 )η z + n 2 (a 2 1 + b 2 1 )η = ηg(ξ n η m ). . separately. An extensive list of exact solutions to equations of this form for various kinetic functions F (ζ)=ζf(ζ)+ag(ζ)–bh(ζ) can be found in the book by Polyanin and Zaitsev (2004 ). Note two important. w n g u w . For f (z)=kz –m and g(z)=–kz n–m , the system describes an nth-order chemical reaction (of order n –m in the component u and of order m in the component w); to n = 2 and m = 1 there. C 1 , T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1365 where C 1 and C 2 are arbitrary constants, and the functions U = U (ξ)andW = W (ξ)are determined by the system of ordinary differential