T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1347 6a. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u w  , ∂w ∂t = a ∂ 2 w ∂x 2 + wg  u w  . This system is a special case of system T10.3.1.6 with b = a and hence it admits the above solutions given in Items 1 ◦ –5 ◦ . In addition, it has some interesting properties and other solutions, which are given below. Suppose u = u(x, t), w = w(x, t) is a solution of the system. Then the functions u 1 = Au( x + C 1 , t + C 2 ), w 1 = Aw( x + C 1 , t + C 2 ); u 2 =exp(λx + aλ 2 t)u(x + 2aλt, t), w 2 =exp(λx + aλ 2 t)w(x + 2aλt, t), where A, C 1 , C 2 ,andλ are arbitrary constants, are also solutions of these equations. 6 ◦ . Point-source solution: u =exp  – x 2 4at  ϕ(t), w =exp  – x 2 4at  ψ(t), where the functions ϕ = ϕ(t)andψ = ψ(t) are determined by the autonomous system of ordinary differential equations ϕ  t =– 1 2t ϕ + ϕf  ϕ ψ  , ψ  t =– 1 2t ψ + ψg  ϕ ψ  . 7 ◦ . Functional separable solution: u =exp  kxt + 2 3 ak 2 t 3 – λt  y(ξ), w =exp  kxt + 2 3 ak 2 t 3 – λt  z(ξ), ξ = x + akt 2 , where k and λ are arbitrary constants, and the functions y = y(ξ)andz = z(ξ) are determined by the autonomous system of ordinary differential equations ay  ξξ +(λ – kξ)y + yf(y/z)=0, az  ξξ +(λ – kξ)z + zg(y/z)=0. 8 ◦ .Letk be a root of the algebraic (transcendental) equation f(k)=g(k). (1) Solution: u = ke λt θ, w = e λt θ, λ = f (k), where the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 . 9 ◦ . Periodic solution: u = Ak exp(–μx)sin(βx – 2aβμt + B), w = A exp(–μx)sin(βx – 2aβμt + B), β =  μ 2 + 1 a f(k), where A, B,andμ are arbitrary constants, and k is a root of the algebraic (transcendental) equation (1). 1348 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 10 ◦ . Solution: u = ϕ(t)exp   g(ϕ(t)) dt  θ(x, t), w =exp   g(ϕ(t)) dt  θ(x, t), where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation ϕ  t =[f (ϕ)–g(ϕ)]ϕ,(2) and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 . To the particular solution ϕ = k = const of equation (2) there corresponds the solution given in Item 8 ◦ . The general solution of equation (2) is written out in implicit form as  dϕ [f(ϕ)–g(ϕ)]ϕ = t + C. 11 ◦ . The transformation u = a 1 U + b 1 W , w = a 2 U + b 2 W , where a n and b n are arbitrary constants (n = 1, 2), leads to an equation of similar form for U and W . 7. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u w  + g  u w  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  u w  + h  u w  . Let k be a root of the algebraic (transcendental) equation g(k)=kh(k). 1 ◦ . Solution with f (k) ≠ 0: u(x, t)=k  exp[f (k)t]θ(x, t)– h(k) f(k)  , w(x, t)=exp[f(k)t]θ(x, t)– h(k) f(k) , where the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 .(1) 2 ◦ . Solution with f (k)=0: u(x, t)=k[θ(x, t)+h(k)t], w(x, t)=θ(x, t)+h(k)t, where the function θ = θ(x, t) satisfies the linear heat equation (1). T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1349 8. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u w  + u w h  u w  , ∂w ∂t = a ∂ 2 w ∂x 2 + wg  u w  + h  u w  . Solution: u = ϕ(t)G(t)  θ(x, t)+  h(ϕ) G(t) dt  , w = G(t)  θ(x, t)+  h(ϕ) G(t) dt  , G(t)=exp   g(ϕ) dt  , where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation ϕ  t =[f (ϕ)–g(ϕ)]ϕ,(1) and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 . The general solution of equation (1) is written out in implicit form as  dϕ [f(ϕ)–g(ϕ)]ϕ = t + C. 9. ∂u ∂t = a ∂ 2 u ∂x 2 +uf 1  w u  +wg 1  w u  , ∂w ∂t = a ∂ 2 w ∂x 2 +uf 2  w u  +wg 2  w u  . Solution: u=exp   [f 1 (ϕ)+ϕg 1 (ϕ)] dt  θ(x, t), w(x, t)=ϕ(t)exp   [f 1 (ϕ)+ϕg 1 (ϕ)] dt  θ(x, t), where the function ϕ = ϕ(t) is described by the separable first-order nonlinear ordinary differential equation ϕ  t = f 2 (ϕ)+ϕg 2 (ϕ)–ϕ[f 1 (ϕ)+ϕg 1 (ϕ)], and the function θ = θ(x, t) satisfies the linear heat equation ∂θ ∂t = a ∂ 2 θ ∂x 2 . 10. ∂u ∂t = a ∂ 2 u ∂x 2 + u 3 f  u w  , ∂w ∂t = a ∂ 2 w ∂x 2 + u 3 g  u w  . Solution: u =(x + A)ϕ(z), w =(x + A)ψ(z), z = t + 1 6a (x + A) 2 + B, where A and B are arbitrary constants, and the functions ϕ = ϕ(z)andψ = ψ(z)are determined by the autonomous system of ordinary differential equations ϕ  zz + 9aϕ 3 f(ϕ/ψ)=0, ψ  zz + 9aϕ 3 g(ϕ/ψ)=0. 1350 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 11. ∂u ∂t = ∂ 2 u ∂x 2 + au – u 3 f  u w  , ∂w ∂t = ∂ 2 w ∂x 2 + aw – u 3 g  u w  . 1 ◦ . Solution with a > 0: u =  C 1 exp  1 2 √ 2ax+ 3 2 at  – C 2 exp  – 1 2 √ 2ax+ 3 2 at  ϕ(z), w =  C 1 exp  1 2 √ 2ax+ 3 2 at  – C 2 exp  – 1 2 √ 2ax+ 3 2 at  ψ(z), z = C 1 exp  1 2 √ 2ax+ 3 2 at  + C 2 exp  – 1 2 √ 2ax+ 3 2 at  + C 3 , where C 1 , C 2 ,andC 3 are arbitrary constants, and the functions ϕ = ϕ(z)andψ = ψ(z)are determined by the autonomous system of ordinary differential equations aϕ  zz = 2ϕ 3 f(ϕ/ψ), aψ  zz = 2ϕ 3 g(ϕ/ψ). 2 ◦ . Solution with a < 0: u =exp  3 2 at  sin  1 2  2|a| x + C 1  U(ξ), w =exp  3 2 at  sin  1 2  2|a| x + C 1  W (ξ), ξ =exp  3 2 at  cos  1 2  2|a| x + C 1  + C 2 , where C 1 and C 2 are arbitrary constants, and the functions U = U(ξ)andW = W (ξ)are determined by the autonomous system of ordinary differential equations aU  ξξ =–2U 3 f(U/W), aW  ξξ =–2U 3 g(U/W). 12. ∂u ∂t = a ∂ 2 u ∂x 2 + u n f  u w  , ∂w ∂t = b ∂ 2 w ∂x 2 + w n g  u w  . If 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). 1 ◦ . Self-similar solution with n ≠ 1: u =(C 1 t + C 2 ) 1 1–n y(ξ), w =(C 1 t + C 2 ) 1 1–n z(ξ), ξ = x + C 3 √ C 1 t + C 2 , where C 1 , C 2 ,andC 3 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations ay  ξξ + 1 2 C 1 ξy  ξ + C 1 n – 1 y + y n f  y z  = 0, bz  ξξ + 1 2 C 1 ξz  ξ + C 1 n – 1 z + z n g  y z  = 0. 2 ◦ . Solution with b = a: u(x, t)=kθ(x, t), w(x, t)=θ(x, t), where k is a root of the algebraic (transcendental) equation k n–1 f(k)=g(k), and the function θ = θ(x, t) satisfies the heat equation with a power-law nonlinearity ∂θ ∂t = a ∂ 2 θ ∂x 2 + g(k)θ n . T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1351 13. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u w  ln u + ug  u w  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  u w  ln w + wh  u w  . Solution: u(x, t)=ϕ(t)ψ(t)θ(x, t), w(x, t)=ψ(t)θ(x, t), where the functions ϕ = ϕ(t)andψ = ψ(t) are determined by solving the first-order autonomous ordinary differential equations ϕ  t = ϕ[g(ϕ)–h(ϕ)+f(ϕ)lnϕ], (1) ψ  t = ψ[h(ϕ)+f (ϕ)lnψ], (2) and the function θ = θ(x, t) is determined by the differential equation ∂θ ∂t = a ∂ 2 θ ∂x 2 + f(ϕ)θ ln θ.(3) The separable equation (1) can be solved to obtain a solution in implicit form. Equa- tion (2) is easy to integrate—with the change of variable ψ = e ζ , it is reduced to a linear equation. Equation (3) admits exact solutions of the form θ =exp  σ 2 (t)x 2 + σ 1 (t)x + σ 0 (t)  , where the functions σ n (t) are described by the equations σ  2 = f (ϕ)σ 2 + 4aσ 2 2 , σ  1 = f (ϕ)σ 1 + 4aσ 1 σ 2 , σ  0 = f (ϕ)σ 0 + aσ 2 1 + 2aσ 2 . This system can be integrated directly, since the first equation is a Bernoulli equation and the second and third ones are linear in the unknown. Note that the first equation has a particular solution σ 2 = 0. Remark. Equation(1) hasa special solutionϕ =k =const, where k isa rootof thealgebraic (transcendental) equation g(k)–h(k)+f(k)lnk = 0. 14. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  w u  – wg  w u  + u √ u 2 + w 2 h  w u  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  w u  + ug  w u  + w √ u 2 + w 2 h  w u  . Solution: u = r(x, t)cosϕ(t), w = r(x, t)sinϕ(t), where the function ϕ = ϕ(t) is determined from the separable first-order ordinary differential equation ϕ  t = g(tan ϕ), and the function r = r(x, t) satisfies the linear equation ∂r ∂t = a ∂ 2 r ∂x 2 + rf(tan ϕ)+h(tan ϕ). (1) 1352 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS The change of variable r = F (t)  Z(x, t)+  h(tan ϕ) dt F (t)  , F (t)=exp   f(tan ϕ) dt  brings (1) to the linear heat equation ∂Z ∂t = a ∂ 2 Z ∂x 2 . 15. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  w u  + wg  w u  + u √ u 2 – w 2 h  w u  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  w u  + ug  w u  + w √ u 2 – w 2 h  w u  . Solution: u = r(x, t)coshϕ(t), w = r(x, t)sinhϕ(t), where the function ϕ = ϕ(t) is determined from the separable first-order ordinary differential equation ϕ  t = g(tanh ϕ), and the function r = r(x, t) satisfies the linear equation ∂r ∂t = a ∂ 2 r ∂x 2 + rf(tanh ϕ)+h(tanh ϕ). (1) The change of variable r = F (t)  Z(x, t)+  h(tanh ϕ) dt F (t)  , F (t)=exp   f(tanh ϕ) dt  brings (1) to the linear heat equation ∂Z ∂t = a ∂ 2 Z ∂x 2 . T10.3.1-3. Arbitrary functions depend on the product of powers of the unknowns. 16. ∂u ∂t = a ∂ 2 u ∂x 2 + uf(u n w m ), ∂w ∂t = b ∂ 2 w ∂x 2 + wg(u n w m ). Solution: u = e m(kx–λt) y(ξ), w = e –n(kx–λt) z(ξ), ξ = βx – γt, where k, λ, β,andγ are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the autonomous system of ordinary differential equations aβ 2 y  ξξ +(2akmβ + γ)y  ξ + m(ak 2 m + λ)y + yf(y n z m )=0, bβ 2 z  ξξ +(–2bknβ + γ)z  ξ + n(bk 2 n – λ)z + zg(y n z m )=0. To the special case k = λ = 0 there corresponds a traveling-wave solution. T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1353 17. ∂u ∂t = a ∂ 2 u ∂x 2 + u 1+kn f  u n w m  , ∂w ∂t = b ∂ 2 w ∂x 2 + 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 3 √ C 1 t + C 2 , where C 1 , C 2 ,andC 3 are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the system of ordinary differential equations ay  ξξ + 1 2 C 1 ξy  ξ + C 1 kn y + y 1+kn f  y n z m  = 0, bz  ξξ + 1 2 C 1 ξz  ξ – C 1 km z + z 1–km g  y n z m  = 0. 18. ∂u ∂t = a ∂ 2 u ∂x 2 +cu ln u +uf(u n w m ), ∂w ∂t = b ∂ 2 w ∂x 2 +cw ln w +wg(u n w m ). Solution: u =exp(Ame ct )y(ξ), w =exp(–Ane ct )z(ξ), ξ = kx – λt, where A, k,andλ are arbitrary constants, and the functions y = y(ξ)andz = z(ξ)are determined by the autonomous system of ordinary differential equations ak 2 y  ξξ + λy  ξ + cy ln y + yf(y n z m )=0, bk 2 z  ξξ + λz  ξ + cz ln z + zg(y n z m )=0. To the special case A = 0 there corresponds a traveling-wave solution. For λ = 0,we have a solution in the form of the product of two functions dependent on time t and the coordinate x. T10.3.1-4. Arbitrary functions depend on the sum of squares of the unknowns. 19. ∂u ∂t = a ∂ 2 u ∂x 2 + uf(u 2 + w 2 ) – wg(u 2 + w 2 ), ∂w ∂t = a ∂ 2 w ∂x 2 + ug(u 2 + w 2 ) + wf(u 2 + w 2 ). 1 ◦ . A periodic solution in the spatial coordinate: u = ψ(t)cosϕ(x, t), w = ψ(t)sinϕ(x, t), ϕ(x, t)=C 1 x +  g(ψ 2 ) dt + C 2 , where C 1 and C 2 are arbitrary constants, and the function ψ = ψ(t) is described by the separable first-order ordinary differential equation ψ  t = ψf(ψ 2 )–aC 2 1 ψ, whose general solution can be represented in implicit form as  dψ ψf(ψ 2 )–aC 2 1 ψ = t + C 3 . . solution. For λ = 0,we have a solution in the form of the product of two functions dependent on time t and the coordinate x. T10.3.1-4. Arbitrary functions depend on the sum of squares of the unknowns. 19. ∂u ∂t =. leads to an equation of similar form for U and W . 7. ∂u ∂t = a ∂ 2 u ∂x 2 + uf  u w  + g  u w  , ∂w ∂t = a ∂ 2 w ∂x 2 + wf  u w  + h  u w  . Let k be a root of the algebraic (transcendental). 2aβμt + B), β =  μ 2 + 1 a f(k), where A, B ,and are arbitrary constants, and k is a root of the algebraic (transcendental) equation (1). 1348 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 10 ◦ . Solution: u