1368 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 7. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf(u 2 + w 2 ) – wg(u 2 + w 2 ), ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = wf(u 2 + w 2 ) + ug(u 2 + w 2 ). 1 ◦ . A periodic solution in y: u = r(x)cos θ(x)+C 1 y + C 2 , w = r(x)sin θ(x)+C 1 y + C 2 , where C 1 and C 2 are arbitrary constants, and the functions r = r(x)andθ = θ(x)are determined by the autonomous system of ordinary differential equations r xx = r(θ x ) 2 + C 2 1 r + rf(r 2 ), rθ xx =–2r x θ x + rg(r 2 ). 2 ◦ . Solution (generalizes the solution of Item 1 ◦ ): u = r(z)cos θ(z)+C 1 y + C 2 , w = r(z)sin θ(z)+C 1 y + C 2 , z = k 1 x + k 2 y, where C 1 , C 2 , k 1 ,andk 2 are arbitrary constants, and the functions r = r(z)andθ = θ(z) are determined by the autonomous system of ordinary differential equations (k 2 1 + k 2 2 )r zz = k 2 1 r(θ z ) 2 + r(k 2 θ z + C 1 ) 2 + rf(r 2 ), (k 2 1 + k 2 2 )rθ zz =–2 (k 2 1 + k 2 2 )θ z + C 1 k 2 r z + rg(r 2 ). 8. ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = uf(u 2 – w 2 ) + wg(u 2 – w 2 ), ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = wf(u 2 – w 2 ) + ug(u 2 – w 2 ). Solution: u = r(z)cosh θ(z)+C 1 y + C 2 , w = r(z)sinh θ(z)+C 1 y + C 2 , z = k 1 x + k 2 y, where C 1 , C 2 , k 1 ,andk 2 are arbitrary constants, and the functions r = r(z)andθ = θ(z) are determined by the autonomous system of ordinary differential equations (k 2 1 + k 2 2 )r zz + k 2 1 r(θ z ) 2 + r(k 2 θ z + C 1 ) 2 = rf(r 2 ), (k 2 1 + k 2 2 )rθ zz + 2 (k 2 1 + k 2 2 )θ z + C 1 k 2 r z = rg(r 2 ). T10.3.4. Systems of the Form ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + F (u, w), ∂ 2 w ∂t 2 = b x n ∂ ∂x x n ∂w ∂x + G(u, w) T10.3.4-1. Arbitrary functions depend on a linear combination of the unknowns. 1. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + uf(bu – cw) + g(bu – cw), ∂ 2 w ∂t 2 = a x n ∂ ∂x x n ∂w ∂x + wf(bu – cw) + h(bu – cw). 1 ◦ . Solution: u = ϕ(t)+cθ(x, t), w = ψ(t)+bθ(x, t), T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1369 where ϕ = ϕ(t)andψ = ψ(t) are determined by the autonomous system of ordinary differential equations ϕ tt = ϕf (bϕ – cψ)+g(bϕ – cψ), ψ tt = ψf(bϕ – cψ)+h(bϕ – cψ), and the function θ = θ(x, t) satisfies linear equation ∂ 2 θ ∂t 2 = a x n ∂ ∂x x n ∂θ ∂x + f(bϕ – cψ)θ. For f = const, this equation can be solved by separation of variables. 2 ◦ . Let us multiply the first equation by b and the second one by –c and add the results together to obtain ∂ 2 ζ ∂t 2 = a x n ∂ ∂x x n ∂ζ ∂x + ζf(ζ)+bg(ζ)–ch(ζ), ζ = bu – cw.(1) This equation will be considered in conjunction with the first equation of the original system ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + uf(ζ)+g(ζ). (2) Equation (1) can be treated separately. Given a solution ζ = ζ(x, t) to equation (1), the function u = u(x, t) can be determined by solving equation (2) and the function w = w(x, t) is found as w =(bu – ζ)/c. Note three important solutions to equation (1): (i) In the general case, equation (1) admits a spatially homogeneous solution ζ = ζ(t). The corresponding solution to the original system is given in Item 1 ◦ in another form. (ii) In the general case, equation (1) admits a steady-state solution ζ = ζ(x). The corresponding exact solutions to equation (2) are expressed as u = u 0 (x)+ e –β n t u n (x) and u = u 0 (x)+ cos(β n t)u (1) n (x)+ sin(β n t)u (2) n (x). (iii) If the condition ζf(ζ)+bg(ζ)–ch(ζ)=k 1 ζ + k 0 holds, equation (1) is linear, ∂ 2 ζ ∂t 2 = a x n ∂ ∂x x n ∂ζ ∂x + k 1 ζ + k 0 , and, hence, can be solved by separation of variables. 2. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + e λu f(λu – σw), ∂ 2 w ∂t 2 = b x n ∂ ∂x x n ∂w ∂x + e σw g(λu – σw). 1 ◦ . Solution: u = y(ξ)– 2 λ ln(C 1 t + C 2 ), w = z(ξ)– 2 σ ln(C 1 t + C 2 ), 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 C 2 1 (ξ 2 y ξ ) ξ + 2C 2 1 λ –1 = aξ –n (ξ n y ξ ) ξ + e λy f(λy – σz), C 2 1 (ξ 2 z ξ ) ξ + 2C 2 1 σ –1 = bξ –n (ξ n z ξ ) ξ + e σz g(λy – σz). 1370 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 2 ◦ . Solution with b = a: u = θ(x, t), w = λ σ θ(x, t)– k σ , where k is a root of the algebraic (transcendental) equation λf(k)=σe –k g(k), and the function θ = θ(x, t) is described by the equation ∂ 2 θ ∂t 2 = a x n ∂ ∂x x n ∂θ ∂x + f(k)e λθ . This equation is solvable for n = 0; for its exact solutions, see Polyanin and Zaitsev (2004). T10.3.4-2. Arbitrary functions depend on the ratio of the unknowns. 3. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x +uf u w , ∂ 2 w ∂t 2 = b x n ∂ ∂x x n ∂w ∂x +wg u w . 1 ◦ . Periodic multiplicative separable solution: u =[C 1 cos(kt)+C 2 sin(kt)]y(x), w =[C 1 cos(kt)+C 2 sin(kt)]z(x), where C 1 , C 2 ,andk are arbitrary constants 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 + k 2 y + yf(y/z)=0, bx –n (x n z x ) x + k 2 z + zg(y/z)=0. 2 ◦ . Multiplicative separable solution: u =[C 1 exp(kt)+C 2 exp(–kt)]y(x), w =[C 1 exp(kt)+C 2 exp(–kt)]z(x), where C 1 , C 2 ,andk are arbitrary constants 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 – k 2 y + yf(y/z)=0, bx –n (x n z x ) x – k 2 z + zg(y/z)=0. 3 ◦ . Degenerate multiplicative separable solution: u =(C 1 t + C 2 )y(x), w =(C 1 t + C 2 )z(x), where the functions y = y(x)andz = z(x) are determined by the system of ordinary differential equations ax –n (x n y x ) x + yf(y/z)=0, bx –n (x n z x ) x + zg(y/z)=0. T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1371 4 ◦ . Multiplicative separable solution: u = x 1–n 2 [C 1 J ν (kx)+C 2 Y ν (kx)]ϕ(t), ν = 1 2 |n – 1|, w = x 1–n 2 [C 1 J ν (kx)+C 2 Y ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, J ν (z)andY ν (z) are Bessel functions, and the functions ϕ = ϕ(t)andψ = ψ(t) are determined by the autonomous system of ordinary differential equations ϕ tt =–ak 2 ϕ + ϕf (ϕ/ψ), ψ tt =–bk 2 ψ + ψg(ϕ/ψ). 5 ◦ . Multiplicative separable solution: u = x 1–n 2 [C 1 I ν (kx)+C 2 K ν (kx)]ϕ(t), ν = 1 2 |n – 1|, w = x 1–n 2 [C 1 I ν (kx)+C 2 K ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, I ν (z)andK ν (z) are modified Bessel functions, and the functions ϕ = ϕ(t)andψ = ψ(t) are determined by the autonomous system of ordinary differential equations ϕ tt = ak 2 ϕ + ϕf (ϕ/ψ), ψ tt = bk 2 ψ + ψg(ϕ/ψ). 6 ◦ . Solution with b = a: u = kθ(x, t), w = θ(x, t), where k is a root of the algebraic (transcendental) equation f(k)=g(k), and the function θ = θ(x, t) is described by the linear Klein–Gordon equation ∂ 2 θ ∂t 2 = a x n ∂ ∂x x n ∂θ ∂x + f(k)θ. 4. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + uf u w + u w h u w , ∂ 2 w ∂t 2 = a x n ∂ ∂x x n ∂u ∂w + wg u w + h u w . Solution: u = kθ(x, t), w = θ(x, t), where k is a root of the algebraic (transcendental) equation f(k)=g(k), and the function θ = θ(x, t) is described by the linear equation ∂ 2 θ ∂t 2 = a x n ∂ ∂x x n ∂θ ∂x + f(k)θ + h(k). 1372 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 5. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x +u k f u w , ∂ 2 w ∂t 2 = b x n ∂ ∂x x n ∂w ∂x +w k g u w . Self-similar solution: u =(C 1 t + C 2 ) 2 1–k y(ξ), w =(C 1 t + C 2 ) 2 1–k 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 C 2 1 ξ 2 y ξξ + 2C 2 1 (k + 1) k – 1 ξy ξ + C 2 1 (k + 1) (k – 1) 2 y = a ξ n (ξ n y ξ ) ξ + y k f y z , C 2 1 ξ 2 z ξξ + 2C 2 1 (k + 1) k – 1 ξz ξ + C 2 1 (k + 1) (k – 1) 2 z = b ξ n (ξ n z ξ ) ξ + z k g y z . T10.3.4-3. Arbitrary functions depend on the product of powers of the unknowns. 6. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + uf(x, u k w m ), ∂ 2 w ∂t 2 = 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 – m 2 λ 2 y + yf(x, y k z m )=0, bx –n (x n z x ) x – k 2 λ 2 z + zg(x, y k z m )=0. T10.3.4-4. Arbitrary functions depend on the u 2 w 2 . 7. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + uf(u 2 + w 2 ) – wg(u 2 + w 2 ), ∂ 2 w ∂t 2 = a x n ∂ ∂x x n ∂w ∂x + wf(u 2 + w 2 ) + ug(u 2 + w 2 ). 1 ◦ . Periodic solution in t: 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(x) are determined by the system of ordinary differential equations ar xx – ar(θ x ) 2 + an x r x + C 2 1 r + rf(r 2 )=0, arθ xx + 2ar x θ x + an x rθ x + rg(r 2 )=0. 2 ◦ .Forn = 0, there is an exact solution of the form u = r(z)cos θ(z)+C 1 t + C 2 , w = r(z)sin θ(z)+C 1 t + C 2 , z = kx – λt. T10.3. NONLINEAR SYSTEMS OF TWO SECOND-ORDER EQUATIONS 1373 8. ∂ 2 u ∂t 2 = a x n ∂ ∂x x n ∂u ∂x + uf(u 2 – w 2 ) + wg(u 2 – w 2 ), ∂ 2 w ∂t 2 = a x n ∂ ∂x x n ∂w ∂x + wf(u 2 – w 2 ) + ug(u 2 – w 2 ). 1 ◦ . 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(x) are determined by the system of ordinary differential equations ar xx + ar(θ x ) 2 + an x r x – C 2 1 r + rf(r 2 )=0, arθ xx + 2ar x θ x + an x rθ x + rg(r 2 )=0. 2 ◦ .Forn = 0, there is an exact solution of the form u = r(z)cosh θ(z)+C 1 t + C 2 , w = r(z)sinh θ(z)+C 1 t + C 2 , z = kx – λt. T10.3.5. Other Systems 1. ax ∂u ∂x + ay ∂u ∂y = ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 – f(u, w), ax ∂w ∂x + ay ∂w ∂y = ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 – g(u, w). Solution: u(x, y)=U(z), w(x, y)=W (z), z = k 1 x + k 2 y, where k 1 and k 2 are arbitrary constants, and the functions U = U(z)andW = W (z)are described by the system of ordinary differential equations azU z =(k 2 1 + k 2 2 )U – f(U, W ), azW z =(k 2 1 + k 2 2 )W – g(U, W). 2. ∂u ∂t = ∂ ∂x f t, u w ∂u ∂x + ug t, u w , ∂w ∂t = ∂ ∂x f t, u w ∂w ∂x + wh t, u w . Solution: u = ϕ(t)exp h(t, ϕ(t)) dt θ(x, τ), w =exp h(t, ϕ(t)) dt θ(x, τ), τ = f(t, ϕ(t)) dt, where the function ϕ = ϕ(t) is described by the ordinary differential equation ϕ t =[g(t, ϕ)–h(t, ϕ)]ϕ, and the function θ = θ(x, τ) satisfies the linear heat equation ∂θ ∂τ = ∂ 2 θ ∂x 2 . 1374 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS T10.4. Systems of General Form T10.4.1. Linear Systems 1. ∂u ∂t = L[u] + f 1 (t)u + g 1 (t)w, ∂w ∂t = L[w] + f 2 (t)u + g 2 (t)w. Here, L is an arbitrary linear differential operator with respect to the coordinates x 1 , , x n (of any order in derivatives), whose coefficients can be dependent on x 1 , , x n , t.Itis assumed that L[const] = 0. Solution: u = ϕ 1 (t)U(x 1 , , x n , t)+ϕ 2 (t)W (x 1 , , x n , t), w = ψ 1 (t)U(x 1 , , x n , t)+ψ 2 (t)W (x 1 , , x n , t), where the two pairs of functions ϕ 1 = ϕ 1 (t), ψ 1 = ψ 1 (t)andϕ 2 = ϕ 2 (t), ψ 2 = ψ 2 (t)are linearly independent (fundamental) solutions to the system of first-order linear ordinary differential equations ϕ t = f 1 (t)ϕ + g 1 (t)ψ, ψ t = f 2 (t)ϕ + g 2 (t)ψ, and the functions U = U(x 1 , , x n , t)andW = W(x 1 , , x n , t) satisfy the independent linear equations ∂U ∂t = L[U], ∂W ∂t = L[W ]. 2. ∂ 2 u ∂t 2 = L[u] + a 1 u + b 1 w, ∂ 2 w ∂t 2 = L[w] + a 2 u + b 2 w. Here, L is an arbitrary linear differential operator with respect to the coordinates x 1 , , x n (of any order in derivatives). Solution: u = a 1 – λ 2 a 2 (λ 1 – λ 2 ) θ 1 – a 1 – λ 1 a 2 (λ 1 – λ 2 ) θ 2 , w = 1 λ 1 – λ 2 θ 1 – θ 2 , where λ 1 and λ 2 are roots of the quadratic equation λ 2 –(a 1 + b 2 )λ + a 1 b 2 – a 2 b 1 = 0, and the functions θ n = θ n (x 1 , , x n , t) satisfy the independent linear equations ∂ 2 θ 1 ∂t 2 = L[θ 1 ]+λ 1 θ 1 , ∂ 2 θ 2 ∂t 2 = L[θ 2 ]+λ 2 θ 2 . T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t 1. ∂u ∂t = L[u] + uf(t, bu – cw) + g(t, bu – cw), ∂w ∂t = L[w] + wf(t, bu – cw) + h(t, bu – cw). Here, L is an arbitrary linear differential operator (of any order) with respect to the spatial variables x 1 , , x n . . x 1–n 2 [C 1 J ν (kx)+C 2 Y ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, J ν (z)andY ν (z) are Bessel functions, and the functions ϕ = ϕ(t )and = ψ(t) are determined by the autonomous system of ordinary differential. x 1–n 2 [C 1 I ν (kx)+C 2 K ν (kx)]ψ(t), where C 1 , C 2 ,andk are arbitrary constants, I ν (z)andK ν (z) are modified Bessel functions, and the functions ϕ = ϕ(t )and = ψ(t) are determined by the autonomous system of ordinary differential. equation ϕ t =[g(t, ϕ)–h(t, ϕ)]ϕ, and the function θ = θ(x, τ) satisfies the linear heat equation ∂θ ∂τ = ∂ 2 θ ∂x 2 . 1374 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS T10.4. Systems of General Form T10.4.1. Linear