T10.4. SYSTEMS OF GENERAL FORM 1375 1 ◦ . Solution: u = ϕ(t)+c exp   f(t, bϕ – cψ) dt  θ(x, t), w = ψ(t)+b exp   f(t, bϕ – cψ) dt  θ(x, t), where ϕ = ϕ(t)andψ = ψ(t) are determined by the system of ordinary differential equations ϕ  t = ϕf (t, bϕ – cψ)+g(t, bϕ – cψ), ψ  t = ψf(t, bϕ – cψ)+h(t, bϕ – cψ), and the function θ = θ(x 1 , , x n , t) satisfi es linear equation ∂θ ∂t = L[θ]. Remark 1. The coefficients of the linear differential operator L can be dependent on x 1 , , x n , t. 2 ◦ . Let us multiply the first equation by b and the second one by –c and add the results together to obtain ∂ζ ∂t = L[ζ]+ζf(t, ζ)+bg(t, ζ)–ch(t, ζ), ζ = bu – cw.(1) This equation will be considered in conjunction with the first equation of the original system ∂u ∂t = L[u]+uf (t, ζ)+g(t, ζ). (2) Equation (1) can be treated separately. Given a solution of this equation, ζ = ζ(x 1 , , x n , t), the function u = u(x 1 , , x n , t) can be determined by solving the linear equation (2) and the function w = w(x 1 , , x n , t) is found as w =(bu – ζ)/c. Remark 2. Let L be a constant-coefficient differential operator with respect to the independent variable x = x 1 and let the condition ∂ ∂t  ζf(t, ζ)+bg(t, ζ)–ch(t, ζ)  = 0 hold true (for example, it is valid if the functions f , g, h are not implicitly dependent on t). Then equation (1) admits an exact, traveling-wave solution ζ = ζ(z), where z = kx – λt with arbitrary constants k and λ. 2. ∂u ∂t = L 1 [u] + uf  u w  , ∂w ∂t = L 2 [w] + wg  u w  . Here, L 1 and L 2 are arbitrary constant-coefficient linear differential operators (of any order) with respect to x. 1 ◦ . Solution: u = e kx–λt y(ξ), w = e 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 system of ordinary differential equations M 1 [y]+λy + yf(y/z)=0, M 2 [z]+λz + zg(y/z)=0, M 1 [y]=e –kx L 1 [e kx y(ξ)], M 2 [z]=e –kx L 2 [e kx z(ξ)]. To the special case k = λ = 0 there corresponds a traveling-wave solution. 2 ◦ . If the operators L 1 and L 2 contain only even derivatives, there are solutions of the form u =[C 1 sin(kx)+C 2 cos(kx)]ϕ(t), w =[C 1 sin(kx)+C 2 cos(kx)]ψ(t); u =[C 1 exp(kx)+C 2 exp(–kx)]ϕ(t), w =[C 1 exp(kx)+C 2 exp(–kx)]ψ(t); u =(C 1 x + C 2 )ϕ(t), w =(C 1 x + C 2 )ψ(t), where C 1 , C 2 ,andk are arbitrary constants. Note that the third solution is degenerate. 1376 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 3. ∂u ∂t = L[u] + uf  t, u w  , ∂w ∂t = L[w] + wg  t, u 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: L[u]=  A k 1 k n (x 1 , , x n , t) ∂ k 1 +···+k n u ∂x k 1 1 ∂x k n n .(1) 1 ◦ . Solution: u = ϕ(t)exp   g(t, ϕ(t)) dt  θ(x 1 , , x n , t), w =exp   g(t, ϕ(t)) dt  θ(x 1 , , x n , t), (2) where the function ϕ = ϕ(t) is described by the first-order nonlinear ordinary differential equation ϕ  t =[f(t, ϕ)–g(t, ϕ)]ϕ,(3) and the function θ = θ(x 1 , , x n , t) satisfi es the linear equation ∂θ ∂t = L[θ]. 2 ◦ . The transformation u = a 1 (t)U + b 1 (t)W , w = a 2 (t)U + b 2 (t)W , where a n (t)andb n (t) are arbitrary functions (n = 1, 2), leads to an equation of the similar form for U and W . Remark. The coefficients of the operator (1) can also depend on the ratio of the unknowns u/w, A k 1 k n = A k 1 k n (x 1 , , x n , t, u/w) (in this case, L will be a quasilinear operator). Then there also exists a solution of the form (2), where ϕ = ϕ(t) is described by the ordinary differential equation (3) and θ = θ(x 1 , , x n , t) satisfies the linear equation ∂θ ∂t = L ◦ [θ], L ◦ = L u/w=ϕ . 4. ∂u ∂t = L[u] + uf  u w  + g  u w  , ∂w ∂t = L[w] + wf  u w  + h  u w  . Here, L is an arbitrary linear differential operator with respect to x 1 , , x n (of any order in derivatives), whose coefficients can be dependent on x 1 , , x n , t: L[u]=  A k 1 k n (x 1 , , x n , t) ∂ k 1 +···+k n u ∂x k 1 1 ∂x k n n , where k 1 + ···+ k n ≥ 1. Let k be a root of the algebraic (transcendental) equation g(k)=kh(k). (1) 1 ◦ . Solution if 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) , T10.4. SYSTEMS OF GENERAL FORM 1377 where the function θ = θ(x 1 , , x n , t) satisfi es the linear equation ∂θ ∂t = L[θ]. (2) 2 ◦ . Solution if 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 1 , , x n , t) satisfi es the linear equation (2). 5. ∂u ∂t = L[u]+uf  t, u w  + u w h  t, u w  , ∂w ∂t = L[w]+wg  t, u w  +h  t, u w  . Solution: u = ϕ(t)G(t)  θ(x 1 , , x n , t)+  h(t, ϕ) G(t) dt  , G(t)=exp   g(t, ϕ) dt  , w = G(t)  θ(x 1 , , x n , t)+  h(t, ϕ) G(t) dt  , where the function ϕ = ϕ(t) is described by the first-order nonlinear ordinary differential equation ϕ  t =[f(t, ϕ)–g(t, ϕ)]ϕ, and the function θ = θ(x 1 , , x n , t) satisfi es the linear equation ∂θ ∂t = L[θ]. 6. ∂u ∂t = L[u] + uf  t, u w  ln u + ug  t, u w  , ∂w ∂t = L[w] + wf  t, u w  ln w + wh  t, u w  . Solution: u(x, t)=ϕ(t)ψ(t)θ(x 1 , , x n , t), w(x, t)=ψ(t)θ(x 1 , , x n , t), where the functions ϕ=ϕ(t)andψ =ψ(t) are determined by solving the ordinary differential equations ϕ  t = ϕ[g(t, ϕ)–h(t, ϕ)+f(t, ϕ)lnϕ], ψ  t = ψ[h(t, ϕ)+f(t, ϕ)lnψ], (1) and the function θ = θ(x 1 , , x n , t) is determined by the differential equation ∂θ ∂t = L[θ]+f(t, ϕ)θ ln θ.(2) Givenasolutiontothefirst equation in (1), the second equation can be solved with the change of variable ψ = e ζ by reducing it to a linear equation for ζ.IfL is a constant-coefficient one-dimensional operator (n = 1)andf = const, then equation (2) has a traveling-wave solution θ = θ(kx – λt). 1378 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 7. F 1  w, ∂w ∂x , , ∂ m w ∂x m , 1 u k ∂w ∂t , 1 u ∂u ∂x , , 1 u ∂ n u ∂x n  =0, F 2  w, ∂w ∂x , , ∂ m w ∂x m , 1 u k ∂w ∂t , 1 u ∂u ∂x , , 1 u ∂ n u ∂x n  =0. Solution: w = W (z), u =[ϕ  (t)] 1/k U(z), z = x + ϕ(t), where ϕ(t) is an arbitrary function, and the functions W (z)andU (z) are determined by the autonomous system of ordinary differential equations F 1  W , W  z , , W (m) z , W  z /U k , U  z /U, , U (n) z /U  = 0, F 2  W , W  z , , W (m) z , W  z /U k , U  z /U, , U (n) z /U  = 0. T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1. ∂ 2 u ∂t 2 = L[u] + uf(t, au – bw) + g(t, au – bw), ∂ 2 w ∂t 2 = L[w] + wf(t, au – bw) + h(t, au – bw). Here, L is an arbitrary linear differential operator (of any order) with respect to the spatial variables x 1 , , x n , whose coefficients can be dependent on x 1 , , x n , t. It is assumed that L[const] = 0. 1 ◦ . Solution: u = ϕ(t)+aθ(x 1 , , x n , t), w = ψ(t)+bθ(x 1 , , x n , t), where ϕ = ϕ(t)andψ = ψ(t) are determined by the system of ordinary differential equations ϕ  tt = ϕf (t, aϕ – bψ)+g(t, aϕ – bψ), ψ  tt = ψf(t, aϕ – bψ)+h(t, aϕ – bψ), and the function θ = θ(x 1 , , x n , t) satisfi es linear equation ∂ 2 θ ∂t 2 = L[θ]+f(t, aϕ – bψ)θ. 2 ◦ . Let us multiply the fi rst equation by a and the second one by –b and add the results together to obtain ∂ 2 ζ ∂t 2 = L[ζ]+ζf(t, ζ)+ag(t, ζ)–bh(t, ζ), ζ = au – bw.(1) This equation will be considered in conjunction with the first equation of the original system ∂ 2 u ∂t 2 = L[u]+uf(t, ζ)+g(t, ζ). (2) T10.4. SYSTEMS OF GENERAL FORM 1379 Equation (1) can be treated separately. Given a solution ζ = ζ(x, t) to equation (1), the function u = u(x 1 , , x n , t) can be determined by solving the linear equation (2) and the function w = w(x 1 , , x n , t) is found as w =(au – ζ)/b. Note three important cases where equation (1) admits exact solutions: (i) Equation (1) admits a spatially homogeneous solution ζ = ζ(t). (ii) Suppose the coeffi cients of L and the functions f , g, h are not implicitly dependent on t. Then equation (1) admits a steady-state solution ζ = ζ(x 1 , , x n ). (iii) If the condition ζf(t, ζ)+bg(t, ζ)–ch(t, ζ)=k 1 ζ + k 0 holds, equation (1) is linear. If the operator L is constant-coefficient, the method of separation of variables can be used to obtain solutions. 2. ∂ 2 u ∂t 2 = L 1 [u] + uf  u w  , ∂ 2 w ∂t 2 = L 2 [w] + wg  u w  . Here, L 1 and L 2 are arbitrary constant-coefficient linear differential operators (of any order) with respect to x. It is assumed that L 1 [const] = 0 and L 2 [const] = 0. 1 ◦ . Solution in the form of the product of two waves traveling at different speeds: u = e kx–λt y(ξ), w = e 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 system of ordinary differential equations γ 2 y  ξξ + 2λγy  ξ + λ 2 y = M 1 [y]+yf(y/z), γ 2 z  ξξ + 2λγz  ξ + λ 2 z = M 2 [z]+zg(y/z), M 1 [y]=e –kx L 1 [e kx y(ξ)], M 2 [z]=e –kx L 2 [e kx z(ξ)]. To the special case k = λ = 0 there corresponds a traveling-wave solution. 2 ◦ . Periodic multiplicative separable solution: u =[C 1 sin(kt)+C 2 cos(kt)]ϕ(x), w =[C 1 sin(kt)+C 2 cos(kt)]ψ(x), where C 1 , C 2 ,andk are arbitrary constants and the functions ϕ = ϕ(x)andψ = ψ(x)are determined by the system of ordinary differential equations L 1 [ϕ]+k 2 ϕ + ϕf(ϕ/ψ)=0, L 2 [ψ]+k 2 ψ + ψg(ϕ/ψ)=0. 3 ◦ . Multiplicative separable solution: u =[C 1 sinh(kt)+C 2 cosh(kt)]ϕ(x), w =[C 1 sinh(kt)+C 2 cosh(kt)]ψ(x), where C 1 , C 2 ,andk are arbitrary constants and the functions ϕ = ϕ(x)andψ = ψ(x)are determined by the system of ordinary differential equations L 1 [ϕ]–k 2 ϕ + ϕf(ϕ/ψ)=0, L 2 [ψ]–k 2 ψ + ψg(ϕ/ψ)=0. 4 ◦ . Degenerate multiplicative separable solution: u =(C 1 t + C 2 )ϕ(x), w =(C 1 t + C 2 )ψ(x), 1380 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS where C 1 and C 2 are arbitrary constants and the functions ϕ = ϕ(x)andψ = ψ(x)are determined by the system of ordinary differential equations L 1 [ϕ]+ϕf(ϕ/ψ)=0, L 2 [ψ]+ψg(ϕ/ψ)=0. Remark 1. The coefficients of L 1 , L 2 and the functions f and g in Items 2 ◦ –4 ◦ can be dependent on x. Remark 2. If L 1 and L 2 contain only even derivatives, there are solutions of the form u =[C 1 sin(kx)+C 2 cos(kx)]U(t), w =[C 1 sin(kx)+C 2 cos(kx)]W (t); u =[C 1 exp(kx)+C 2 exp(–kx)]U(t), w =[C 1 exp(kx)+C 2 exp(–kx)]W (t); u =(C 1 x + C 2 )U(t), w =(C 1 x + C 2 )W (t), where C 1 , C 2 ,andk are arbitrary constants. Note that the third solution is degenerate. 3. ∂ 2 u ∂t 2 = L[u] + uf  t, u w  , ∂ 2 w ∂t 2 = L[w] + wg  t, u 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 the coordinates. Solution: u = ϕ(t)θ(x 1 , , x n ), w = ψ(t)θ(x 1 , , x n ), where the functions ϕ = ϕ(t)andψ = ψ(t) are described by the nonlinear system of second-order ordinary differential equations ϕ  tt = aϕ + ϕf(t, ϕ/ψ), ψ  tt = aψ + ψg(t, ϕ/ψ), a is an arbitrary constant, and the function θ = θ(x 1 , , x n ) satisfies the linear steady-state equation L[θ]=aθ. 4. ∂ 2 u ∂t 2 = L[u] + uf  u w  + g  u w  , ∂ 2 w ∂t 2 = L[w] + wf  u w  + h  u 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. Solution: u = kθ(x 1 , , x n , t), w = θ(x 1 , , x n , t), where k is a root of the algebraic (transcendental) equation g(k)=kh(k) and the function θ = θ(x, t) satisfies the linear equation ∂ 2 θ ∂t 2 = L[θ]+f(k)w + h(k). 5. ∂ 2 u ∂t 2 = L[u] + au ln u + uf  t, u w  , ∂ 2 w ∂t 2 = L[w] + aw ln w + wg  t, u 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 the coordinates. T10.4. SYSTEMS OF GENERAL FORM 1381 Solution: u = ϕ(t)θ(x 1 , , x n ), w = ψ(t)θ(x 1 , , x n ), where the functions ϕ = ϕ(t)andψ = ψ(t) are described by the nonlinear system of second-order ordinary differential equations ϕ  tt = aϕ ln ϕ + bϕ + ϕf(t, ϕ/ψ), ψ  tt = aψ ln ψ + bψ + ψg(t, ϕ/ψ), b is an arbitrary constant, and the function θ = θ(x 1 , , x n ) satisfies the steady-state equation L[θ]+aθ ln θ – bθ = 0. T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in t 1. ∂u m ∂t = L[u m ] + u m f(t, u 1 – b 1 u n , , u n–1 – b n–1 u n ) + g m (t, u 1 – b 1 u n , , u n–1 – b n–1 u n ), m =1, , n. The system involves n + 1 arbitrary functions f, g 1 , , g n that depend on n arguments; L is an arbitrary linear differential operator with respect to the spatial variables 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 m = ϕ m (t)+exp   f(t, ϕ 1 – b 1 ϕ n , , ϕ n–1 – b n–1 ϕ n ) dt  θ(x 1 , , x n , t). Here, the functions ϕ m = ϕ m (t) are determined by the system of ordinary differential equations ϕ  m = ϕ m f(t, ϕ 1 – b 1 ϕ n , , ϕ n–1 – b n–1 ϕ n )+g m (t, ϕ 1 – b 1 ϕ n , , ϕ n–1 – b n–1 ϕ n ), where m = 1, , n, the prime denotes the derivative with respect to t, and the function θ = θ(x 1 , , x n , t) satisfies the linear equation ∂θ ∂t = L[θ]. 2. ∂u m ∂t = L[u m ] + u m f m  t, u 1 u n , , u n–1 u n  + u m u n g  t, u 1 u n , , u n–1 u n  , ∂u n ∂t = L[u n ] + u n f n  t, u 1 u n , , u n–1 u n  + g  t, u 1 u n , , u n–1 u n  . Here, m = 1, , n – 1 and the system involves n + 1 arbitrary functions f 1 , , f n , g that depend on n arguments; L is an arbitrary linear differential operator with respect to the spatial variables x 1 , , x n (of any order in derivatives), whose coefficients can be dependent on x 1 , , x n , t. It is assumed that L[const] = 0. . functions (n = 1, 2), leads to an equation of the similar form for U and W . Remark. The coefficients of the operator (1) can also depend on the ratio of the unknowns u/w, A k 1 k n = A k 1 k n (x 1 ,. C 2 )ψ(x), 1380 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS where C 1 and C 2 are arbitrary constants and the functions ϕ = ϕ(x )and = ψ(x)are determined by the system of ordinary differential. The coefficients of L 1 , L 2 and the functions f and g in Items 2 ◦ –4 ◦ can be dependent on x. Remark 2. If L 1 and L 2 contain only even derivatives, there are solutions of the form u =[C 1 sin(kx)+C 2 cos(kx)]U(t),