Handbook of mathematics for engineers and scienteists part 193 pot

1312 NONLINEAR MATHEMATIC AL PHYSICS EQUATIONS 5 ◦ . Solution (A, B,andC are arbitrary constants): w(x, t)=ψ(z)exp  i(Axt – 2 3 A 2 t 3 + Bt + C)  , z = x – At 2 , where the function ψ = ψ(z) is determined by the ordinary differential equation ψ  zz + f(|ψ|)ψ –(Az + B)ψ = 0. 6 ◦ . Solutions: w(x, t)= 1 √ C 1 t exp  iϕ(x, t)  , ϕ(x, t)= (x + C 2 ) 2 4t +  f  |C 1 t| –1/2  dt + C 3 , where C 1 , C 2 ,andC 3 are arbitrary real constants. 7 ◦ . Solution: w(x, t)=u(x)exp  iϕ(x, t)  , ϕ(x, t)=C 1 t + C 2  dx u 2 (x) + C 3 , where C 1 , C 2 ,andC 3 are arbitrary real constants, and the function u = u(x) is determined by the autonomous ordinary differential equation u  xx – C 1 u – C 2 2 u –3 + f(|u|)u = 0. 8 ◦ . There is an exact solution of the form w(x, t)=u(z)exp  iAt + iϕ(z)  , z = kx + λt, where A, k,andλ are arbitrary real constants. T9.2. Hyperbolic Equations T9.2.1. Nonlinear Wave Equations of the Form ∂ 2 w ∂t 2 = a ∂ 2 w ∂x 2 + f(w) 1. ∂ 2 w ∂t 2 = ∂ 2 w ∂x 2 + aw + bw n . 1 ◦ . Traveling-wave solutions for a > 0: w(x, t)=  2b sinh 2 z a(n + 1)  1 1–n , z = 1 2 √ a (1 – n)(x sinh C 1 t cosh C 1 )+C 2 if b(n + 1)>0, w(x, t)=  – 2b cosh 2 z a(n + 1)  1 1–n , z = 1 2 √ a (1 – n)(x sinh C 1 t cosh C 1 )+C 2 if b(n + 1)<0, where C 1 and C 2 are arbitrary constants. 2 ◦ . Traveling-wave solutions for a < 0 and b(n + 1)>0: w(x, t)=  – 2b cos 2 z a(n + 1)  1 1–n , z = 1 2  |a| (1 – n)(x sinh C 1 t cosh C 1 )+C 2 . 3 ◦ .Fora = 0, there is a self-similar solution of the form w = t 2 1–n F (z), where z = x/t. 4 ◦ . For other exact solutions of this equation, see equation T9.2.1.7 with f(w)=aw + bw n . T9.2. HYPERBOLIC EQUATIONS 1313 2. ∂ 2 w ∂t 2 = ∂ 2 w ∂x 2 + aw n + bw 2n–1 . Solutions: w(x, t)=  a(1 – n) 2 2(n + 1) (x sinh C 1 t cosh C 1 + C 2 ) 2 – b(n + 1) 2an  1 1–n , w(x, t)=  1 4 a(1 – n) 2  (t + C 1 ) 2 –(x + C 2 ) 2  – b an  1 1–n , where C 1 and C 2 are arbitrary constants. 3. ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 + be βw . 1 ◦ . Traveling-wave solutions: w(x, t)= 1 β ln  2(B 2 – a 2 A 2 ) bβ(Ax + Bt + C) 2  , w(x, t)= 1 β ln  2(a 2 A 2 – B 2 ) bβ cosh 2 (Ax + Bt + C)  , w(x, t)= 1 β ln  2(B 2 – a 2 A 2 ) bβ sinh 2 (Ax + Bt + C)  , w(x, t)= 1 β ln  2(B 2 – a 2 A 2 ) bβ cos 2 (Ax + Bt + C)  , where A, B,andC are arbitrary constants. 2 ◦ . Functional separable solutions: w(x, t)= 1 β ln  8a 2 C bβ  – 2 β ln   (x + A) 2 – a 2 (t + B) 2 + C   , w(x, t)=– 2 β ln  C 1 e λx  2bβ 2aλ sinh(aλt + C 2 )  , w(x, t)=– 2 β ln  C 1 e λx  –2bβ 2aλ cosh(aλt + C 2 )  , w(x, t)=– 2 β ln  C 1 e aλt  –2bβ 2aλ sinh(λx + C 2 )  , w(x, t)=– 2 β ln  C 1 e aλt  2bβ 2aλ cosh(λx + C 2 )  , where A, B, C, C 1 , C 2 ,andλ are arbitrary constants. 3 ◦ . General solution: w(x, t)= 1 β  f(z)+g(y)  – 2 β ln     k  exp  f(z)  dz – bβ 8a 2 k  exp  g(y)  dy     , z = x – at, y = x + at, where f = f (z)andg = g(y) are arbitrary functions and k is an arbitrary constant. 1314 NONLINEAR MATHEMATIC AL PHYSICS EQUATIONS 4. ∂ 2 w ∂t 2 = ∂ 2 w ∂x 2 + ae βw + be 2βw . 1 ◦ . Traveling-wave solutions: w(x, t)=– 1 β ln  aβ C 2 1 – C 2 2 + C 3 exp(C 1 x + C 2 t)+ 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 t)  , w(x, t)=– 1 β ln  aβ C 2 2 – C 2 1 +  a 2 β 2 + bβ(C 2 2 – C 2 1 ) C 2 2 – C 2 1 sin(C 1 x + C 2 t + C 3 )  , where C 1 , C 2 ,andC 3 are arbitrary constants. 2 ◦ . For other exact solutions of this equation, see equation T9.2.1.7 with f (w)=ae βw + be 2βw . 5. ∂ 2 w ∂t 2 = a ∂ 2 w ∂x 2 + b sinh(λw). Sinh-Gordon equation. It arises in some areas of physics. 1 ◦ . Traveling-wave solutions: w(x, t)= 2 λ ln  tan bλ(kx + μt + θ 0 ) 2  bλ(μ 2 – ak 2 )  , w(x, t)= 4 λ arctanh  exp bλ(kx + μt + θ 0 )  bλ(μ 2 – ak 2 )  , where k, μ,andθ 0 are arbitrary constants. It is assumed that bλ(μ 2 – ak 2 )>0 in both formulas. 2 ◦ . Functional separable solution: w(x, t)= 4 λ arctanh  f(t)g(x)  ,arctanhz = 1 2 ln 1 + z 1 – z , where the functions f = f(t)andg = g(x) are determined by the first-order autonomous ordinary differential equations (f  t ) 2 = Af 4 + Bf 2 + C, a(g  x ) 2 = Cg 4 +(B – bλ)g 2 + A, where A, B,andC are arbitrary constants. 3 ◦ . For other exact solutions of this equation, see equation T9.2.1.7 with f(w)=b sinh(λw). 6. ∂ 2 w ∂t 2 = a ∂ 2 w ∂x 2 + b sin(λw). Sine-Gordon equation. It arises in differential geometry and various areas of physics. 1 ◦ . Traveling-wave solutions: w(x, t)= 4 λ arctan  exp  bλ(kx + μt + θ 0 )  bλ(μ 2 – ak 2 )  if bλ(μ 2 – ak 2 )>0, w(x, t)=– π λ + 4 λ arctan  exp  bλ(kx + μt + θ 0 )  bλ(ak 2 – μ 2 )  if bλ(μ 2 – ak 2 )<0, where k, μ,andθ 0 are arbitrary constants. The first expression corresponds to a single- soliton solution. T9.2. HYPERBOLIC EQUATIONS 1315 2 ◦ . Functional separable solutions: w(x, t)= 4 λ arctan  μ sinh(kx + A) k √ a cosh(μt + B)  , μ 2 = ak 2 + bλ > 0; w(x, t)= 4 λ arctan  μ sin(kx + A) k √ a cosh(μt + B)  , μ 2 = bλ – ak 2 > 0; w(x, t)= 4 λ arctan  γ μ e μ(t+A) + ak 2 e –μ(t+A) e kγ(x+B) + e –kγ(x+B)  , μ 2 = ak 2 γ 2 + bλ > 0, where A, B, k,andγ are arbitrary constants. 3 ◦ .AnN-soliton solution is given by (a = 1, b =–1,andλ = 1) w(x, t) = arccos  1 – 2  ∂ 2 ∂x 2 – ∂ 2 ∂t 2  (ln F )  , F =det  M ij  , M ij = 2 a i + a j cosh  z i + z j 2  , z i = x – μ i t + C i  1 – μ 2 i , a i =  1 – μ i 1 + μ i , where μ i and C i are arbitrary constants. 4 ◦ . For other exact solutions of the original equation, see equation T9.2.1.7 with f (w)= b sin(λw). 5 ◦ . The sine-Gordon equation is integrated by the inverse scattering method; see the book by Novikov et al. (1984). 7. ∂ 2 w ∂t 2 = ∂ 2 w ∂x 2 + f(w). Nonlinear Klein–Gordon equation. 1 ◦ . Suppose w = w(x, t) is a solution of the equation in question. Then the functions w 1 = w( x + C 1 , t + C 2 ), w 2 = w(x cosh β + t sinh β, t cosh β + x sinh 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). 2 ◦ . Traveling-wave solution in implicit form:   C 1 + 2 λ 2 – k 2  f(w) dw  –1/2 dw = kx + λt + C 2 , where C 1 , C 2 , k,andλ are arbitrary constants. 3 ◦ . Functional separable solution: w = w(ξ), ξ = 1 4 (t + C 1 ) 2 – 1 4 (x + C 2 ) 2 , where C 1 and C 2 are arbitrary constants, and the function w = w(ξ) is determined by the ordinary differential equation ξw  ξξ + w  ξ – f(w)=0. 1316 NONLINEAR MATHEMATIC AL PHYSICS EQUATIONS T9.2.2. Other Nonlinear Wave Equations 1. ∂ 2 w ∂t 2 = a ∂ ∂x  w ∂w ∂x  . 1 ◦ . Solutions: w(x, t)= 1 2 aA 2 t 2 + Bt + Ax + C, w(x, t)= 1 12 aA –2 (At + B) 4 + Ct+ D + x(At + B), w(x, t)= 1 a  x + A t + B  2 , w(x, t)=(At + B) √ Cx + D, w(x, t)=  A(x + aλt)+B + aλ 2 , where A, B, C, D,andλ are arbitrary constants. 2 ◦ . Generalized separable solution quadratic in x: w(x, t)= 1 at 2 x 2 +  C 1 t 2 + C 2 t 3  x + aC 2 1 4t 2 + C 3 t + C 4 t 2 + 1 2 aC 1 C 2 t 3 + 1 54 aC 2 2 t 8 , where C 1 , , C 4 are arbitrary constants. 3 ◦ . Solution: w = U(z)+4aC 2 1 t 2 + 4aC 1 C 2 t, z = x + aC 1 t 2 + aC 2 t, where C 1 and C 2 are arbitrary constants and the function U(z) is determined by the first- order ordinary differential equation (U – aC 2 2 )U  z – 2C 1 U = 8C 2 1 z + C 3 . 4 ◦ . See also equation T9.2.2.5 with f(w)=aw. 2. ∂ 2 w ∂t 2 = a ∂ ∂x  w n ∂w ∂x  + bw k . There are solutions of the following forms: w(x, t)=U(z), z = λx + βt (traveling-wave solution); w(x, t)=t 2 1–k V (ξ), ξ = xt k–n–1 1–k (self-similar solution). 3. ∂ 2 w ∂t 2 = ∂ ∂x  ae λw ∂w ∂x  , a >0. 1 ◦ . Additive separable solutions: w(x, t)= 1 λ ln |Ax + B| + Ct + D, w(x, t)= 2 λ ln |Ax + B| – 2 λ ln | A √ at+ C|, w(x, t)= 1 λ ln(aA 2 x 2 + Bx + C)– 2 λ ln(aAt + D), T9.2. HYPERBOLIC EQUATIONS 1317 w(x, t)= 1 λ ln(Ax 2 + Bx + C)+ 1 λ ln  p 2 aA cos 2 (pt + q)  , w(x, t)= 1 λ ln(Ax 2 + Bx + C)+ 1 λ ln  p 2 aA sinh 2 (pt + q)  , w(x, t)= 1 λ ln(Ax 2 + Bx + C)+ 1 λ ln  –p 2 aA cosh 2 (pt + q)  , where A, B, C, D, p,andq are arbitrary constants. 2 ◦ . There are solutions of the following forms: w(x, t)=F (z), z = kx + βt (traviling-wave solution); w(x, t)=G(ξ), ξ = x/t (self-similar solution); w(x, t)=H(η)+2(k – 1)λ –1 ln t, η = xt –k ; w(x, t)=U(ζ)–2λ –1 ln |t|, ζ = x + k ln |t|; w(x, t)=V (ζ)–2λ –1 t, η = xe t , where k and β are arbitrary constants. 4. ∂ 2 w ∂t 2 = a x n ∂ ∂x  x n ∂w ∂x  + f(w), a >0. To n = 1 and n = 2 there correspond nonlinear waves with axial and central symmetry, respectively. Functional separable solution: w = w(ξ), ξ =  ak(t + C) 2 – kx 2 , where w(ξ) is determined by the ordinary differential equation w  ξξ +(1 + n)ξ –1 w  ξ = (ak) –1 f(w). 5. ∂ 2 w ∂t 2 = ∂ ∂x  f(w) ∂w ∂x  . This equation is encountered in wave and gas dynamics. 1 ◦ . Traveling-wave solution in implicit form: λ 2 w –  f(w) dw = A(x + λt)+B, where A, B,andλ are arbitrary constants. 2 ◦ . Self-similar solution: w = w(ξ), ξ = x + A t + B , where the function w(ξ) is determined by the ordinary differential equation  ξ 2 w  ξ )  ξ =  f(w)w  ξ   ξ , which admits the first integral  ξ 2 – f(w)  w  ξ = C. To the special case C = 0 there corresponds the solution in implicit form: ξ 2 = f (w). 3 ◦ . The equation concerned can be reduced to a linear equation; see Item 3 ◦ of equa- tion T9.3.3.3, where one should set g(w)=–1 and y = t. 1318 NONLINEAR MATHEMATIC AL PHYSICS EQUATIONS T9.3. Elliptic Equations T9.3.1. Nonlinear Heat Equations of the Form ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = f(w) 1. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = aw + bw n . 1 ◦ . Traveling-wave solutions for a > 0: w(x, y)=  2b sinh 2 z a(n + 1)  1 1–n , z = 1 2 √ a (1 – n)(x sin C 1 + y cos C 1 )+C 2 if b(n + 1)>0, w(x, y)=  – 2b cosh 2 z a(n + 1)  1 1–n , z = 1 2 √ a (1 – n)(x sin C 1 + y cos C 1 )+C 2 if b(n + 1)<0, where C 1 and C 2 are arbitrary constants. 2 ◦ . Traveling-wave solutions for a < 0 and b(n + 1)>0: w(x, y)=  – 2b cos 2 z a(n + 1)  1 1–n , z = 1 2  |a| (1 – n)(x sin C 1 + y cos C 1 )+C 2 . 3 ◦ .Fora = 0, there is a self-similar solution of the form w = x 2 1–n F (z), where z = y/x. 4 ◦ . For other exact solutions of this equation, see equation T9.3.1.7 with f(w)=aw + bw n . 2. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = aw n + bw 2n–1 . Solutions: w(x, y)=  a(1 – n) 2 2(n + 1) (x sin C 1 + y cos C 1 + C 2 ) 2 – b(n + 1) 2an  1 1–n , w(x, y)=  1 4 a(1 – n) 2  (x + C 1 ) 2 +(y + C 2 ) 2  – b an  1 1–n , where C 1 and C 2 are arbitrary constants. 3. ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = ae βw . This equation occurs in combustion theory and is a special case of equation T9.3.1.7 with f(w)=ae βw . . solution of the form w(x, t)=u(z)exp  iAt + iϕ(z)  , z = kx + λt, where A, k ,and are arbitrary real constants. T9.2. Hyperbolic Equations T9.2.1. Nonlinear Wave Equations of the Form ∂ 2 w ∂t 2 =. n)(x sinh C 1 t cosh C 1 )+C 2 . 3 ◦ .Fora = 0, there is a self-similar solution of the form w = t 2 1–n F (z), where z = x/t. 4 ◦ . For other exact solutions of this equation, see equation T9.2.1.7. n)(x sin C 1 + y cos C 1 )+C 2 . 3 ◦ .Fora = 0, there is a self-similar solution of the form w = x 2 1–n F (z), where z = y/x. 4 ◦ . For other exact solutions of this equation, see equation T9.3.1.7