Any solution of the Poisson equationΔw = C is also a solution of the original equation these are “inviscid” solutions... and Degasperis, A., Spectral Transform and Solitons: Tolls to Sol
Trang 1T9.5 H IGHER -O RDER E QUATIONS 1333
6◦ The Boussinesq equation is solved by the inverse scattering method Any rapidly decaying function F = F (x, y; t) as x → +∞ and satisfying simultaneously the two linear
equations
1
√
3
∂F
∂t + ∂
2F
∂x2 –
∂2F
∂y2 =0, ∂
3F
∂x3 +
∂3F
∂y3 =0
generates a solution of the Boussinesq equation in the form
w=12 d
dx K (x, x; t), where K(x, y; t) is a solution of the linear Gel’fand–Levitan–Marchenko integral equation
K (x, y; t) + F (x, y; t) +
∞
x K (x, s; t)F (s, y; t) ds =0
Time t appears here as a parameter.
∂y
∂
∂x (Δw) – ∂w
∂x
∂
∂y (Δw) = νΔΔw, Δw = ∂2w
∂x2 + ∂
2w
∂y2
There is a two-dimensional stationary equation of motion of a viscous incompressible fluid — it is obtained from the Navier–Stokes equation by the introduction of the stream
function w.
1◦ Suppose w(x, y) is a solution of the equation in question Then the functions
w1= –w(y, x),
w2= w(C1x + C2, C1y + C3) + C4,
w3= w(x cos α + y sin α, –x sin α + y cos α),
where C1, , C4and α are arbitrary constants, are also solutions of the equation.
2◦ Any solution of the Poisson equationΔw = C is also a solution of the original equation
(these are “inviscid” solutions)
3◦ Solutions in the form of a one-variable function or the sum of functions with different
arguments:
w (y) = C1y3+ C
2y2+ C
3y + C4,
w (x, y) = C1x2+ C
2x + C3y2+ C
4y + C5,
w (x, y) = C1exp(–λy) + C2y2+ C
3y + C4+ νλx,
w (x, y) = C1exp(λx) – νλx + C2exp(λy) + νλy + C3,
w (x, y) = C1exp(λx) + νλx + C2exp(–λy) + νλy + C3,
where C1, , C5and λ are arbitrary constants.
4◦ Generalized separable solutions:
w (x, y) = A(kx + λy)3+ B(kx + λy)2+ C(kx + λy) + D,
w (x, y) = Ae–λ(y+kx) + B(y + kx)2+ C(y + kx) + νλ(k2+1)x + D,
w (x, y) =6νx (y + λ)–1+ A(y + λ)3+ B(y + λ)–1+ C(y + λ)–2+ D,
Trang 21334 NONLINEARMATHEMATICALPHYSICSEQUATIONS
w (x, y) = (Ax + B)e–λy + νλx + C,
w (x, y) =
A sinh(βx) + B cosh(βx)
e–λy+ ν
λ (β2+ λ2)x + C,
w (x, y) =
A sin(βx) + B cos(βx)
e–λy+ ν
λ (λ2– β2)x + C,
w (x, y) = Ae λy+βx + Be γx + νγy + ν
λ γ (β – γ)x + C, γ =
λ2+ β2,
where A, B, C, D, k, β, and λ are arbitrary constants.
5◦ Generalized separable solution linear in x:
w (x, y) = F (y)x + G(y), (1)
where the functions F = F (y) and G = G(y) are determined by the autonomous system of
fourth-order ordinary differential equations
F
y F yy – F F yyy = νF yyyy , (2)
G
y F yy – F G yyy = νG yyyy. (3) Equation (2) has the following particular solutions:
F = ay + b,
F =6ν (y + a)–1,
F = ae–λy+ λν, where a, b, and λ are arbitrary constants.
Let F = F (y) be a solution of equation (2) (F const) Then the corresponding general solution of equation (3) can be written in the form
G=
U dy + C4, U = C1U1+ C2U2+ C3
U2
U1
Φ dy – U1
U2
Φ dy
,
where C1, C2, C3, and C4are arbitrary constants, and
U1=
F
yy if F yy 0,
F if F yy ≡ 0, U2= U1
Φ dy
U2 1 , Φ = exp
–1
ν
F dy
6◦ There is an exact solution of the form (generalizes the solution of Item5◦):
w (x, y) = F (z)x + G(z), z = y + kx, kis any number
7◦ Self-similar solution:
w=
F (z) dz + C1, z= arctan
x y
,
where the function F is determined by the first-order autonomous ordinary differential
equation 3ν (F z )2–2F3+12νF2+ C
2F + C3=0 (C1, C2, and C3are arbitrary constants)
8◦ There is an exact solution of the form (generalizes the solution of Item7◦):
w = C1ln|x|+
V (z) dz + C2, z= arctan
x y
Trang 3
R EFERENCES FOR C HAPTER T9 1335
References for Chapter T9
Ablowitz, M J and Segur, H., Solitons and the Inverse Scattering Transform, Society for Industrial and
Applied Mathematics, Philadelphia, 1981.
Andreev, V K., Kaptsov, O V., Pukhnachov, V V., and Rodionov, A A., Applications of Group-Theoretical
Methods in Hydrodynamics, Kluwer Academic, Dordrecht, 1999.
Bullough, R K and Caudrey, P J (Editors), Solitons, Springer-Verlag, Berlin, 1980.
Burde, G I., New similarity reductions of the steady-state boundary-layer equations, J Physica A: Math Gen.,
Vol 29, No 8, pp 1665–1683, 1996.
Calogero, F and Degasperis, A., Spectral Transform and Solitons: Tolls to Solve and Investigate Nonlinear
Evolution Equations, North-Holland, Amsterdam, 1982.
Cariello, F and Tabor, M., Painlev´e expansions for nonintegrable evolution equations, Physica D, Vol 39,
No 1, pp 77–94, 1989.
Clarkson, P A and Kruskal, M D., New similarity reductions of the Boussinesq equation, J Math Phys.,
Vol 30, No 10, pp 2201–2213, 1989.
Dodd, R K., Eilbeck, J C., Gibbon, J D., and Morris, H C., Solitons and Nonlinear Wave Equations,
Academic Press, London, 1982.
Galaktionov, V A., Invariant subspaces and new explicit solutions to evolution equations with quadratic
nonlinearities, Proc Roy Soc Edinburgh, Vol 125A, No 2, pp 225–448, 1995.
Gardner, C S., Greene, J M., Kruskal, M D., and Miura, R M., Method for solving the Korteweg–de
Vries equation, Phys Rev Lett., Vol 19, No 19, pp 1095–1097, 1967.
Goursat, E., A Course of Mathematical Analysis, Vol 3, Part 1 [Russian translation], Gostekhizdat, Moscow,
1933.
Grundland, A M and Infeld, E., A family of non-linear Klein-Gordon equations and their solutions, J Math.
Phys., Vol 33, pp 2498–2503, 1992.
Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solutions, Phys Rev.
Lett., Vol 27, p 1192, 1971.
Ibragimov, N H (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol 1, Symmetries,
Exact Solutions and Conservation Laws, CRC Press, Boca Raton, 1994.
Kawahara, T and Tanaka, M., Interactions of traveling fronts: an exact solution of a nonlinear diffusion
equation, Phys Lett., Vol 97, p 311, 1983.
Kersner, R., On some properties of weak solutions of quasilinear degenerate parabolic equations, Acta Math.
Academy of Sciences, Hung., Vol 32, No 3–4, pp 301–330, 1978.
Novikov, S P., Manakov, S V., Pitaevskii, L B., and Zakharov, V E., Theory of Solitons The Inverse
Scattering Method, Plenum Press, New York, 1984.
Pavlovskii, Yu N., Investigation of some invariant solutions to the boundary layer equations [in Russian],
Zhurn Vychisl Mat i Mat Fiziki, Vol 1, No 2, pp 280–294, 1961.
Polyanin, A D and Zaitsev, V F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd
Edition, Chapman & Hall/CRC Press, Boca Raton, 2004.
Polyanin, A D and Zaitsev, V F., Handbook of Nonlinear Partial Differential Equations, Chapman &
Hall/CRC Press, Boca Raton, 2004.
Samarskii, A A., Galaktionov, V A., Kurdyumov, S P., and Mikhailov, A P., Blow-Up in Problems for
Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995.
Svirshchevskii, S R., Lie–B¨acklund symmetries of linear ODEs and generalized separation of variables in
nonlinear equations, Phys Lett A, Vol 199, pp 344–348, 1995.
Trang 5Chapter T10
Systems of Partial
Differential Equations
T10.1 Nonlinear Systems of Two First-Order Equations
∂t = buw.
General solution:
t (t)
aϕ (x) + bψ(t), w= –
ϕ
x (x)
aϕ (x) + bψ(t), where ϕ(x) and ψ(t) are arbitrary functions.
∂t = bu k.
General solution:
w = ϕ(x) + E(x)
ψ (t) – 1
2ak
E (x) dx
–1
,
u=
1
b
∂w
∂t
1/k
, E (x) = exp
ak
ϕ (x) dx
,
where ϕ(x) and ψ(t) are arbitrary functions.
∂t = bu k w.
General solution:
u=
–ψ t (t)
bnψ (t) – akϕ(x)
1/k
, w=
ϕ
x (x)
bnψ (t) – akϕ(x)
1/n
,
where ϕ(x) and ψ(t) are arbitrary functions.
∂t = u k g(w).
1◦ First integral:
∂w
∂x = kg(w)
f (w)
g (w) dw + θ(x)g(w), (1)
where θ(x) is an arbitrary function The first integral (1) may be treated as a first-order ordinary differential equation in x On finding its general solution, one should replace the constant of integration C with an arbitrary function of time ψ(t), since w is dependent on
x and t.
1337
Trang 61338 SYSTEMS OFPARTIALDIFFERENTIALEQUATIONS
2◦ To the special case θ(x) = const in (1) there correspond special solutions of the form
w = w(z), u = [ψ (t)]1/kv (z), z = x + ψ(t) (2)
involving one arbitrary function ψ(t), with the prime denoting a derivative The functions
w (z) and v(z) are described by the autonomous system of ordinary differential equations
v
z = f (w)v,
w
z = g(w)v k,
the general solution of which can be written in implicit form as
dw
g (w)[kF (w) + C1] = z + C2, v = [kF (w) + C1]
1/k, F (w) =
f (w)
g (w) dw.
∂x = f (a1u + b1w), ∂w
∂t = g(a2u + b2w).
LetΔ = a1b2– a2b1≠ 0
Additive separable solution:
u= 1
Δ[b2ϕ (x) – b1ψ (t)], w=
1
Δ[a1ψ (t) – a2ϕ (x)],
where the functions ϕ(x) and ψ(t) are determined by the autonomous ordinary differential
equations
b2
Δϕ x = f (ϕ),
a1
Δψ t = g(ψ).
Integrating yields
b2
Δ
dϕ
f (ϕ) = x + C1,
a1
Δ
dψ
g (ψ) = t + C2.
∂t = g(au + bw).
Solution:
u = b(k1x – λ1t ) + y(ξ), w = –a(k1x – λ1t ) + z(ξ), ξ = k2x – λ2t,
where k1, k2, λ1, and λ2 are arbitrary constants, and the functions y(ξ) and z(ξ) are
determined by the autonomous system of ordinary differential equations
k2y
ξ + bk1= f (ay + bz), –λ2z
ξ + aλ1= g(ay + bz).
∂t = ug(au – bw) + wh(au – bw) + r(au – bw).
Here, f (z), g(z), h(z), and r(z) are arbitrary functions.
Generalized separable solution:
u = ϕ(t) + bθ(t)x, w = ψ(t) + aθ(t)x.
Here, the functions ϕ = ϕ(t), ψ = ψ(t), and θ = θ(t) are determined by a system involving
one algebraic (transcendental) and two ordinary differential equations:
bθ = f (aϕ – bψ),
aθ
t = bθg(aϕ – bψ) + aθh(aϕ – bψ),
ψ
t = ϕg(aϕ – bψ) + ψh(aϕ – bψ) + r(aϕ – bψ).
Trang 7T10.1 N ONLINEAR S YSTEMS OF T WO F IRST -O RDER E QUATIONS 1339
∂x = f (au – bw) + cw, ∂w
∂t = ug(au – bw) + wh(au – bw) + r(au – bw).
Here, f (z), g(z), h(z), and r(z) are arbitrary functions.
Generalized separable solution:
u = ϕ(t) + bθ(t)e λx, w = ψ(t) + aθ(t)e λx, λ= ac
b
Here, the functions ϕ = ϕ(t), ψ = ψ(t), and θ = θ(t) are determined by a system involving
one algebraic (transcendental) and two ordinary differential equations:
f (aϕ – bψ) + cψ =0,
ψ
t = ϕg(aϕ – bψ) + ψh(aϕ – bψ) + r(aϕ – bψ),
aθ
t = bθg(aϕ – bψ) + aθh(aϕ – bψ).
∂x = e λu f (λu – σw), ∂w
∂t = e σw g(λu – σw).
Solutions:
u = y(ξ) – 1
λ ln(C1t + C2), w = z(ξ) – 1
σ ln(C1t + C2), ξ= x + C3
C1t + C2, where the functions y(ξ) and z(ξ) are determined by the system of ordinary differential
equations
y
ξ = e λy f (λy – σz), –C1ξz
ξ–
C1
σ = e σz g (λy – σz).
∂x = u k f (u n w m), ∂w
∂t = w s g(u n w m).
Self-similar solution with s≠ 1and n≠ 0:
u = t n(s–1) m y (ξ), w = t– s–11 z (ξ), ξ = xt m(k–1) n(s–1), where the functions y(ξ) and z(ξ) are determined by the system of ordinary differential
equations
y
ξ = y k f (y n z m),
m (k –1)ξz ξ – nz = n(s –1)z s g (y n z m).
∂x = u k f (u n w m), ∂w
∂t = wg(u n w m).
1◦ Solution:
u = e mt y (ξ), w = e–nt z (ξ), ξ = e m(k–1) t x,
where the functions y(ξ) and z(ξ) are determined by the system of ordinary differential
equations
y
ξ = y k f (y n z m),
m (k –1)ξz ξ – nz = zg(y n z m).