Equations of the Form ∂w =a∂ w 4 +F x, t, w, ∂w
10.1.1. Equations of the Form ∂w =a∂ w 4 +F x, t, w, ∂w
∂t
∂x
∂x
∂w 4 ∂ w
1. = a ∂t
+ bw ln w + f (t)w.
∂x 4
1 ◦ . Generalized traveling-wave solution:
e bt + e bt
w(x, t) = exp 4 Ae bt x + Be bt aA −
e bt f (t) dt ,
where
A and B are arbitrary constants.
2 ◦ . Solution:
w(x, t) = exp − Ae bt + e bt e bt f (t) dt ϕ(z), z = x + λt, where
A and λ are arbitrary constants, and the function ϕ = ϕ(z) is determined by the autonomous ordinary differential equation
aϕ ′′′′ zzzz − λϕ ′ z + bϕ ln ϕ = 0,
whose order can be reduced by one.
3 ◦ . The substitution
w(x, t) = exp − e bt e bt f (t) dt u(x, t)
leads to the simpler equation
2. = a + f (t)w ln w + [g(t)x + h(t)]w.
∂t
∂x 4
This is a special case of equation 11.1.2.5 with n = 4. ∂w
+ f (w).
∂x 4
∂x
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
1 = w(x + C 1 e bt , t+C 2 ),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Generalized traveling-wave solution:
w = w(z), z=x+C 1 e − bt ,
where the function w(z) is determined by the ordinary differential equation
aw ′′′′ zzzz +( bz + c)w ′ z + f (w) = 0.
This equation describes the evolution of nonlinear waves in a dispersive medium; see Rudenko and Robsman (2002).
1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function
1 w(C 1 x + aC 1 C 2 t+C 3 , C 1 t+C 4 )+ C 2 ,
where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solutions:
x+C 1
w(x, t) = −
at + C 2 120 b
w(x, t) = −
a(x + aC 1 t+C 2 ) 3
The first solution is degenerate and the second one is a traveling-wave solution.
3 ◦ . Traveling-wave solution in implicit form:
1 1 /3 3 Z C w+C 2 dη
2 2 /3 = C 1 x + aC 1 C 2 t+C 3 9 . a 0 (1 −
With C 1 = 1 and C 2 = C 3 = 0, we have the stationary solution obtained in Rudenko and Robsman (2002).
4 ◦ . Traveling-wave solution (generalizes the second solution of Item 2 ◦ and the solution of Item 3 ◦ ):
w = w(ξ), ξ = x − λt,
where the function w(ξ) is determined by the third-order autonomous ordinary differential equation
C and λ are arbitrary constants.
5 ◦ . Self-similar solution:
w(x, t) = t −3 /4
u(η), −1 η = xt /4 ,
where the function u(η) is determined by the ordinary differential equation
1 bu 3 ′′′′
ηηηη = auu η ′ + 4 ηu ′ η + 4 u.
6 ◦ . Solution:
w(x, t) = U (ζ) + 2C 2 1 t, ζ = x + aC 1 t + C 2 t,
where C 1 and C 2 are arbitrary constants and the function U (ζ) is determined by the third-order ordinary differential equation
bU ζζζ ′′′ 1 − 2 2 aU + C 2 U = −2C 1 ζ+C 3 .
7 ◦ . Solution:
z = ϕ(t)x + ψ(t). Here, the functions ϕ(t) and ψ(t) are defined by
ϕ(t) = (4At + C 1 ) −1 /4 , ψ(t) = C 2 (4 At + C 1 3 ) −1 /4 + C 3 (4 At + C 1 ) /4 ,
where
F (z) is determined by the ordinary differential equation
1 , A, C C 2 , and C 3 are arbitrary constants, and the function
bF zzzz ′′′′ − aF F z ′
−2 AF + 3
z = 0.
the independent variables). The equation admits a formal series solution of the form
w(x, t) =
3 w n ( t)[x − ϕ(t)] n [ .
x − ϕ(t)] n=0
The series coefficients w n = w n ( t) are expressed as w 0 = −120, w 1 = w 2 = 0, w 3 =− ϕ ′ ( t), w 4 = w 5 = 0, w 6 = ψ(t),
n−6
( n + 1)(n − 6)(n − 13 n + 60)w n = ( m − 3)w n−m w m + w ′ n−4 ,
m=6
where ϕ(t) and ψ(t) is an arbitrary function. This solution has a singularity at x = ϕ(t).
9 ◦ . If a = b = −1, the equation also admits the formal series solution
t n−1 n−1 X
w(x, t) = 2 +
n=1
k=0
1 where 1 A
0 is an arbitrary constant and the other coefficients can be expressed in terms of A 0 with recurrence relations. This solution can be generalized with the help of translations in the independent ✂✁ variables.
The solutions of Items 8 ◦ and 9 ◦ were obtained by V. G. Baydulov (private communication, 2002).
∂w ∂w
5. = aw
+ f (t).
∂t ∂x
∂x 4
The transformation
w = u(z, t) +
f (τ ) dτ ,
z=x+a
( t − τ )f (τ ) dτ ,
where t 0 is any, leads to an equation of the form 10.1.1.4:
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = w(x + bC 1 e ct + C 2 , t+C 3 )+ C 1 ce ct ,
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
where C 1 and C 2 are arbitrary constants and the function U (z) is determined by the autonomous ordinary differential equation
aU zzzz ′′′′ + bU U z ′ − C 2 U z ′ + cU = 0.
If C 1 = 0, we have a traveling-wave solution.
3 ◦ . There is a degenerate solution linear in x:
w(x, t) = ϕ(t)x + ψ(t).
7. = a ∂t
+[ f (t) ln w + g(t)]
∂x 4
∂x
Generalized traveling-wave solution:
w(x, t) = exp[ϕ(t)x + ψ(t)],
f (t) dt + C 1 , 3 ψ(t) = ϕ(t) [ g(t) + aϕ ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
1 = C 1 w(C 1 x + 2bC 1 C 2 t+C 3 , C 1 t+C 4 )+ C 2 x + bC 2 t+C 5 ,
where C 1 , ...,C 5 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solution:
w(x, t) = C 1 t+C 2 +
θ(z) dz,
z = x + λt,
where C 1 , C 2 , and λ are arbitrary constants, and the function θ(z) is determined by the third-order autonomous ordinary differential equation
aθ 2 ′′′
zzz + bθ − λθ − C 1 = 0.
To C 1 = 0 there corresponds a traveling-wave solution.
3 ◦ . Self-similar solution:
w(x, t) = t −1 /2
u(ζ), −1 ζ = xt /4 ,
where the function u(ζ) is determined by the ordinary differential equation
au ′′′′ ζζζζ + b(u ′ 2 ζ 1 ) + 4 ζu ′
ζ + 2 u = 0.
4 ◦ . There is a degenerate solution quadratic in x:
w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t).
∂w 2 ∂ 4 w ∂w
9. = a 4 + b + f (t).
∂t ∂x
∂x
The substitution w = U (x, t) +
f (t) dt leads to a simpler equation of the form 10.1.1.8:
+ b + cw + f (t).
w(x, t) = Ae − e ct f (t) dt + θ(z), z = x + λt, where
ct + e ct
A and λ are arbitrary constants, and the function θ(z) is determined by the autonomous ordinary differential equation
aθ 2
zzzz ′′′′ + bθ z ′ − λθ ′ z + cθ = 0.
w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t).
3 ◦ . The substitution
w = U (x, t) + e ct
e − ct f (t) dt
leads to the simpler equation