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