Equations of the Form ∂w n = a∂ w n + f (w) ∂w

11.1.3. Equations of the Form ∂w n = a∂ w n + f (w) ∂w

∂t

∂x

∂x

Preliminary remarks. Equations of this form admit traveling-wave solutions:

w = w(z),

z = x + λt,

where λ is an arbitrary constant and the function w(z) is determined by the (n−1)st-order autonomous ordinary differential equation (

C is an arbitrary constant)

aw ( n−1)

Generalized Burgers–Korteweg–de Vries equation .

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w 1 = C 1 n−1 w(C 1 x + bC 1 C 2 t+C 3 , C n 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) = −

b(t + C 2 ) a(2n − 2)!

+ C 1 . The first solution is degenerate and the second one is a traveling-wave solution (a special case of the

w(x, t) = (−1) n

b(n − 1)! ( x + bC 1 t+C 2 ) n−1

solution of Item 3 ◦ ).

3 ◦ . Traveling-wave solution:

w = w(ξ), ξ = x + λt,

where λ is an arbitrary constant and the function w(ξ) is determined by the (n−1)st-order autonomous ordinary differential equation

+ 2 bw = λw + C.

aw ( n−1) 1 2

4 ◦ . Self-similar solution:

1− n

w(x, t) = t n u(η),

η = xt − n ,

where the function u(η) is determined by the ordinary differential equation

w(x, t) = U (ζ) + 2C 2

1 t,

ζ = x + bC 1 t + C 2 t,

where C 1 and C 2 are arbitrary constants and the function U (ζ) is determined by the (n − 1)st-order ordinary differential equation

1 aU 2

( n−1)

+ 2 bU − C 2 U = 2C 1 ζ+C 3 .

z = ϕ(t)x + ψ(t).

bϕ

Here, the functions ϕ(t) and ψ(t) are defined by

1 ϕ(t) = (Ant + C 1 ) − n ,

n−1

ψ(t) = C 2 ( Ant + C 1 )

+ C 3 ( Ant + C 1 ) − n +

A 2 ( n − 1) where

F (z) is determined by the ordinary differential equation

A, B, C 1 , C 2 , and C 3 are arbitrary constants, and the function

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

w 1 = C 1 n−1 w(C 1 k x+C 2 , C nk 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Self-similar solution:

1− n

w(x, t) = t nk U (z),

z = xt − n ,

where the function U = U (z) is determined by the ordinary differential equation

1 ◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

1 x+C 2 , C n 1 t+C 3 w(C n−1 )+ ln C 1 ,

where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.

2 ◦ . Solution:

1− n

w(x, t) = U (z) +

where the function U = U (z) is determined by the ordinary differential equation

Generalized traveling-wave solution: x+C 2 a 1 c

w(x, t) = exp

b where C 1 and C 2 are arbitrary constants.

C 1 − bt

b(n − 2) ( C 1 − bt) n−1

5. =a

+ [b arcsinh(kw) + c]

Generalized traveling-wave solution:

1 x+C 2 a 1 c

w(x, t) = sinh

bt) 2 n −

b where C 1 and C 2 are arbitrary constants.

1 − bt

b(2n − 1) C 1 −

+ [b arccosh(kw) + c]

Generalized traveling-wave solution:

1 x+C 2 a 1 c

w(x, t) = cosh

b where C 1 and C 2 are arbitrary constants.

C 1 − bt

b(2n − 1) ( C 1 − bt) 2 n

+ [b arcsin(kw) + c]

∂x 2 n+1

∂x

Generalized traveling-wave solution:

1 x+C 2 a(−1) n

w(x, t) = sin

C 1 − bt b(2n − 1) ( C 1 − bt) 2 n −

b where C 1 and C 2 are arbitrary constants.

2 n+1 + [b arccos(kw) + c]

∂t ∂x

∂x

Generalized traveling-wave solution:

1 x+C 2 a(−1) n

w(x, t) =

cos

b where C 1 and C 2 are arbitrary constants.

C 1 − bt b(2n − 1) ( C 1 − bt) n