Equations of the Form ∂w m =a∂ w ∂w + bw k

1.1.11. Equations of the Form ∂w m =a∂ w ∂w + bw k

∂t

∂x

∂x

◮ Equations of this form admit traveling-wave solutions w = w(kx + λt). ∂w

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions

= w −1 1 C 1 w( P C 1 x+C 2 , C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . Generalized separable solutions linear and quadratic in x:

w(x, t) = C 1 x + (aC 2 1 + b)t + C 2 ,

( x+C 2 ) 2

w(x, t) = − −1 C

3 | t+C 1 | /3 3 + b(t + C 1 ),

6 a(t + C 1 )

where C 1 , C 2 , and C 3 are arbitrary constants. The first solution is degenerate.

3 ◦ . Traveling-wave solution in implicit form: Z

4 ◦ . For other solutions, see equation 1.1.11.11 with m = 1 and k = 0. ∂w

1 ◦ . Suppose w(x, t) is a solution of this equation. Then the functions

w −1

1 = C 1 w( C 1 x+C 2 , C 1 t+C 3 ),

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . The transformation

2 −1 ∂u b ∂ 1 ∂u

x=−

, w(x, t) = −

leads to the equation ∂ −1

u ∂t ∂y 2 It follows that any solution u = u(x, t) of the linear heat equation

(2) generates a solution (1) of the original nonlinear equation. ◆☎❖

Reference : V. A. Dorodnitsyn and S. R. Svirshchevskii (1983).

Functional separable solutions: −1 b /2

w(x, t) = ◗ C 1 ( x+C 2 ) 2 + C 3 exp(2 aC 1 t) −

aC 1 where C 1 , C 2 , and C 3 are arbitrary constants.

ab > 0, the transformation

w(x, t) = exp( ◗

3 λx)z(ξ, t), ξ=

exp( ◗ 2 λx),

3 a leads to a simpler equation of the form 1.1.10.4:

References : N. H. Ibragimov (1994), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

2 ◦ . For

ab < 0, the transformation z(ξ, t) 1 1 b /2

w(x, t) =

tan( λx),

cos ( λx)

also leads to equation 1.1.10.4.

3 ◦ . Multiplicative separable solution:

w(x, t) = (t + C) 3 /4 u(x),

where

C is an arbitrary constant, and the function u = u(x) is determined by the autonomous ordinary differential equation

−4 /3 ′ ) ′ + a(u −1 u

bu /3 − 3 4 u = 0.

4 ◦ . See also equation 1.1.12.6 with b = c = 0. ∂w

1 ◦ . Functional separable solution: w(x, t) = a 2 /3 3 Ax 3 −2 + /3 f 2 ( t)x 2 + f 1 ( t)x + f 0 ( t) .

Here,

f 2 ( t) = 3 ϕ(t) dt + 3B, f 1 ( t) =

ϕ(t) dt + B +

ϕ(t),

2 ϕ t ( 9 t), A 6 A 36 A where the function ϕ(t) is defined implicitly by

f 0 ( t) =

2 ϕ(t) dt + B +

2 ϕ(t)

ϕ(t) dt + B +

Z ( C 1 − 108 A 2 abϕ − 8ϕ 3 ) −1 /2 dϕ = t + C 2 ,

and

A, B, C 1 , and C 2 are arbitrary constants.

2 ◦ . For other solutions, see equations 1.1.11.9 and 1.1.11.11 with m = −3/2.

3 ◦ . The substitution w=u −2 /3 leads to an equation of the form 1.1.9.7:

∂u 2 ∂ 2 u 2 ∂u 3 = au 2 − a − b.

∂t

∂x

3 ∂x

This is a special case of equation 1.1.12.5 with b = 0. See also equation 1.1.11.11 with m = −4/3 and k = 7/3.

w(x, t) = e bt

w(x, t) = e

x+

a ab

w(x, t) = e bt

bm ( x − A)

2 a(m + 2)(B − e bmt

2 bmt

bm 2 ( x + B) 2 m

w(x, t) =

w(x, t) = e A|e

2 a(m + 2) e bmt + B where

A, B, C, and λ are arbitrary constants.

2 ◦ . By the transformation

w(x, t) = e bt

v(x, τ ), τ=

e bmt + const

bm

the original equation can be reduced to an equation of the form 1.1.10.7:

❯☎❱ 3 ◦ . See also equation 1.1.11.11 with k = 1.

Reference : L. K. Martinson and K. B. Pavlov (1972).

1 ◦ . Multiplicative separable solution ( a = b = 1, m > 0):

 2( m + 1) cos 2 (

πx/L) /m

for | x| ≤ ,

w(x, t) =

2 where L = 2π(m + 1) 1 /2 /m. Solution (1) describes a blow-up regime that exists on a limited time ❯☎❱ interval t ❲ [0, t 0 ). The solution is localized in the interval | x| < L/2.

References : N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii (1976), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

2 ◦ . Multiplicative separable solution:

Ae µx + Be − µx

+ 1 /m D

w(x, t) =

a(m + 1) where

A(m + 2) 2 b(m + 2)

A, C, and λ are arbitrary constants, ab(m + 1) < 0.

w(x, t) = (mλt + C) −1 /m ϕ(x),

where the function ϕ = ϕ(x) is determined by the ordinary differential equation

(2) Equation (2) has the following solution in implicit form:

A and B are arbitrary constants.

4 ◦ . Functional separable solution [it is assumed that ab(m + 1) < 0]:

w(x, t) =

f (t) + g(t)e λx

where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations

bmf 2 ,

bm(m + 2)

f g.

m+1

Integrating yields

f (t) = (C 1 − bmt) −1 , g(t) = C 2 ( C 1 − bmt) − m+1 , where C 1 and C 2 are arbitrary constants.

m+2

5 ◦ . Functional separable solution (

A and B are arbitrary constants):

− b w(x, t) =

f (t) + g(t)(Ae λx + Be λx )

where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations

2 4 bmAB 2 bm(m + 2)

On eliminating t from this system, one obtains a homogeneous first-order equation:

The substitution ζ = f /g leads to a separable equation. Integrating yields the solution of equation (5) in the form

f= ❳

g 4AB + C 1 g − m+2 2 ,

C 1 is any.

Substituting this expression into the second equation of system (4), one obtains a separable equation for g = g(t).

6 ◦ . The functional separable solutions

1 w(x, t) = /m f (t) + g(t) cosh(λx) ,

1 w(x, t) = /m f (t) + g(t) sinh(λx)

are special cases of formula (3) with A= 1 1 1 2 1 , B= 2 and A= 2 , B=− 2 , respectively.

b w(x, t) =

f (t) + g(t) cos(λx + C)

where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations

2 = bm 2 bm(m + 2)

❨☎❩ which coincides with system (4) for AB = 1 4 .

References for equation 1.1.11.8: M. Bertsch, R. Kersner, and L. A. Peletier (1985), V. A. Galaktionov and S. A. Posash- kov (1989), V. F. Zaitsev and A. D. Polyanin (1996).

This is a special case of equation 1.1.11.11 with k = 1 − m. Functional separable solution:

w(x, t) =

A, B, and C are arbitrary constants.

Reference : R. Kersner (1978).

This is a special case of equation 1.1.11.11 with m = 2n and k = 1 − n. Generalized traveling-wave solution:

1 bn 2 2 a(n + 1) w(x, t) =

n where C 1 and C 2 are arbitrary constants.

This is a special case of equation 1.6.15.2 with f (w) = aw m and g(w) = bw k . For b = 0, see Subsection 1.1.10.

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

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

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation.

2 ◦ . A space-homogeneous solution and a stationary solution are given by (the latter is written out in implicit form):

w(t) =

(1 − k)bt + C 1− k if k ≠ 1,

w m+k+1

A, B, and C are arbitrary constants.

w = w(z), z= ❭ x + λt,

where the function w(z) is determined by the autonomous ordinary differential equation

(1) The substitution

a(w m w ′ z ) ′ z − λw z ′ + bw k = 0.

a u(w) = w m w ′

brings (1) to the Abel equation

(2) The book by Polyanin and Zaitsev (2003) presents exact solutions of equation (2) with m+k=

uu ′ w − u = −abλ −2 w m+k .

4 ◦ . Self-similar solution for k ≠ 1:

1 k−m−1

w=t 1− k u(ξ), ξ = xt 2(1− k) ,

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

a(u m u ′ ξ )

m−k+1

Reference : V. A. Dorodnitsyn (1982).