Equations of the Form ∂ + f (w) ∂w =∂ g(w) ∂w

3.3.5. Equations of the Form ∂ + f (w) ∂w =∂ g(w) ∂w

∂t 2 ∂t

∂x

∂x

◮ Equations of this form admit traveling-wave solutions w = w(kx + λt); to k = 0 there corresponds

a homogeneous solution dependent on t alone and to λ = 0, a stationary solution dependent on x alone. For g(w) = const, such equations are encountered in the theory of electric field and nonlinear Ohm laws, where w is the electric field strength.

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

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

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

2 ◦ . Traveling-wave solutions in implicit form:

Z ( n + 1)dw

where C 1 , C 2 , and a are arbitrary constants (a ≠ 0, ❘ 1).

Example 1. Traveling-wave solution for n ≠ 0, −1:

C and a are arbitrary constants (a ≠ 0, ❙ 1 ).

Example 2. Traveling-wave solutions for n = 1:

w = 2C 1 tanh( C 1 ξ+C 2 ),

a(x − at) ,

w = −2C 1 tan( C 1 ξ+C 2 ),

where C 1 , C 2 , and

a are arbitrary constants; a ≠ 0, ❙ 1 .

3 ◦ . Self-similar solution:

w=t −1 /n

x ϕ(ξ), ξ= ,

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

(1 − 2 ξ ) ϕ ′′ + ϕ n 2( n + 1)

1 n+1 n+1

Z dx w = ϕ(x) t+C 1 + C 2 . ϕ 2 ( x)

Here, the function ϕ = ϕ(x) is determined by the autonomous ordinary differential equation ϕ ′′ xx = ϕ 2 , which has a particular solution ❚✝❯ ϕ = 6(x + C) −2 .

References : Y. P. Emech and V. B. Taranov (1972), N. H. Ibragimov (1994, 1995), A. D. Polyanin and V. F. Zaitsev (2002).

1 ◦ . Self-similar solution:

w(x, t) = u(z)t −1 /n ,

z = xt −1 ,

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

n 2 ( z 2 − b)u

zz ′′ + nz(2n + 2 − nau ) u z ′ + u(1 + n − nau ) = 0.

2 ◦ . Passing to the new independent variables τ = at and z = aβ −1 /2 x, we arrive at an equation of the form 3.3.5.1:

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

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

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

2 ◦ . Self-similar solution:

w(x, t) = u(z)t −1 /n ,

z = xt ( k−2n)/(2n) ,

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

4 2 bn 2 u k − (2 n − k) 2 z 2 u ′′

zz +4 bkn 2 u k−1 u ′ z + (2 n − k)(k − 4 − 4n + 2nau n ) zu ′ z =4 u(1 + n − anu n ).

Reference : N. H. Ibragimov (1994).

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

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

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

2 ◦ . Traveling-wave solutions:

w(x, t) = − ln

w(x, t) = − ln

( ax − t) + C 1 ,

where C 1 , C 2 , and a are arbitrary constants (a ≠ ❱ 1).

t + Cx

x+t a

w(x, t) = − ln

w(x, t) = − ln

C and a are arbitrary constants.

References : Y. P. Emech and V. B. Taranov (1973), N. H. Ibragimov (1994, 1995).

1 −1 w(x, t) = u(z) − ln t, z = xt , λ

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

λ(z 2 − b)u ′′

λu

zz + λz(2 − ae λu ) u z ′ +1− ae = 0.

2 ◦ . Passing to the new variables τ = at, z = aβ −1 /2 x, and u = λw, we arrive at an equation of the form 3.3.5.4:

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

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

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

2 ◦ . Solution:

w(x, t) = u(z) − ln t, z = xt λ−2µ)/(2µ) ,

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

( µ − 2λ) 2 z 2 −4 bλ 2 e µu u ′′ −4 bµλ 2 zz e µu ( u ′ z ) 2

+( µ − 2λ)(µ − 4λ + 2aλe λu ) zu ′ z +4 λ(1 − ae λu ) = 0.

Reference : N. H. Ibragimov (1994).