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

h 1.1.13. Equations of the Form ∂w i =∂ f (w) ∂w + g(w)

∂t

∂x

∂x

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

This is a special case of equation 1.6.15.1 with f (w) = aw 2 + bw.

Solutions:

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

1 t+C 2 − , a

x+C 1 b

w(x, t) = √

C 2 −4 at 2 a

where C 1 and C 2 are arbitrary constants. The first solution is of the traveling-wave type and the second one is self-similar.

This is a special case of equation 1.6.15.1 with f (w) = a(w 2 + b 2 ) −1 .

1 ◦ . Solutions (

A and B are arbitrary constants):

w(x) = b tan(Ax + B),

w(x, t) = ❞

−1 bx A − 2ab /2 −2 t−x 2 ,

2 w(x, t) = Ab exp(ab −1 t − x) 1− −2 /2 A exp 2( ab t − x) .

2 ◦ . Traveling-wave solution in implicit form: ak 2 1 2 2 A w

λ(kx + λt) + B = 2 2 ln | w + A| − ln( w + b )+

arctan ,

A + b 2 b b where

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

leads to the equation

which is a special case of 8.1.2.12 with

F (t, ξ, η) = ab −2 ( ξ − η). Equation (2) has multiplicative separable solutions

A sin(λx) + B cos(λx)

A, B, C, and λ are arbitrary constants. Formulas (1) and (3) provide two solutions of the original equation. ❣☎❤

Reference : P. W. Doyle and P. J. Vassiliou (1998); see also Example 10 in Subsection S.5.3.

z − arctan ψ(z)

2 t+C 2 , b

where

C is an arbitrary constant and the function ψ = ψ(z) is determined by the ordinary differential equation

ψ z ′ = (1 + ψ 2 )

Here the function z = z(x, t) is defined implicitly.

5 ◦ . Solution:

C at

w = b tan ϕ(z) + arctan ψ(z) +

C is an arbitrary constant and the functions ϕ(z) and ψ(z) are determined by the system of ordinary differential equations

, ψ z = (1 + ψ )

Here the function ❣☎❤ z = z(x, t) is defined implicitly.

References : I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989), N. H. Ibragimov (1994).

This is a special case of equation 1.6.15.1 with f (w) = aw 2 n + bw n .

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

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

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

1 /n

w(x, t) =

2 C 1 nx + 2aC 2

1 nt + C 2 −

where C 1 and C 2 are arbitrary constants.

3 ◦ . Self-similar solution:

1 x+C /n

1 b 2 a(n + 1)

w(x, t) =

Generalized traveling-wave solution:

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

n where C 1 and C 2 are arbitrary constants. ∂w

Generalized traveling-wave solutions:

1 b /n dt

w(x, t) = ϕ(t)( x+C 1 )+ ϕ(t) ϕ(t) dt + snϕ(t)

ϕ(t)

ϕ(t) = C 2 e −2 cnt a(n + 1) − 2 ,

cn

where C 1 and C 2 are arbitrary constants.

h 1.1.14. Equations of the Form ∂w i

f (w) ∂w +g x, t, w, ∂w

∂t

∂x

∂x

∂x

For m = 0, see equation 1.1.11.4. For m ≠ 0, the original equation can be reduced to a simpler equation 1.1.10.4 that corresponds to the case b = 0 [see equation 1.6.13.1 with f (x) = bx m ].

This is a special case of equation 1.6.13.8 with m = −2, f (t) = b, and g(t) = c. The transformation (

A and B are arbitrary constants)

w(x, t) = e ct

u(z, τ ), z = x + bt + A, τ=B−

e −2 ct

leads to an equation of the form 1.1.10.3:

Reference : V. A. Dorodnitsyn and S. R. Svirshchevskii (1983); the case b = 0 was treated.

The transformation

u(z, t) = w(x, t) + b,

z = x + ct

leads to an equation of the form 1.1.10.3:

This is a special case of equation 1.1.14.16 with n = 1. Degenerate solution linear in x:

ax + b ln |t + C 1 |+ C 2

w(x, t) =

( t+C 1 )

where C 1 and C 2 are arbitrary constants. ∂w

This is a special case of equation 1.1.14.16 with n = 2.

1 ◦ . Traveling-wave solution in implicit form:

Z w 2 dw

2 b 2 = x + λt + C 2 , aw +2 λw + C 1

where C 1 , C 2 , and λ are arbitrary constants.

2 ◦ . Degenerate solution linear in x:

w(x, t) = (x + C 1 ) f (t).

Here, C 1 is an arbitrary constant, and the function f = f (t) is determined by the ordinary differential equation

af 2 =2 3 bf ,

whose solution can be represented in implicit form:

1 ◦ . Degenerate solution linear in x:

w(x, t) = f (t)x + g(t),

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

af 2 =2 3 bf ,

af g = 2bf 2 g + cf 2 .

The solution of the first equation can be found in 1.1.14.5, Item 2 ◦ . The second equation is easy to integrate, since it is linear in g.

2 ◦ . Traveling-wave solution in implicit form:

where C 1 , C 2 , and λ are arbitrary constants.

This is a special case of equation 1.6.13.2 with f (t) = bt n . ∂w

This is a special case of equation 1.6.13.2 with f (t) = be λt . ∂w

This is a special case of equation 1.6.13.3 with f (t) = bt n . ∂w

This is a special case of equation 1.6.13.3 with f (t) = be λt . ∂w

This is a special case of equation 1.6.13.4 with f (t) = bt n and g(t) = ct k . ∂w

This is a special case of equation 1.6.13.4 with f (t) = be λt and g(t) = ce µt . ∂w

This is a special case of equation 1.6.13.5 with f (t) = ct n and g(t) = st k . ∂w

This is a special case of equation 1.6.13.6 with f (x) = bx n . ∂w

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

w 1 = C 1 −2 w( ♦ C 1 n x+C 2 e bt , t+C 3 ),

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

2 ◦ . Generalized traveling-wave solutions:

−1 /n

w(x, t) = ♦

where C 1 and C 2 are arbitrary constants.

3 ◦ . Generalized traveling-wave solutions:

w = w(z),

z= ♦ x + Ce bt ,

where

C is an arbitrary constant and the function w(z) is determined by the ordinary differential equation

a(w n w z ′ ) ′ z − bzw z ′ = 0.

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

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

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

2 ◦ . Traveling-wave solution in implicit form:

w n dw

2 b 2 = x + λt + C 2 , aw +2 λw + C 1

where C 1 , C 2 , and λ are arbitrary constants.

3 ◦ . Self-similar solution for n ≠ 2:

w(x, t) = u(z)t 1 /(n−2) , z = xt −( n−1)/(n−2) ,

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

Generalized traveling-wave solution:

a ln |t + C |

/n

w(x, t) =

b(t + C 1 ) b 2 n(t + C 1 ) b

where C 1 and C 2 are arbitrary constants. ∂w

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

w 1 = C 1 w(C 2 x+C 3 , C n 1 2 C 2 t+C 4 ), where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . Functional separable solution for

w(x, t) = n

ψ(t) dt + B, ϕ 1 ( t) =

ψ(t) dt + B + ψ(t),

3 A 2 Aβn

1 3 1 1 ϕ 0 ( t) =

12 2 β 2 A t n 2 where the function ψ = ψ(t) is defined implicitly by

2 ψ(t) dt + B +

ψ(t)

ψ(t) dt + B +

A, B, C 1 , and C 2 are arbitrary constants, β = a + 1; A ≠ 0, n ≠ 0, a > −1.

♣☎q

Reference : G. A. Rudykh and E. I. Semenov (1998).

3 ◦ . Functional separable solution for b= 1 4 n(a − 3) − a − 1:

X 4 1 /n

w(x, t) = n

ϕ k ( t)x k

k=0

Here, the functions ϕ k = ϕ k ( t) are determined by the system of ordinary differential equations ϕ ′ 0 =− 3 4 βϕ 2 1 +2 βϕ 2 ϕ 0 ,

where r☎s β = n(a + 1); the prime denotes a derivative with respect to t.

Reference : G. A. Rudykh and E. I. Semenov (1998).

4 ◦ . There are exact solutions of the following forms:

w(x, t) = F (z), z = Ax + Bt; w(x, t) = (At + B) −1 /n G(x);

w(x, t) = t β H(ξ), ξ = xt − βn+1 2 ; w(x, t) = e −2 t U (η), η = xe nt ; w(x, t) = (At + B) −1 /n

V (ζ), ζ = x + C ln(At + B), where

A, B, C, and β are arbitrary constants. The first solution is of the traveling-wave type, the second is a solution in multiplicative separable form, and the third is self-similar.

Generalized traveling-wave solution:

1 b /n w(x, t) = ϕ(t)x + (st + C 1 ) ϕ(t) + ϕ(t) ϕ(t) dt

where C 1 is an arbitrary constant and the function ϕ(t) is determined by the first-order separable ordinary differential equation

a(n + 1)

ϕ 3 + cϕ 2 .