Burgers–Korteweg–de Vries Equation and Other Equations

9.1.7. Burgers–Korteweg–de Vries Equation and Other Equations

Burgers–Korteweg–de Vries equation . It describes nonlinear waves in dispersive-dissipative media with instabilities, waves arising in thin films flowing down an inclined surface, changes of the concentration of substances in chemical reactions, etc. ✭✂✮

References : Y. Kuramoto and T. Tsuzuki (1976), B. J. Cohen, J. A. Krommes, W. M. Tang, and M. N. Rosenbluth (1976), V. Ya. Shkadov (1977), J. Topper and T. Kawahara (1978), G. I. Sivashinsky (1983).

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

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

2 ◦ . Degenerate solution:

x+C 1

w(x, t) =

t+C 2

3 ◦ . Traveling-wave solutions:

12 b 2 b b 6 b 3

w(x, t) = C 1 −

w(x, t) = C 1 −

w(x, t) = C 1 +

5 a 5 a 125 a 2 ✯✂✰ where C 1 and C 2 are arbitrary constants.

25 a (1 + C 2 e z ) 2

Reference : N. A. Kudryashov (1990 a).

4 ◦ . Traveling-wave solutions:

12 b 2 b 6 b 3

w(x, t) = C 1 ξ ϕ(ξ),

ξ=C 2 exp

x+

25 a 5 a 125 a 2 5 a where the function ϕ(ξ) is defined implicitly by

and C 1 , C 2 , and C 3 are arbitrary constants. For the upper sign, the inversion of this relation leads to

the classical Weierstrass elliptic function, ✯✂✰ ϕ(ξ) = ℘(ξ + C 3 , 0, 1).

Reference : N. A. Kudryashov (1990 a).

5 ◦ . Solution:

w(x, t) = U (ζ) + 2C 2

1 t,

ζ=x−C 1 t + C 2 t,

where C 1 and C 2 are arbitrary constants and the function U (ζ) is determined by the second-order

ordinary differential equation ( C 3 is an arbitrary constant) aU ζζ ′′ − bU ζ ′ + 1 2 U 2 + C 2 U = −2C 1 ζ+C 3 . To the special case C 1 = 0 there corresponds a traveling-wave solution.

Modified Harry Dym equation .

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

1 = C 1 C 2 w(C 1 x+C 3 , C 2 t+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, is also a solution of the equation.

2 ◦ . The transformation u=w −1 /2 , τ = at leads to an equation of the form 9.1.4.8:

3 ◦ . The equation is invariant under the transformation

d ¯x = w dx + [a(w −3 /2 w x ) x ] dt, d¯t = dt, w = 1/w. ¯

Some integrable nonlinear third-order equations of the form 9.1.7.4 Type of generated equation

Form of generated equation

Solvable equation of the form (3)

∂t = ∂ 2 ∂z 2 u a 3 ∂z Korteweg–de Vries

∂ Linear equation 3

∂t = ∂z 2 u 3 ∂z − 2 u 2 ∂z Modified Korteweg–de Vries

Functional separable solution in implicit form:

f (w) dw = at − bx + C 1 x + C 2 x+C 3 ,

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

+ g(w)

1 ◦ . Traveling-wave solution:

w = w(z),

z = kx + λt,

where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation (

C is an arbitrary constant)

g(w) dw. The substitution U (w) = f (w)w ′ z leads to a first-order separable equation.

k 3 [ f (w)w ′ z ] ′ z + kG(w) − λw + C = 0,

G(w) =

2 ◦ . The transformation ∂z

∂z dz = w dx + [ f (w)w x ] x + G(w) dt, dτ = dt, u = 1/w

dz = dx + dt (1) ∂x

∂t leads to an equation of the similar form

Φ( u) = 3 f , Ψ( u) = g − G , G(w) =

g(w) dw.

The inverse of transformation (1) is written out as

dx =

dz −

w[wf (w)w z ] z + G(w) dτ , dt = dτ , w = 1/u.

Table 3 lists some solvable equations of the form (2) generated by known solvable third-order equations. Equation (2) can be reduced to the form (see equation 9.1.7.5)

Z v= 3 wf (w) dw, ϕ(v) = w f (w), ψ(v) = wg(w) − G(w).

Some integrable nonlinear third-order equations of the form 9.1.7.5; k = (8/a) 1 /2

Type of generated equation

Form of generated equation

Solvable equation of the form (3)

3 Linear equation 3 ∂w = a ∂ w

∂t = kU 3 /2 ∂ U ∂z 3 − bU ∂U ∂z Modified Korteweg–de Vries

Korteweg–de Vries

3 ◦ . Conservation laws:

D t ( w) + D x −[ f (w)w x ] x − G(w) = 0,

F (w)[f (w)w x ] x + 2 [ f (w)w x ] − Ψ( w) = 0, where

D t Φ( w) + D x −

, G(w) =

g(w) dw, F (w) =

F (w)g(w) dw. ∂w

+ g(w)

1 ◦ . Traveling-wave solution:

w = w(z),

z = x + λt,

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

C is an arbitrary constant)

λ − g(w)

which is easy to integrate.

2 ◦ . Conservation law:

D t ϕ(w) + D x − w xx − ψ(w) = 0,

where

Z dw

Z g(w)

3 ◦ . The transformation

∂z ∂z dz = ϕ(w) dx + w xx + ψ(w) dt, dτ = dt, U=

ϕ(w) dw

dz = dx + dt (2) ∂x

∂t leads to an equation of the similar form

The functions

F (U ) and G(U ) in (3) are defined parametrically by

ϕ(w) dw, where ϕ(w) and ψ(w) are defined in (1).

F (U ) = f (w)ϕ 3 ( w), G(U ) = g(w)ϕ(w) − ψ(w), U=

Table 4 presents some solvable equations of the form (3) generated by known solvable third-order equations.

4 ◦ . The substitution ϕ= leads to an equation of the form 9.1.7.4:

where the functions

F and G are given by

Z dw

F(ϕ) = f (w), G(ϕ) = g(w), ϕ=

+ g(w) + ax

+ h(w).

∂x 3

∂x

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

1 = w(x + C 1 e at , 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 + Ce at ,

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

where

f (w)w ′′′ zzz +[ g(w) + az]w ′ z + h(w) = 0.

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

w 1 = C 1 −2 w(C 1 x+C 2 , C 1 3 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:

w = U (ξ),

ξ = x + λt,

where λ is an arbitrary constant and the function U = U (ξ) is determined by the second-order autonomous ordinary differential equation

bU ξξ ′′ + a ln |U | + λU = C 1 .

3 ◦ . Self-similar solution:

w=t 2 /3

u(z), −1 z = xt /3 ,

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

buu 2 ′′′

zzz − 1 3 2 zuu ′ z + au ′ z + 3 u = 0.