Fifth-Order Equations

11.1.1. Fifth-Order Equations

This is a special case of equation 11.1.3.1 with n = 5 and b = −1. ∂w

This is a special case of equation 11.1.3.2 with n = 5. ∂w

This is a special case of equation 11.1.3.3 with n = 5. ∂w

This is a special case of equation 11.1.3.4 with n = 5. ∂w

∂w

5. =a

+ (b arcsinh w + c)

This is a special case of equation 11.1.3.5 with n = 2 and k = 1. ∂w

∂w

6. =a ∂t

+ (b arccosh w + c)

∂x 5

∂x

This is a special case of equation 11.1.3.6 with n = 2 and k = 1. ∂w

∂w

7. =a

5 + (b arcsin w + c)

∂t ∂x

∂x

This is a special case of equation 11.1.3.7 with n = 2 and k = 1. ∂w

∂w

8. =a

+ (b arccos w + c)

This is a special case of equation 11.1.3.8 with n = 2 and k = 1.

Kawahara’s equation . It describes magnetoacoustic waves in plasma and long water waves under ice cover. ✂✁

References : T. Kawahara (1972), A. V. Marchenko (1988).

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

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

where C 1 , C 2 , and C 3 are arbitrary constants, are also solutions of the equation (either plus or minus signs are taken in the first formula).

2 ◦ . Degenerate solution:

x+C 1

w(x, t) =

t+C 2

3 ◦ . Traveling-wave solutions:

r 105 a 5 a w(x, t) =

ab > 0; 169 b cosh z

4 +2 C 1 , 1 z= 2 kx − (18bk + C 1 k)t + C 2 , k=

4 +2 C 1 1 5 , a z= 2 kx − (18bk + C 1 k)t + C 2 , k= if

w(x, t) =

ab > 0; 169 b sinh z

13 b

r 105 a 1 5 a w(x, t) =

ab < 0, 169 b cos z

4 +2 C 1 ,

z= 2 kx − (18bk + C 1 k)t + C 2 , k= − if

13 b where ✂✁ C 1 and C 2 are arbitrary constants.

Reference : N. A. Kudryashov (1990 a, the first solution was obtained).

4 ◦ . Traveling-wave solution for a = 0:

1680 b

w(x, t) =

( x+C 1 t+C 2 ) 4

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 fourth-order

ordinary differential equation ( C 3 is an arbitrary constant)

1 bU 2

ζζζζ ′′′′ − aU ζζ ′′ − 2 U − C 2 U = 2C 1 ζ+C 3 .

The special case C 1 = 0 corresponds to a traveling-wave solution.

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

w 1 = w(x − aC 1 e kt + C 2 , t+C 3 )+ C 1 ke kt ,

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

where C 1 and C 2 are arbitrary constants and the function U (z) is determined by the autonomous ordinary differential equation

cU (5) z − bU zzz ′′′ − aU U z ′ − C 2 U z ′ + kU = 0.

If C 1 = 0, we have a traveling-wave solution.

3 ◦ . There is a degenerate solution linear in x:

w(x, t) = ϕ(t)x + ψ(t).

This equation describes long water waves with surface tension (Olver, 1984).

1 ◦ . Traveling-wave solutions:

a 2 a 6 5 k 3 + a 4 k + a 1 k+C 2

w(x, t) = A + C 1 exp( kx + C 2 t),

w(x, t) = A + C 1 sinh( kx + C 2 t+C 3 ), k= ☎

, A=−

a 2 a 6 5 + 4 k 3 a k + a 1 k+C 2

w(x, t) = A + C 1 cosh( kx + C 2 t+C 3 ), k= ☎

A=−

2 5 a 3 a 6 k − a 4 k + a 1 k+C 2

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

k= ☎

A=

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

2 ◦ . There are traveling-wave solutions of the following forms:

w(x, t) = A +

cosh z

cosh 2 z

w(x, t) = A +

sinh

z 2 sinh z sinh z

w(x, t) = A + B

cosh z

cosh z

B + C sinh z + D cosh z

w(x, t) = A +

E + cosh z) 2

where z = kx + λt + const, and the constants A, B, C, D, E, k, and λ are identified by substituting these solutions into the original equation. ✆✂✝

References : N. A. Kudryashov and M. B. Sukharev (2001), P. Saucez, A. Vande Wouwer, W. E. Schiesser, and P. Zegeling (2003).