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).
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