Equations of the Form ∂w +a∂ w 3 +f w, ∂w =0
9.1.5. Equations of the Form ∂w +a∂ w 3 +f w, ∂w =0
∂t
∂x
∂x
◮ For f (w, u) = bu 2 and f (w, u) = bu 3 , see equations 9.1.4.3 and 9.1.4.5, respectively. Equations of this form admit traveling-wave solutions, w = w(kx + λt).
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
w 1 = w x + bC 1 e − ct + C 2 , t+C 3 + cC 1 e − ct , where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
2 ◦ . Solution:
z = x + bC 1 e ct + C 2 t, where C 1 and C 2 are arbitrary constants, and the function U (z) is determined by the autonomous ordinary differential equation
w(x, t) = U (z) + C −
1 e ct ,
aU zzz ′′′ +( bU + C 2 ) U z ′ + cU = 0. To the special case C 1 = 0 there corresponds a traveling-wave solution.
1 ◦ . Generalized separable solutions for
ab < 0:
C 2 1 b w(x, t) =
2 exp( λx + λ t) −
a where C 1 and C 2 are arbitrary constants.
( t+C 1 )
b(t + C 1 )
2 ◦ . Generalized separable solution for
ab < 0:
1 1 1 1 1 1 b w(x, t) =
cosh( λx + λ 3 t+C 3 )−
2 bt + C 1 bt + C 2 2 bt + C 1 bt + C 2 a
where C 1 , C 2 , and C 3 are arbitrary constants.
3 ◦ . Generalized separable solution for
ab > 0:
w(x, t) =
✩✂✪ where C 1 , C 2 , and C 3 are arbitrary constants.
References : V. A. Galaktionov and S. A. Posashkov (1989), A. D. Polyanin and V. F. Zaitsev (2002).
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
1 = 3− C 1 k w(C k
1 x+C 2 , C 1 t+C 3 ),
where C 1 , C 2 , and C 3 are arbitrary constants, is also a solution of the equation.
2 ◦ . Degenerate solution linear in x:
w = x(kbt) −1 /k +
Ct −1 /k .
3 ◦ . Self-similar solution:
k−3
w=t 3 k
U (z),
z = xt 3 ,
where the function U = U (z) is determined by the ordinary differential equation
k−3
1 U− zU ′ + bU (U ′ ) k + aU ′′′ = 0.
zzz
Degenerate solution quadratic in x:
w(x, t) =
Reference : W. I. Fushchich, N. I. Serov, and T. K. Akhmerov (1991).
Generalized traveling-wave solutions:
k(kb 1 ) −3 /k ( k−3)/k b 2
exp 2 − a(2b
C is an arbitrary constant.
Reference : W. I. Fushchich, N. I. Serov, and T. K. Akhmerov (1991).
3 +( b 1 arcsin w+b 2 )(1 – w 2 ) 2 ∂t = 0. ∂x ∂x Generalized traveling-wave solutions:
) −3 k(kb 1 /k ( k−3)/k b 2
C is an arbitrary constant.
Reference : W. I. Fushchich, N. I. Serov, and T. K. Akhmerov (1991).
∂w ∂ 3 w
3 +( b 1 arcsinh w+b )(1 + w 2 2 ) 2 ∂t = 0. ∂x ∂x Generalized traveling-wave solutions:
k(kb 1 ) −3 /k
1 /k t) x if k ≠ 2, w=
sinh
t ( k−3)/k b − 2 +
C is an arbitrary constant.
Reference : W. I. Fushchich, N. I. Serov, and T. K. Akhmerov (1991).
9.1.6. Equations of the Form ∂w +a∂ w
+F x, t, w, ∂w =0
∂t
∂x 3
∂x
1. = a f (w).
1 ◦ . Suppose w(x, t) is a solution of this equation. Then the function
1 = w(x + C 1 e bt , t+C 2 ),
where C 1 and C 2 are arbitrary constants, is also a solution of the equation.
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
+ f (t)w
+ g(t)w = 0.
Suppose w(x, t) is a solution of this equation. Then the function
w 1 = wx+C 1 ψ(t) + C 2 , t − C 1 ϕ(t),
where
ϕ(t) = exp −
g(t) dt ,
ψ(t) =
f (t)ϕ(t) dt,
is also a solution of the equation ( C 1 and C 2 are arbitrary constants).
+[ f (t) ln w + g(t)]
∂x 3
∂x
Generalized traveling-wave solution:
w(x, t) = exp[ϕ(t)x + ψ(t)],
where
ψ(t) = ϕ(t) [ g(t) + aϕ ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.* ∂w
ϕ(t) = − 2 f (t) dt + C
∂w
4. = a
+[ f (t) arcsinh(kw) + g(t)]
Generalized traveling-wave solution:
1 w(x, t) = sinh ϕ(t)x + ψ(t) , k
where
ψ(t) = ϕ(t) [ g(t) + aϕ ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants. ∂w
ϕ(t) = − 2 f (t) dt + C
∂w
5. = a ∂t
+[ f (t) arccosh(kw) + g(t)]
∂x 3
∂x
Generalized traveling-wave solution:
1 w(x, t) = cosh ϕ(t)x + ψ(t) , k
ψ(t) = ϕ(t) [ g(t) + aϕ 2 ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants.
f (t) dt + C 1 ,
* The constant a in equations 9.1.6.2 to 9.1.6.6 and their solutions can be replaced by an arbitrary function of time, a = a(t).
+[ f (t) arcsin(kw) + g(t)]
Generalized traveling-wave solution:
1 w(x, t) = sin ϕ(t)x + ψ(t) , k
where
ψ(t) = ϕ(t) [ g(t) − aϕ 2 ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants. ∂w
7. = a +[ f (t) arccos(kw) + g(t)]
∂t
∂x 3 ∂x
Generalized traveling-wave solution:
1 w(x, t) = cos ϕ(t)x + ψ(t) , k
where
ψ(t) = ϕ(t) [ g(t) − aϕ 2 ( t)] dt + C 2 ϕ(t), and C 1 and C 2 are arbitrary constants. ∂w
+ b + cw + f (t).
∂x 3
∂x
1 ◦ . Degenerate solution quadratic in x:
w(x, t) = ϕ(t)x 2 + ψ(t)x + χ(t),
where the functions ϕ k = ϕ k ( t) satisfy an appropriate system of ordinary differential equations.
2 ◦ . Solution:
w(x, t) = Ae ct + e ct e − ct f (t) dt + θ(z), z = x + λt, where
A and λ are arbitrary constants, and the function θ(z) is determined by the autonomous ordinary differential equation
aθ 2 ′′′
zzz + bθ ′ z − λθ ′ z + cθ = 0.
3 ◦ . The substitution
w = U (x, t) + e ct
e − ct f (t) dt
leads to the simpler equation