454 Chapter 15. Wave-wave nonlinearities
15.6 Ion acoustic wave soliton
The ion acoustic wave dispersion relation is
ω^2 =
k^2 c^2 s
1+k^2 λ^2 D
(15.165)
which has the forward propagating solution
ω=kcs
(
1+k^2 λ^2 D
)− 1 / 2
(15.166)
or, for smallkλD,
ω=kcs−k^3 λ^2 Dcs/ 2 (15.167)
where the last term is small. Since∂/∂x→ikand∂/∂t→−iωthe reverse substitution
k→−i∂/∂xandω→i∂/∂tcan be invoked so that the forward propagating ion acoustic
wave can be written as a partial differential equation for say, the ion velocity
i
∂ui
∂t
=−i
∂ui
∂x
cs−(−i)^3
∂^3 ui
∂x^3
λ^2 Dcs
2
. (15.168)
After multiplying by−i,this gives the dispersive forward propagating wave equation
∂ui
∂t
=−cs
∂ui
∂x
−
λ^2 Dcs
2
∂^3 ui
∂x^3
. (15.169)
This wave equation was derived using a linearized version of the ionfluid equation of
motion, namely
mi
∂ui
∂t
=qiE. (15.170)
If the complete nonlinear ion equation had been used instead, the ionfluid equation of
motion would contain a convective nonlinear term and be
mi
(
∂ui
∂t
+ui
∂ui
∂x
)
=qiE. (15.171)
This suggests that inclusion of convective ion nonlinearity corresponds to making the gen-
eralization
∂ui
∂t
→
∂ui
∂t
+ui
∂ui
∂x
(15.172)
and so the forward propagating ion acoustic wave equation with inclusion of ion convective
nonlinearity is
∂ui
∂t
+(ui+cs)
∂ui
∂x
+
λ^2 Dcs
2
∂^3 ui
∂x^3
=0. (15.173)
This suggests defining a new variable
U=ui+cs (15.174)
which differs only by a constant from the ionfluid velocity. The forward propagating
nonlinear ion acoustic wave equation can thus be re-written as
∂U
∂t
+U
∂U
∂x
+
λ^2 Dcs
2
∂^3 U
∂x^3