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
∂xcs−(−i)^3∂^3 ui
∂x^3λ^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