15.6 Ion acoustic wave soliton 455We now introduce dimensionless variables by normalizing lengths to the Debye length,
velocities tocsand time toωpi=cs/λde.In this case
χ=U
cs
, ξ=x
λD
τ=ωpit (15.176)so the wave equation becomes
∂χ
∂τ+χ
∂χ
∂ξ+
1
2
∂^3 χ
∂ξ^3=0; (15.177)
this is called the (KdV) equation (Korteweg 1895) and modern interest in this equation
was stimulated with the discovery of a general solution to the soliton problem by Gardner,
Greene, Kruskal and Miura (1967).
Because the linear wave is forward traveling with unity velocity inthese dimensionless
variables, it is reasonable to postulate that the nonlinear solution has theforward propagat-
ing form
χ=χ(ξ−Vτ) (15.178)
whereVis of order unity. A special solution can be found by introducing the wave-frame
position variable
η=ξ−Vτ (15.179)
so the lab frame space and time derivatives can be written as
∂
∂ξ=
∂
∂η
∂
∂τ= −V
∂
∂η. (15.180)
Substitution of these into the wave equation gives an ordinary differentialequation in the
wave-frame,
−Vdχ
dη+
1
2
dχ^2
dη+
1
2
d^3 χ
dη^3=0 (15.181)
wheredhas been used instead of∂because the equation is an ordinary differential equation.
A solution is now sought that vanishes at both plus and minus infinity;such a solution is
called a solitary wave. To find this solution, Eq.(15.181) is integrated using the boundary
condition thatχvanishes at infinity to obtain
−Vχ+χ^2
2+
1
2
d^2 χ
dη^2=0. (15.182)
Multiplying by the integration factordχ/dηallows this to be recast as
d
dη(
−
Vχ^2
2+
χ^3
6+
1
4
(
dχ
dη) 2 )
=0 (15.183)
and then integrating gives
dχ
dη=χ√(
2 V−
2
3
χ