15.5 Digging a hole in the plasma via ponderomotive force 453By letting
g=√
2Ω
coshθ(15.153)
it is seen that
dg=−√
2Ω
cosh^2 θsinhθdθ (15.154)in which case
θ=−
√
Ωξ+δ (15.155)
whereδis an arbitrary constant. Thus, the solution is
χ(ξ,τ)=√
2ΩeiΩτ
cosh(√
Ωξ−δ) (15.156)
which is localized in space. For largeΩthe oscillation frequency and amplitude both
increase, and the localization is more pronounced.
Propagating envelope soliton
One may ask whether the above solution can be generalized to a propagatingsolution,
i.e., canξbe replaced byξ−vtwherevis a velocity? Making only this replacement is
clearly inadequate and so a solution of the form
χ=g(ξ−vt)eiΩτ+ih(ξ,τ) (15.157)is assumed whereh(ξ,τ)is an unknown function to be determined. Substitution of this
assumed solution into Eq.(15.147) gives
−ivg′−Ωg−g∂h
∂τ+g^3 +g′′+2i∂h
∂ξg′−(
∂h
∂ξ) 2
g=0. (15.158)Setting the imaginary part to zero gives
v=2
∂h
∂ξ(15.159)
which can be integrated to give
h=vξ
2+f(τ) (15.160)wheref(τ)is to be determined. The real part of the equation becomes
−Ωg−g∂h
∂τ+g^3 +g′′−v^2
4g=0. (15.161)If we set
∂h
∂τ
=−
v^2
4(15.162)
then the terms linear ingvanish,f(τ)=−v^2 τ/ 4 and so
h(ξ,τ)=vξ
2−
v^2
4
τ. (15.163)Thus, Eq.(15.161) reverts to Eq.(15.149) which is solved as in Eqs.(15.150)-(15.156) to
give the propagating envelope soliton
χ(ξ,τ)=√
2Ωexp(
iΩτ+ivξ/ 2 −iv^2 τ/ 4)
cosh(√
Ω(ξ−vt)−δ