Fundamentals of Plasma Physics

456 Chapter 15. Wave-wave nonlinearities

where the boundary condition at infinity has been used again. This can be written as

d(χ/3)

21 /^2

χ

3

√

V−

χ

3

=dη. (15.185)

The substitution
χ
3

=

V

cosh^2 θ

(15.186)

allows simplification of Eq.(15.185) to

d

(χ

3

)

=−

2 Vsinhθ

cosh^3 θ

dθ. (15.187)

Thus, Eq.(15.185) becomes

−

2 Vsinhθ

cosh^3 θ

dθ

21 /^2

V

cosh^2 θ

√

V−

V

cosh^2 θ

=dη (15.188)

or

−

√

2

V

dθ=dη

which can be integrated to give

θ=

√

V

2

(ξ 0 −η) (15.189)

whereξ 0 is a constant. The propagating solitary wave solution to Eq. (15.177) is therefore

χ(η)=

3 V

cosh^2

(√

V

2

(ξ 0 −η)

) (15.190)

or

χ(ξ,τ)=

3 V

cosh^2

(√

V

2

(ξ 0 −(ξ−Vτ))

). (15.191)

This solution, called a soliton, has the following properties:

1. It vanishes when|ξ|→∞as required.


2. The spatial profile consists of a solitary pulse centered around the positionξ=ξ 0 +Vτ


3. The pulse width scales asV−^1 /^2 and the pulse height scales asVso that larger am-
   plitudes are sharper and propagate faster.
   An important property of solitons is that they obey a form of superposition principle
   even though they are essentially nonlinear. In particular, a fast soliton can overtake a slow
   soliton such that after the collision or interaction, both the fast andslow solitons retain
   their identity. The underlying mathematical theory explaining this surprising behavior is
   so complex as to be beyond the scope of this text (the interested reader shouldconsult