13 Nonlinear Waves

In linear theory, the wave amplitude is assumed to be sufficiently small to ignorecontributions of terms of second order and higher (i.e., nonlinear terms) in waveamplitude. In such a case, it is sufficient to consider only one Fourier componentat a time, and a general solution can be expressed by appropriate superpositionof Fourier components.

However, when the wave amplitude becomes larger, the linear approximationbreaks down and nonlinear effects must be taken into account. Linear theorypredicts exponential growth of unstable waves, but nonlinear effects cause sat-uration and limit the wave amplitude at a finite level.

13.1 Ion Acoustic Soliton and Shock Wave

Consider the idealized potential profile of an ion acoustic shock wave. The waveis travelling to the left with velocity u0. In the frame moving with the wave,the potential profile ϕ(x) is constant in time, and the plasma is impinging onthe wave from the left with a velocity u0. For simplicity, assume Ti = 0 and letthe electrons be Maxwellian. From energy conservation, the velocity of the ionsin the shock wave is

ui = u0

√1 − 2eϕ

Mu20

.

Taking the density on the left to be n0, the ion density in the shock is

ni =n0u0

ui= n0

(1 − 2eϕ

Mu20

)−1/2

.

Using the normalized parameters

ni

n0= n,

u0

cs= u,

x

λD= ξ, ωpit = τ,

eϕ

Te= φ

equations are expressed as follows.continuity equation

∂n

∂τ+

∂

∂ξ(nu) = 0

ion equation of motion

∂u

∂τ+ u

∂u

∂ξ+

∂φ

∂ξ= 0

electron equation of motion leads to Boltzmann distribution

ne

n0= eφ

Poisson equation∂2φ

∂ξ2= eφ − n.

57

Rewriting∂2φ

∂ξ2= eφ −

(1 − 2φ

u2

)−1/2

= −dV (φ)dφ

This equation can be regarded as the equation of motion for a particle movingin the potential V (φ), with the potential φ playing the role of position and theposition ξ playing the role of time, respectively. The quasi-potential V (φ) iscalled the Sagdeev potential.

Fig. 1. The Sagdeev potential.

The function V (φ) can be found by integrating with the boundary conditionV (φ = 0) = 0,

V (φ) = 1 − eφ + u2

(1 −

√1 − 2φ

u2

).

A particle will make a single excursion to positive φ and return to φ = 0. Sucha potential structure is called a soliton. It is a potential and density disturbancepropagating to the left with velocity u0.

If there is dissipation, the potential will oscillate in space about some positivevalue of φ, and never return to φ = 0. Reflection of ions from the shock fronthas the same effect as dissipation.

Expanding n = 1+n′, and transforming to a coordinate system moving withvelocity cs, y = ξ − τ ,

∂n′

∂τ+

∂

∂y(u − n′ + n′u) = 0

∂u

∂τ+ u

∂u

∂y+

∂

∂y(φ − u) = 0

−∂2φ

∂y2= n′ − φ − φ2

2− · · ·

58

Fig. 2. The potential of a soliton moving to the left. In the wave frame, ionsstream into the shock front from the left.

Fig. 3. Typical potential structure of an ion-acoustic shock moving to the left.

59

The 1st order solution, with the boundary condition that φ, u, n′ → 0 as|y| → ∞, is u = n′ = φ. The 2nd order equation is

∂u

∂τ+ u

∂u

∂y+

12

∂3u

∂y3= 0.

This equation is called the Korteweg-de Vries (K-dV) equation. The secondterm is the convective term v · ∇v which leads to wave steepening. The thirdterm arises from wave dispersion, i.e., k dependence of the phase velocity. TheK-dV equation has a localized solution (soliton)

u = u∞ + Bsech2

(x − ct

δ

)where

c = u∞ +B

3, δ =

√6B

.

The nonlinearity produces harmonic components (2ω, 2k), (3ω, 3k), . . .. Lowharmonics grow because they satisfy the linear dispersion relation, and causewave steepening. High harmonics are suppressed because they deviate fromthe linear dispersion relation ω = kcs. A stable soliton solution is obtained bybalancing the steepening due to nonlinearity and the dispersion effect of higherorder terms.

Fig. 4. A large amplitude ion acoustic wave steepens so that the leading edgehas a steeper slope than the trailing edge.

13.2 Ponderomotive Force

Equation of motion is

mdv

dt= q

[E(r) + v × B(r)

].

60

Take E(r) = Es(r) cos ωt, and expand r = r0 + δr1 + · · ·, E(r) = E(r0) + (δr1 ·∇)E(r0) + · · ·. The first order equation can be solved to give

mdv1

dt= qE(r0)

v1 =q

mωEs(r0) sinωt =

dδr1

dt

δr1 = − q

mω2Es(r0) cos ωt,

The second order equation is

mdv2

dt= q

[(δr1 · ∇)E(r0) + v1 × B1

].

From ∇× E = −∂B/∂t,

B1 =1ω∇× Es(r0) sin ωt

which can be substituted in the equation of motion and time averaged to give

m

⟨dv2

dt

⟩= − q2

mω2

12

[(Es · ∇)Es + Es × (∇× Es)

]= −1

4q2

mω2∇E2

s .

The nonlinear force is given by

FNL = −14

n0q2

mω2∇E2

s = −ω2

p

ω2∇

⟨ϵ0E

2⟩

2

where⟨E2

⟩= E2

s/2. This force is called the ponderomotive force. The pondero-motive force works in the direction to expel particles from a region of strongelectric field, and is stronger for particles with lighter mass and for lower fre-quencies. This is a nonlinear force because it depends on the square of theelectric field strength.

A laser beam of finite diameter travelling through a plasma causes a radiallyoutward ponderomotive force. This force moves plasma out of the beam so thatωp is lower and the dielectric constant ϵ and the index of refraction n are higherinside the beam than outside, since

n =√

ϵ =

√1 −

ω2pe

ω2.

The plasma acts as a convex lens, and focuses the beam to a smaller diameter.

13.3 Caviton and Envelope Soliton

In case of electron plasma wave, higher harmonic components do not satisfy thelinear dispersion relation, so generation of harmonics is not important. If thelinear wave is represented by A cos(kx − ωt), the second order components are

A2 cos2(kx − ωt) =12A2 [cos(2kx − 2ωt) + 1] .

61

Fig. 5. Self-focusing of a laser beam is caused by the ponderomotive force.

The first term in the bracket describes second harmonic generation, which inthis case is unimportant. The second term is the k = 0, ω = 0 component anddescribes the slow evolution of nearly uniform modulation.

The wave equation for electron plasma wave can be expressed in linear ap-proximation as

∂2

∂t2uex + ω2

peuex − 3v2te

∂2

∂x2uex = 0.

Note that ω2pe is proportional to the electron density and v2

te is proportional tothe electron temperature, and that the third term can be ignored in the limitk → 0. Assume that the plasma density is modulated from n0 to n0 + δne(x, t),and that the time variation of δne is much slower than plasma oscillation. Insuch a case

∂2

∂t2uex − 3v2

te

∂2

∂x2uex + ω2

pe

[1 +

δne(x, t)n0

]uex = 0.

Writing the solution as uex = ℜ[u(x, t) exp(−iωpet)] and ignoring ∂2u/∂t2 com-pared to −ω2

peu, the following equation is obtained

i∂

∂tu(x, t) +

32

v2te

ωpe

∂2

∂x2u(x, t) − ωpe

2δne(x, t)

n0u(x, t) = 0.

This equation has the form of Schrodinger equation with the third term as thenonlinear potential arising from the ponderomotive force. As will be shownlater in this sub-section, δne/n0 is proportional to |u(x, t)|2. This equation iscalled the nonlinear Schodinger equation. The solution that corresponds to aparticle trapped in the potential well represents a standing wave trapped in thedensity well. Because the soliton solution of this equation is oscillating at highfrequency, it is called the envelope soliton.

The slow evolution of the density modulation δne is described by the follow-ing equation, obtained by averaging the electron equation of motion over thefast oscillation (ωpe),

me∂uex

∂t+ meuex

∂

∂xuex = − 1

ne

∂pe

∂x+ e

∂ϕ

∂x.

The first term on the left hand side is the electron inertia term, and can beignored when the time evolution is slow. The second term is the nonlinear term

62

Fig. 6. An envelope soliton.

which can be expressed as

∂

∂x

(me

2u2

ex

)=

∂

∂x

(me|u(x, t)|2

)and represents the ponderomotive force. The first term on the right hand sidecan be rewritten as −(Te/n0)∂δne/∂x, so the equation above can be expressedas

∂

∂x

(me|u(x, t)|2

)= − ∂

∂x

(Te

n0δne − eϕ

).

This equation can be solved with the boundary condition δne, ϕ → 0 as u(x, t) →0 to give

δne

n0=

eϕ − me|u(x, t)|2

Te.

Finally, the ion equation of motion must be solved to obtain ϕ. Since ionscannot respond to high frequency electron plasma oscillation, linearization canbe used,

mi∂ui

∂t= −e

∂ϕ

∂x− 1

n0

∂

∂x(niTi).

If the time evolution is slow enough, the ion inertia term can be ignored, and

eϕ = −Tiδni

n0.

Therefore,δne

n0= −me|u(x, t)|2

Te + Ti.

Substituting this relationship in the potential term of the Schrodinger equationyields

i∂

∂tu(x, t) +

32

v2te

ωpe

∂2

∂x2u(x, t) +

ωpe

2me

Te + Ti|u(x, t)|2u(x, t) = 0.

Since the potential depends on the wave intensity, this equation is called thenonlinear Schrodinger equation.

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