1. The KdV equation for ion-acoustic waves in plasmas. Ion-acoustic waves
are low-frequency electrostatic waves in a plasma consisting of electrons and ions.
We consider the case with a single ion species.
Consider the following system of one-dimensional equations
∂n
∂t +
∂
∂z (nv) = 0
∂v
∂t + v
∂v
∂z = −
e
m
∂ϕ
∂z
∂
2ϕ
∂z2
=
e
ε0
N0 exp
eϕ
κTe
− n
Here n denotes the ion density, v is the ion velocity, e is the electron charge, m is
the mass of an ion, ϕ is the electrostatic potential, ε0 is the vacuum permittivity,
N0 is the equilibrium density of the ions, κ is Boltzmann’s constant, and Te is the
electron temperature.
(a) Verify that cs =
q
κTe
m
, λDe =
qε0κTe
N0e
2
, and ωpi =
qN0e
2
ε0m
have dimensions of
velocity, length and frequency, respectively. These quantities are known as the
ion acoustic speed, the Debye wavelength for the electrons, and the ion plasma
frequency.(b) Nondimensionalize the above system, using
n = N0n
∗
, v = csv
∗
, z = λDez
∗
, t =
t
∗
ωpi
, ϕ =
κTe
e
ϕ
∗
.(c) You have obtained the system
∂n
∂t +
∂
∂z (nv) = 0
∂v
∂t + v
∂v
∂z = −
∂ϕ
∂z
∂
2ϕ
∂z2
= e
ϕ − n
for the dimensionless variables. Note that we have dropped the ∗’s, to ease the
notation. Find the linear dispersion relation for this system, linearized around
the trivial solution n = 1, v = 0, and ϕ = 0.(d) Rewrite the system using the “stretched variables”
ξ = ϵ
1/2
(z − t), τ = ϵ
3/2
t
Given that we are looking for low-frequency waves, explain how these variables
are inspired by the dispersion relation.(e) Expand the dependent variables as
n = 1 + ϵn1 + ϵ
2n2 + . . .
v = ϵv1 + ϵ
2
v2 + . . .
ϕ = ϵϕ1 + ϵ
2ϕ2 + . . .
Using that all disturbances return to their equilibrium values as ξ → ±∞,
τ → ∞, find a governing equation which determines how ϕ1 depends on ξ and
τ .2. Obtaining the KdV equation from the NLS equation. We have shown that
the NLS equation may be used to describe the slow modulation of periodic wave
trains of the KdV equation. In this problem we show that the KdV equation
describes the dynamics of long-wave solutions of the NLS equation.
Consider the defocusing NLS equation
iat = −axx + |a|
2
a.
(a) Let
a(x, t) = e
i
R
V dxρ
1/2
.
Derive a system of equations for the phase function V (x, t) and for the amplitude function ρ(x, t), by substituting this form of a(x, t) in the NLS equation,
dividing out the exponential, and separating real and imaginary parts. Write
your equations in the form ρt = . . ., and Vt = . . .. Due to their similarity with
the equations of hydrodynamics, this new form of the NLS equation is referred
to as its hydrodynamic form.(b) Find the linear dispersion relation for the hydrodynamic form of the defocusing NLS equation, linearized around the trivial solution V = 0, ρ = 1.
In other words, we are examining perturbations of the so-called Stokes wave
solution of the NLS equation, which is given by a signal of constant amplitude.(c) Rewrite the system using the “stretched variables”
ξ = ϵ(x − βt), τ = ϵ
3
t
Given that we are looking for long waves, explain how these variables are inspired by the dispersion relation. What should the value of β be?(d) Expand the dependent variables as
V = ϵ
2V1 + ϵ
4V2 + . . .
ρ = 1 + ϵ
2
ρ1 + ϵ
4
ρ2 + . . .
Using that all disturbances return to their equilibrium values as ξ → ±∞,
τ → ∞, find a governing equation which determines how V1 depends on ξ and
τ . This equation should be equivalent to the KdV equation.3. Consider the previous problem, but with the focusing NLS equation
iat = −axx − |a|
2
a.
The method presented in the previous problem does not allow one to describe the
dynamics of long-wave solutions of the focusing NLS equation using the KdV equation. How does this show up in the calculations?4. The mKdV equation considered in the text is known as the focusing mKdV equation,
because of the behavior of its soliton solutions. This behavior is similar to that of the
focusing NLS equation. In this problem, we study the defocusing mKdV equation
4ut = −6u
2ux + uxxx.
You have already seen that you can scale the coefficients of this equation to your
favorite values, except for the ratio of the signs of the two terms on the right-hand
side.
(a) Examine, using the potential energy method and phase plane analysis, the
traveling-wave solutions.(b) If you have found any homoclinic or heteroclinic connections, find the explicit
form of the profiles corresponding to these connections.5. Consider the so-called Derivative NLS equation (DNLS)
bt + αb|b|
2
x
− ibxx = 0.
This equation arises in the description of quasi-parallel waves in space plasmas.
Here b(x, t) is a complex-valued function.
(a) Using a polar decomposition
b(x, t) = B(x, t)e
iθ(x,t)
,
with B and θ real-valued functions, and separating real and imaginary parts
(after dividing by the exponential), show that you obtain the system
Bt + 3αB2Bx +
1
B
(B
2
θx)x = 0,
θt + αB2
θx + θ
2
x −
1
B
Bxx = 0(b) Assuming a traveling-wave envelope, B(x, t) = R(z), with z = x − vt and
constant v, show that θ(x, t) = Φ(z) − Ωt, with constant Ω, is consistent
with these equations. You can show (but you don’t have to) that assuming
a traveling-wave amplitude results in only this possibility for θ(x, t). At this
point, we have reduced the problem of finding solutions with traveling envelope to that of finding two one-variable functions R(z) and Φ(x). The problem
also depends on two parameters v (envelope speed) and Ω (frequency like).(c) Substituting these ansatz in the first equation of the above system, show that
Φ
′ =
C + vs − 3s
2
2s
,
where C is a constant and s = αR2/2.(d) Lastly, by substituting your results in the second equation of the system, show
that s(z) satisfies
1
2
s
′2 + V (s) = E,
the equation for the motion of a particle with potential V (s). Find the expression for V (s) and for E.6. Consider example 5.2 in the notes. Check that y = x
2/t and t
1/2
q are both scaling
invariant. Find the ordinary differential equation satisfied by G(y), for similarity
solutions of the form q(x, t) = t
−1/2G(y). Show that this results in the same similarity solutions as in the example.7. One way to write the Toda Lattice is
dan
dt = an(bn+1 − bn),
dbn
dt = 2(a
2
n − a
2
n−1
),
where an, bn, n ∈ Z, are functions of t.
(a) Find a scaling symmetry of this form of the Toda lattice, i.e., let2 an = αAn,
bn = βBn, t = γτ , and determine relations between α, β and γ so that the
equations for the Toda lattice in the (An, Bn, t) variables are identical to those
using the (an, bn, τ ) variables.(b) Using this scaling symmetry, find a two-parameter family of similarity solutions of the Toda lattice. If necessary, find relations among the parameters
that guarantee the solutions you found are real for all n and for t > 0.8. Consider the equation
ut = 30u
2ux + 20uxuxx + 10uuxxx + u5x,
which we will encounter more in later chapters, due to its relation to the KdV
equation. Show that it has a scaling symmetry.
When we look for the scaling symmetry of the KdV equation, we have two equations
for three unknowns: we have three quantities (x, t, u) to scale, and after normalizing one coefficient to 1, two remaining terms that need to remain invariant. Thus it
is no surprise that we find a one-parameter family of scaling symmetries. The above
equation has two more terms, and it should be clear that some “luck” is needed in
order for there to be a scaling symmetry.9. Consider a Modified KdV equation
ut − 6u
2ux + uxxx = 0.
(a) Find its scaling symmetry(b) Using the scaling symmetry, write down an ansatz for any similarity solutions
of the equation.(c) Show that your ansatz is compatible with u = (3t)
−1/3w(z), with z = x/(3t)
1/3
.(d) Use the above form of u to find an ordinary differential equation for w(z). This
equation will be of third order. It can be integrated once (do this) to obtain
a second-order equation. The second-order equation you obtain this way is
known as the second of the Painlev´e equations. We will see more about these
later.
573, AMATH, Problem, solved
[SOLVED] Amath 573, problem set 3
$25
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