[SOLVED] Amath 573, problem set 2

1. Problem 1: The Benjamin-Ono equation
ut + uux + Huxx = 0
is used to describe internal water waves in deep water. Here Hf(x, t) is
the spatial Hilbert transform of f(x, t):
Hf(x, t) = 1
π
−
Z ∞
−∞
f(z, t)
z − x
dz
and −
R
denotes the Cauchy principle value integral. Write down the linear
dispersion relationship for this equation linearized about the zero solution.2. Problem 2: Derive the linear dispersion relationship for the one-dimensional
surface water wave problem by linearizing around the trivial solution
ζ(x, t) = 0, ϕ(x, z, t) = 0 :
(a) ∇2ϕ = 0 at −h < z < ζ(x, t)
(b) ϕz = 0 at z = −h
(c) ζt + ϕxζx = ϕz at z = ζ(x, t)
(d) ϕt + gζ +
1
2
(ϕ
2
x + ϕ
2
z
) = T
ζxx
(1+ζ
2
x
)
3/2 at z = ζ(x, t)
Here z = ζ(x, t) is the surface of the water wave, ϕ(x, z, t) is the velocity
potential so that v = ∇ϕ is the velocity of the water, g is the acceleration
of gravity, and T > 0 is the coefficient of surface tension3. Problem 3: Having found that for the surface water wave problem without
surface tension the linear dispersion relationship is ω
2 = gk tanh kh, find
the group velocities for the case of long waves in shallow water (kh small),
and for the case of deep water (kh big).4. Problem 4: Whitham wrote down what is now known as the Whitham
equation to incorporate the full effect of water-wave dispersion for waves
in shallow water by modifying the KdV equation ut+vux+uux+γuxxx = 0
(where we have included the transport term) to
ut + uux +
Z ∞
−∞
K(x − y)uy(y, t)dy = 0
where
K(x) = 1
2π
−
Z ∞
−∞
c(k)e
ikxdk
p
and c(k) is the postive phase speed for the water-wave problem: c(k) =
g tanh(kh)/k.
(a) What is the linear dispersion relation of the Whitham equation?
(b) Show that the dispersion relation of the KdV equation is an approximation to that of the Whitham equation for long waves, i.e., for
k → 0. What are v and γ?
Note that using this process of ”Whithamization”, one could construct a
KdV-like equation (i.e., an equation with the KdV nonlinearity) that has
any desired dispersion relation. Similar procedures can be followed for
other equations, like the NLS equation, etc.5. Problem 5: Consider the linear free Schr¨odinger (”free”, because there’s
no potential) equation
iψt + ψxx = 0
where −∞ < x < ∞, t > 0, ψ → 0 as |x| → ∞. With ψ(x, 0) = ψ0(x)
such that R ∞
−∞ |ψ0|
2dx < ∞.
(a) Using the Fourier transform, write down the solution of this problem.
(b) Using the Method of Stationary Phase, find the dominant behavior
as t → ∞ of the solution, along lines of constant x/t.
(c) With ψ0(x) = e
−x
2
, the integral can be worked out exactly. Compare
(graphically or otherwise) this exact answer with the answer you get
from the Method of Stationary Phase. Use the lines x/t = 1 and
x/t = 2 to compare.
(d) Use your favorite numerical integrator (write your own, or use maple,
mathematica or matlab) to compare (graphically or other) with the
exact answer and the answer you get from the Method of Stationary
Phase.6. Problem 6: Everything that we have done for continuous space equations
also works for equations with a discrete space variable. Consider the
discrete linear Schr¨odinger equation:
i
dψn
dt +
1
h
2
(ψn+1 − 2ψn + ψn−1) = 0
where h is a real constant, n is any integer, t > 0, ψn0 as |n| → ∞, and
ψn(0) = ψn,0 is given.
(a) The discrete analogue of the Fourier transform is given by
ψn(t) = 1
2πi I
|z|=1
ψˆ(z, t)z
n−1
dz
and its inverse
ψˆ(z, t) = X∞
m=−∞
ψm(t)z
−m
Show that these two transformations are indeed inverses of each
other.
(b) The dispersion relation of a semi-discrete problem is obtained by
looking for solutions of the form ψn = z
ne
−iωt. Show that for the
semi-discrete Schr¨odinger eqaution
ω(z) = −
(z − 1)2
zh2
How does this compare to the dispersion relation of the continuous
space problem? Specifically, demonstrate that you recover the dispersion relationship for the continuous problem as h → 0.