[SOLVED] Amath 573 solitons and nonlinear waves homework 5

1. Show that
X =

−iζ q
±q
∗
iζ 
, T =

−iζ2 ∓
i
2
|q|
2
qζ +
i
2
qx
±q
∗
ζ ∓
i
2
q
∗
x
iζ2 ±
i
2
|q|
2

are Lax Pairs for the Nonlinear Schr¨odinger equations
iqt = −
1
2
qxx ± |q|
2
q.
Here the top (bottom) signs of one matrix correspond to the top (bottom) signs of
the other. In other words, show that the X, T with the top (bottom) sign are a
Lax pair for the Nonlinear Schr¨odinger equation with the top (bottom) sign.2. Let ψn = ψn(t), n ∈ Z. Consider the difference equation
ψn+1 = Xnψn,
and the differential equation
∂ψn
∂t = Tnψn.
What is the compatibility condition of these two equations? Using this result, show
that
Xn =

z qn
q
∗
n
1/z 
, Tn =

iqnq
∗
n−1 −
i
2
(1/z − z)
2 i
z
qn−1 − izqn
−izq∗
n−1 +
i
z
q
∗
n −iq∗
n
qn−1 +
i
2
(1/z − z)
2

is a Lax Pair for the semi-discrete equation
i
∂qn
∂t = qn+1 − 2qn + qn−1 − |qn|
2
(qn+1 + qn−1)
Note that this is a discretization of the NLS equation. It is known as the AblowitzLadik lattice. It is an integrable discretization of NLS. For numerical purposes, it
is far superior in many ways to the “standard” discretization of NLS:
i
∂qn
∂t = qn+1 − 2qn + qn−1 − 2|qn|
2
qn.3. For the KdV equation ut + 6uux + uxxx = 0 with initial condition u(x, 0) = 0 for
x ∈ (−∞, −L) ∪ (L, ∞), and u(x, 0) = d for x ∈ (−L, L), with L and d both
positive, consider the forward scattering problem.
• Find a(k), for all time t.
• Knowing that the number of solitons emanating from the initial condition is
the number of zeros of a(k) on the positive imaginary axis (i.e., k = iκ, with
κ > 0), discuss how many solitons correspond to the given initial condition,
depending on the value of 2L
2d. You might want to use Maple, Mathematica
or Matlab for this.
• What happens for d < 0?
• In the limit L → 0, but 2dL = α, u(x, 0) → αδ(x). What happens to a(k)
when you take this limit? Discuss.5. The Liouville equation. Consider the horribly nonlinear1 PDE
uxy = e
u
,
known as Liouville’s equation. Consider the transformation
vx = −ux +
√
2e
(u−v)/2
,
vy = uy −
√
2e
(u+v)/2
,
where u(x, y) satisfies Liouville’s equation above.
(a) Find an equation satisfied by v(x, y): vxy = . . .. Your right-hand side cannot
have any u’s. Those should all be eliminated.(b) Write down the general solution for v(x, y) from the equation you obtained.(c) Use this solution for v in your B¨acklund transformation and solve for u, obtaining the general solution of the Liouville equation!6. The sine-Gordon equation. Consider the sine-Gordon equation
uxt = sin u,
also horribly nonlinear.
(a) Show that the transformation
vx = ux + 2 sin
u + v
2
,
vt = −ut − 2 sin
u − v
2
,
is an auto-B¨acklund transformation for the sine-Gordon equation. In other
words, v satisfies the same equation as u.(b) Let u(x, t) be the simplest2
solution of the sine-Gordon equation. With this
u(x, y) solve the auto-B¨acklund transformation for v(x, t), to find a more complicated solution of the sine-Gordon equation. Congratulations! You just
found the one-soliton solution of the sine-Gordon equation