[SOLVED] Amath 573 solitons and nonlinear waves homework 6

1. Show that N¯(x, k) is analytic in the open lower-half plane, Imk < 0, by showing
N¯(x, k) and ∂N/∂k ¯ are bounded there. What are the conditions you need for this
to be true?2. Recall the second member of the stationary KdV hierarchy from HW4:
30u
2ux + 20uxuxx + 10uuxxx + u5x + c1(6uux + uxxx) + c0ux = 0.
Integrating once, it can be rewritten as
(10u
3 + 10u
2
x + 10uuxx − 5u
2
x + uxxxx) + c1(3u
2 + uxx) + c0u + c−1 = 0. (1)
This is an ordinary differential equation for u as a function of x. You already know
that the two soliton is a solution of this. You know this equation can be written as
δT2
δu = 0 ⇐⇒
δ
δu (F2 + c1F1 + c0F0 + c−1F−1) = 0, (2)
where Fk, k = −1, 0, . . . are the conserved quantities of the KdV equation. These
conserved quantities are in involution,
{Fj
, Fk} = 0 ⇒ {Tj
, Fk} = 0,
for j, k = −1, 0, . . .. From HW4, you know that this implies (using j = 2)
δT2
δu
d
dx
δFk
δu =
dHk
dx , k = 0, 1. (3)
For some functions H0 and H1. Thus, H0 and H1 are conserved quantities of (1)1
.
(a) Find H0 and H1 explicitly.(b) Check explicitly, by taking an x derivative, that H0 and H1 are conserved
along solutions of (1).3. The Ostrovsky equation is used to model weakly nonlinear long waves in a
rotating frame. It is given by
(ηt + ηηx + ηxxx)x = γη,
with γ ̸= 0. In what follows, we assume that as |x| → ∞, η and its derivatives
approach zero as fast as we need them to.
• Show that R ∞
−∞ ηdx = 0. In other words, not only is R ∞
−∞ ηdx conserved, but
its value is fixed at zero.• Using this result, show that R ∞
−∞ η
2dx is a conserved quantity. Do this by
rewriting the equation in evolution form, with an indefinite integral on the
right-hand side.• Use the definition of the variational derivative to verify that the Ostrovsky
equation is Hamiltonian with Poisson operator ∂x and Hamiltonian
H =
1
2
Z ∞
−∞ 
η
2
x −
1
3
η
3 − γϕ2

dx,
where ϕx = η.4. Using the Painlev´e test, discuss the integrability of
ut = u
pux + uxxx.