[SOLVED] Amath 573, problem set 1

1. Problem 1.2: The KdV equation
ut = uux + uxxx
is often written with different coefficients. By using a scaling transformation on all variables (dependent and independent), show that the choice
of the coefficients is irrelevant: by choosing a suitable scaling, we can use
any coefficients we please. Can you say the same for the modified KdV
(mKdV) equation
ut = u
2ux + uxxx?2. Problem 1.4: (Use a symbolic computing software for this problem.) Consider the KdV equation ut + uux + uxxx = 0. Show that
u = 12∂
2
x
ln 
1 + e
k1x−k
3
1
t+α

is a one-soliton solution of the equation (i.e., rewrite it in sech2
form).
Now check that
u = 12∂
2
x
ln
1 + e
k1x−k
3
1
t+α + e
k2x−k
3
2
t+β +

k1 − k2
k1 + k2
2
e
k1x−k
3
1
t+α+k2x−k
3
2
t+β
!
is also a solution of the equation. It is a two-soliton solution of the equation, as we will verify later. By changing t, we can see how the two
solitons interact. With α = 0 and β = 1, examine the following 3 regions
of parameter space:
(a) k1
k2
>
√
3
(b) √
3 >
k1
k2
>
q
(3 + √
5)/2
(c) k1
k2
<
q
(3 + √
5)/2.
Discuss the different types of collisions. Here ”examine” and ”discuss” are
supposed to be interpreted in an experimental sense: play around with this
solution and observe what happens. The results you observe are the topic
of the second part of Lax’s seminal paper.3. Problem 1.5: The Cole-Hopf transformation. Show that every nonzero solution of the heat equation θt = νθxx gives rise to a solution of
the dissipative Burgers equation ut + uux = νuxx, through the mapping
u = −2νθx/θ.4. Problem 1.6: From the previous problem, you know that every solution
of the heat equation θt = νθxx gives rise to a solution of the dissipative
Burger’s equation ut + uux = νuxx, through the mapping u = −2νθx/θ.
(a) Check that θ = 1 + αe−kx+νk2
t
is a solution of the heat equation.
What solution of Burgers’ equation does it correspond to? Describe
this solution qualitatively (velocity, amplitude, steepness, etc) in
terms of its parameters.
(b) Check that θ = 1 + αe−k1x+νk2
1
t + βe−k2x+νk2
2
t
is a solution to the
heat equation. What solution of burgers’ equation does it correspond
to? Describe the dynamics of this solution, i.e., how does it change
in time?