Problems on Travelling Wave Solutions
1. Find any travelling wave solutions to the so called ”modified KdV equation”.
/ t uÝx, tÞ + 6u 2 / x uÝx, tÞ + / xxx uÝx, tÞ = 0
where u, / x u and / xx u ¸ 0 as | x| ¸ K
2. Find any travelling wave solutions to the elastic medium equation
/ tt uÝx, tÞ = / xx uÝx, tÞ + / x uÝx, tÞ / xx uÝx, tÞ + / xxxx uÝx, tÞ
where / x u, / xx u and / xxx u ¸ 0 as | x| ¸ K
3. Find any travelling wave solutions to the equation
/ t uÝx, tÞ = / xx uÝx, tÞ + sin uÝx, tÞ
where u, / x u and / xx u ¸ 0 as | x| ¸ K
4. Determine whether there are any travelling wave solutions
sÝx, tÞ = SÝx ? ctÞ,
iÝx, tÞ = IÝx ? ctÞ,
for the reaction diffusion system
/ t sÝx, tÞ = ?sÝx, tÞ 2 iÝx, tÞ
/ t iÝx, tÞ = / xx iÝx, tÞ + sÝx, tÞ 2 iÝx, tÞ ? K iÝx, tÞ
where
SÝzÞ ¸ 1, IÝzÞ ¸ 0 as z ¸ +K, S L ÝzÞ, I L ÝzÞ ¸ 0 as z ¸ ?K
5. Analyze the travelling wave solutions for
/ t uÝx, tÞ + u / x uÝx, tÞ ? J / xx uÝx, tÞ + K / xxx uÝx, tÞ = 0
by converting the nonlinear ODE to a dynamical system and looking for homoclinic or
heteroclinic orbits.
Analyze the interaction between the parameters J and K by determining the relative
magnitudes where oscillating solutions exist and for what values the solutions cease to
oscillate.
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6. Analyze the travelling wave solutions for
/ t uÝx, tÞ + sgnÝuÞ / x uÝx, tÞ + K / xxx uÝx, tÞ = 0
Suppose uÝx, tÞ = vÝx ? ctÞ where v L , v” ¸ 0 but vÝzÞ ¸ const as | z| ¸ K
7. Find any travelling wave solutions to the equation
where
/ t uÝx, tÞ = / xx uÝx, tÞ + uÝu ? aÞÝu ? 1Þ
0 < a < 1.
Show that the associated dynamical system has three critical points and that for wave
speeds below a certain critical value, two of the critical points are stable foci which leads to
TW solutions with both positive and negative values (which are nonphysical in some
contexts). Show that for c 2 > 4Ýa + 1Þ the TW solutions assume only positive values and
that the heteroclinic orbits of the dynamical system in this case are true TW solutions for
the PDE.
8.Find any travelling wave solutions to the equation
/ t uÝx, tÞ = / xx uÝx, tÞ + uÝu ? aÞÝu ? bÞÝu ? 1Þ
where 0 < a < b < 1.
Assume whatever conditions at infinity you need to get a TWS but examine particularly how
the sign of c (the wave speed) is affected by the relative amounts of area contained under
the loops of fÝuÞ = uÝu ? aÞÝu ? bÞÝu ? 1Þ.
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