QuestionQuestion

1. Consider the PDE
auₜ + buuₓ + cuₓₓₓ = 0 (1)
for u(x,t) with positive constants (a, b, c). Find a rescaling of x and t that transforms
(1) into the standard KdV equation
uₜ + 6uuₓ + uₓₓₓ = 0. (2)

2. Recall the Cole-Hopf transformation for the viscous Burgers equation
uₜ + 6uuₓ + uₓₓₓ = 0, (3)
which relates u(x,t) to a function ϕ(x,t) > 0 that solves the heat equation
ϕₜ = kϕₓₓ (4)
What kind of solution does the Green's function on the real line of (4) correspond to in terms of u?

3. Find an integral expression for the general solution to the initial-value problem of (3) on the real line by combining the Cole-Hopf transformation with the Green's function for (4).

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