Scaling Properties of the PDE Sample Clauses

Scaling Properties of the PDE. We will vary the value of α and see how the solution behaves around the singularity. Before we look at specific α, we will get some intuition as to what should occur to the solution to (3.2.3). Naturally, we choose the self–similar coordinate change of y = (1 − t)α , τ = − log(1 − t) and write v(y, τ ) = u(l, t), where u satisfies (3.2.3). Using the chain rule, one concludes that v satisfies ∂v −(1−2α)τ ∂2v ∂v 2 ∂v ∂τ = e ∂y2 − αy∂y − α sech 2g(y)v + αG(y) . ∂y Thus if α < 1 , the diffusion term exponentially vanishes as τ → ∞ and only the drift has an effect. However, if α > 1 , the diffusion term exponentially explodes as τ → ∞ and so this was the wrong scaling. In this case, we let y = √1 − t , τ = − log(1 − t) and write v(y, τ ) = u(l, t) to see that v satisfies (omitting the arguments of G and sech 2g for typographic clarity) ∂v ∂2v y ∂v 2 −(α− )τ ∂v ∂τ = ∂y2 − 2 ∂y − α sech 2gv + αe 2 G . ∂y ∂y and so the G∂v term exponentially vanishes as τ → ∞ and only the diffusion takes effect. We note that the α = 1 case yields ∂v ∂2v y ∂v 1 2 1 ∂v ∂τ = ∂y2 − 2 ∂y − 2 sech 2g(y)v + G(y) 2 ∂y which is arguably the critical case of α. From these scaling arguments we expect that if α < 1 , the solution u will vanish at the singularity, whereas for α > 1 we expect that this is not the case. These expectations are explained via the scaling above, along with the “shearing” effect of the drift in the case of α < 1 , which is not present in the α > 1 case.