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Cole–Hopf transformation

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The Cole–Hopf transformation is a change of variables that allows to transform a special kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit formula for fairly general solutions of the PDE in terms of the initial datum and the heat kernel.

Consider the following PDE: u t − a Δ u + b ‖ ∇ u ‖ 2 = 0 , u ( 0 , x ) = g ( x ) {\displaystyle u_{t}-a\Delta u+b\|\nabla u\|^{2}=0,\quad u(0,x)=g(x)} where x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , a , b {\displaystyle a,b} are constants, Δ {\displaystyle \Delta } is the Laplace operator, ∇ {\displaystyle \nabla } is the gradient, and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the Euclidean norm in R n {\displaystyle \mathbb {R} ^{n}} . By assuming that w = ϕ ( u ) {\displaystyle w=\phi (u)} , where ϕ ( ⋅ ) {\displaystyle \phi (\cdot )} is an unknown smooth function, we may calculate: w t = ϕ ′ ( u ) u t , Δ w = ϕ ′ ( u ) Δ u + ϕ ″ ( u ) ‖ ∇ u ‖ 2 {\displaystyle w_{t}=\phi '(u)u_{t},\quad \Delta w=\phi '(u)\Delta u+\phi ''(u)\|\nabla u\|^{2}} Which implies that: w t = ϕ ′ ( u ) u t = ϕ ′ ( u ) ( a Δ u − b ‖ ∇ u ‖ 2 ) = a Δ w − ( a ϕ ″ + b ϕ ′ ) ‖ ∇ u ‖ 2 = a Δ w {\displaystyle {\begin{aligned}w_{t}=\phi '(u)u_{t}&=\phi '(u)\left(a\Delta u-b\|\nabla u\|^{2}\right)\\&=a\Delta w-(a\phi ''+b\phi ')\|\nabla u\|^{2}\\&=a\Delta w\end{aligned}}} if we constrain ϕ {\displaystyle \phi } to satisfy a ϕ ″ + b ϕ ′ = 0 {\displaystyle a\phi ''+b\phi '=0} . Then we may transform the original nonlinear PDE into the canonical heat equation by using the transformation:

w ( u ) = e − b u / a {\displaystyle w(u)=e^{-bu/a}}

This is the Cole-Hopf transformation. With the transformation, the following initial-value problem can now be solved: w t − a Δ w = 0 , w ( 0 , x ) = e − b g ( x ) / a {\displaystyle w_{t}-a\Delta w=0,\quad w(0,x)=e^{-bg(x)/a}} The unique, bounded solution of this system is: w ( t , x ) = 1 ( 4 π a t ) n / 2 ∫ R n e − ‖ x − y ‖ 2 / 4 a t − b g ( y ) / a d y {\displaystyle w(t,x)={1 \over {(4\pi at)^{n/2}}}\int _{\mathbb {R} ^{n}}e^{-\|x-y\|^{2}/4at-bg(y)/a}dy} Since the Cole–Hopf transformation implies that u = − ( a / b ) log ⁡ w {\displaystyle u=-(a/b)\log w} , the solution of the original nonlinear PDE is: u ( t , x ) = − a b log ⁡ [ 1 ( 4 π a t ) n / 2 ∫ R n e − ‖ x − y ‖ 2 / 4 a t − b g ( y ) / a d y ] {\displaystyle u(t,x)=-{a \over {b}}\log \left}

The complex form of the Cole-Hopf transformation can be used to transform the Schrödinger equation to the Madelung equation.

Applications

References

  1. Evans, Lawrence C. (2010). Partial Differential Equations. Graduate Studies in Mathematics. Vol. 19 (2nd ed.). American Mathematical Society. pp. 206–207.
  2. Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Die Naturwissenschaften (in German). 40 (3–4): 322–326. doi:10.1007/BF01504657. ISSN 1434-6001.
  3. Cole, Julian D. (1951). "On a quasi-linear parabolic equation occurring in aerodynamics". Quarterly of Applied Mathematics. 9 (3): 225–236. doi:10.1090/qam/42889. ISSN 0033-569X.
  4. Hopf, Eberhard (1950). "The partial differential equation ut + uux = μxx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.
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