Liouville–Bratu–Gelfand equation

For Liouville's equation in differential geometry, see Liouville's equation.

In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville, Gheorghe Bratu and Israel Gelfand. The equation reads

[math]\displaystyle{ \nabla^2 \psi + \lambda e^\psi = 0 }[/math]

The equation appears in thermal runaway as Frank-Kamenetskii theory, astrophysics for example, Emden–Chandrasekhar equation. This equation also describes space charge of electricity around a glowing wire and describes planetary nebula.

Liouville's solution

In two dimension with Cartesian Coordinates [math]\displaystyle{ (x,y) }[/math], Joseph Liouville proposed a solution in 1853 as

[math]\displaystyle{ \lambda e^\psi (u^2 + v^2 + 1) ^2 = 2 \left[\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2\right] }[/math]

where [math]\displaystyle{ f(z)=u + i v }[/math] is an arbitrary analytic function with [math]\displaystyle{ z=x+iy }[/math]. In 1915, G.W. Walker found a solution by assuming a form for [math]\displaystyle{ f(z) }[/math]. If [math]\displaystyle{ r^2=x^2+y^2 }[/math], then Walker's solution is

[math]\displaystyle{ 8 e^{-\psi} = \lambda \left[\left(\frac{r}{a}\right)^n + \left(\frac{a}{r}\right)^n\right]^2 }[/math]

where [math]\displaystyle{ a }[/math] is some finite radius. This solution decays at infinity for any [math]\displaystyle{ n }[/math], but becomes infinite at the origin for [math]\displaystyle{ n\lt 1 }[/math] , becomes finite at the origin for [math]\displaystyle{ n=1 }[/math] and becomes zero at the origin for [math]\displaystyle{ n\gt 1 }[/math]. Walker also proposed two more solutions in his 1915 paper.

Radially symmetric forms

If the system to be studied is radially symmetric, then the equation in [math]\displaystyle{ n }[/math] dimension becomes

[math]\displaystyle{ \psi'' + \frac{n-1}{r}\psi' + \lambda e^\psi=0 }[/math]

where [math]\displaystyle{ r }[/math] is the distance from the origin. With the boundary conditions

[math]\displaystyle{ \psi'(0)=0, \quad \psi(1) = 0 }[/math]

and for [math]\displaystyle{ \lambda\geq 0 }[/math], a real solution exists only for [math]\displaystyle{ \lambda \in [0,\lambda_c] }[/math], where [math]\displaystyle{ \lambda_c }[/math] is the critical parameter called as Frank-Kamenetskii parameter. The critical parameter is [math]\displaystyle{ \lambda_c=0.8785 }[/math] for [math]\displaystyle{ n=1 }[/math], [math]\displaystyle{ \lambda_c=2 }[/math] for [math]\displaystyle{ n=2 }[/math] and [math]\displaystyle{ \lambda_c=3.32 }[/math] for [math]\displaystyle{ n=3 }[/math]. For [math]\displaystyle{ n=1, \ 2 }[/math], two solution exists and for [math]\displaystyle{ 3\leq n\leq 9 }[/math] infinitely many solution exists with solutions oscillating about the point [math]\displaystyle{ \lambda=2(n-2) }[/math]. For [math]\displaystyle{ n\geq 10 }[/math], the solution is unique and in these cases the critical parameter is given by [math]\displaystyle{ \lambda_c=2(n-2) }[/math]. Multiplicity of solution for [math]\displaystyle{ n=3 }[/math] was discovered by Israel Gelfand in 1963 and in later 1973 generalized for all [math]\displaystyle{ n }[/math] by Daniel D. Joseph and Thomas S. Lundgren.

The solution for [math]\displaystyle{ n=1 }[/math] that is valid in the range [math]\displaystyle{ \lambda \in [0,0.8785] }[/math] is given by

[math]\displaystyle{ \psi = -2 \ln \left[e^{-\psi_m/2}\cosh \left(\frac{\sqrt{\lambda}}{\sqrt 2}e^{-\psi_m/2}r\right)\right] }[/math]

where [math]\displaystyle{ \psi_m=\psi(0) }[/math] is related to [math]\displaystyle{ \lambda }[/math] as

[math]\displaystyle{ e^{\psi_m/2} = \cosh \left(\frac{\sqrt{\lambda}}{\sqrt 2}e^{-\psi_m/2}\right). }[/math]

The solution for [math]\displaystyle{ n=2 }[/math] that is valid in the range [math]\displaystyle{ \lambda \in [0,2] }[/math] is given by

[math]\displaystyle{ \psi = \ln \left[\frac{64e^{\psi_m}}{(\lambda e^{\psi_m}r^2+8)^2}\right] }[/math]

where [math]\displaystyle{ \psi_m=\psi(0) }[/math] is related to [math]\displaystyle{ \lambda }[/math] as

[math]\displaystyle{ (\lambda e^{\psi_m}+8)^2 - 64 e^{\psi_m} =0. }[/math]