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


 * $$\nabla^2 \psi + \lambda e^\psi = 0$$

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 $$(x,y)$$, Joseph Liouville proposed a solution in 1853 as


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

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


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

where $$a$$ is some finite radius. This solution decays at infinity for any $$n$$, but becomes infinite at the origin for $$n<1$$, becomes finite at the origin for $$n=1$$ and becomes zero at the origin for $$n>1$$. 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 $$n$$ dimension becomes


 * $$\psi'' + \frac{n-1}{r}\psi' + \lambda e^\psi=0$$

where $$r$$ is the distance from the origin. With the boundary conditions


 * $$\psi'(0)=0, \quad \psi(1) = 0$$

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

The solution for $$n=1$$ that is valid in the range $$\lambda \in [0,0.8785]$$ is given by


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

where $$\psi_m=\psi(0)$$ is related to $$\lambda$$ as


 * $$e^{\psi_m/2} = \cosh \left(\frac{\sqrt{\lambda}}{\sqrt 2}e^{-\psi_m/2}\right).$$

The solution for $$n=2$$ that is valid in the range $$\lambda \in [0,2]$$ is given by


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

where $$\psi_m=\psi(0)$$ is related to $$\lambda$$ as


 * $$ (\lambda e^{\psi_m}+8)^2 - 64 e^{\psi_m} =0.$$