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 $f(z)=u+iv$ 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

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

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

$(\lambda e^{\psi _{m}}+8)^{2}-64e^{\psi _{m}}=0.$