Liouville–Bratu–Gelfand equation

In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville,[1] G. Bratu[2] and Israel Gelfand.[3] The equation reads

${\displaystyle \nabla ^{2}\psi +\lambda e^{\psi }=0}$

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

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[4] and describes planetary nebula.

Liouville's solution[5]

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

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

where ${\displaystyle f(z)=u+iv}$ is an arbitrary analytic function with ${\displaystyle z=x+iy}$. In 1915, G.W. Walker[6] found a solution by assuming a form for ${\displaystyle f(z)}$. If ${\displaystyle r^{2}=x^{2}+y^{2}}$, then Walker's solution is

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

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

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

where ${\displaystyle r}$ is the distance from the origin. With the boundary conditions

${\displaystyle \psi '(0)=0,\quad \psi (1)=0}$

and for ${\displaystyle \lambda \geq 0}$, a real solution exists only for ${\displaystyle \lambda \in [0,\lambda _{c}]}$, where ${\displaystyle \lambda _{c}}$ is the critical parameter called as Frank-Kamenetskii parameter. The critical parameter is ${\displaystyle \lambda _{c}=0.8785}$ for ${\displaystyle n=1}$, ${\displaystyle \lambda _{c}=2}$ for ${\displaystyle n=2}$ and ${\displaystyle \lambda _{c}=3.32}$ for ${\displaystyle n=3}$. For ${\displaystyle n=1,\ 2}$, two solution exists and for ${\displaystyle 3\leq n\leq 9}$ infinitely many solution exists with solutions oscillating about the point ${\displaystyle \lambda =2(n-2)}$. For ${\displaystyle n\geq 10}$, the solution is unique and in these cases the critical parameter is given by ${\displaystyle \lambda _{c}=2(n-2)}$. Multiplicity of solution for ${\displaystyle n=3}$ was discovered by Israel Gelfand in 1963 and in later 1973 generalized for all ${\displaystyle n}$ by Daniel D. Joseph and Thomas S. Lundgren[7].

References

1. Liouville, J. "Sur l’équation aux différences partielles ${\displaystyle {\frac {d^{2}\log \lambda }{dudv}}\pm {\frac {\lambda }{2a^{2}}}=0}$." Journal de mathématiques pures et appliquées (1853): 71–72. http://sites.mathdoc.fr/JMPA/PDF/JMPA_1853_1_18_A3_0.pdf
2. Bratu, G. "Sur les équations intégrales non linéaires." Bulletin de la Société Mathématique de France 42 (1914): 113–142.http://archive.numdam.org/article/BSMF_1914__42__113_0.pdf
3. Gelfand, I. M. "Some problems in the theory of quasilinear equations." Amer. Math. Soc. Transl 29.2 (1963): 295–381. http://www.mathnet.ru/links/aa75c5d339030f17940afb64e17793d8/rm7290.pdf
4. Richardson, Owen Willans. The emission of electricity from hot bodies. Longmans, Green and Company, 1921.
5. Bateman, Harry. "Partial differential equations of mathematical physics." Partial Differential Equations of Mathematical Physics, by H. Bateman, Cambridge, UK: Cambridge University Press, 1932 (1932).
6. Walker, George W. "Some problems illustrating the forms of nebulae." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 91.631 (1915): 410-420.https://www.jstor.org/stable/pdf/93512.pdf?refreqid=excelsior%3Af4a4cc9656b8bbd9266f9d32587d02b1
7. Joseph, D. D., and T. S. Lundgren. "Quasilinear Dirichlet problems driven by positive sources." Archive for Rational Mechanics and Analysis 49.4 (1973): 241-269.