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
∇ 2 ψ + λ e ψ = 0 {\displaystyle \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 Source: In two dimension with Cartesian Coordinates ( x , y ) {\displaystyle (x,y)} , Joseph Liouville proposed a solution in 1853 as
λ e ψ ( u 2 + v 2 + 1 ) 2 = 2 [ ( ∂ u ∂ x ) 2 + ( ∂ u ∂ y ) 2 ] {\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 f ( z ) = u + i v {\displaystyle f(z)=u+iv} is an arbitrary analytic function with z = x + i y {\displaystyle z=x+iy} . In 1915, G.W. Walker found a solution by assuming a form for f ( z ) {\displaystyle f(z)} . If r 2 = x 2 + y 2 {\displaystyle r^{2}=x^{2}+y^{2}} , then Walker's solution is
8 e − ψ = λ [ ( r a ) n + ( a r ) n ] 2 {\displaystyle 8e^{-\psi }=\lambda \left[\left({\frac {r}{a}}\right)^{n}+\left({\frac {a}{r}}\right)^{n}\right]^{2}}
where a {\displaystyle a} is some finite radius. This solution decays at infinity for any n {\displaystyle n} , but becomes infinite at the origin for n < 1 {\displaystyle n<1} , becomes finite at the origin for n = 1 {\displaystyle n=1} and becomes zero at the origin for n > 1 {\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 n {\displaystyle n} dimension becomes
ψ ″ + n − 1 r ψ ′ + λ e ψ = 0 {\displaystyle \psi ''+{\frac {n-1}{r}}\psi '+\lambda e^{\psi }=0}
where r {\displaystyle r} is the distance from the origin. With the boundary conditions
ψ ′ ( 0 ) = 0 , ψ ( 1 ) = 0 {\displaystyle \psi '(0)=0,\quad \psi (1)=0}
and for λ ≥ 0 {\displaystyle \lambda \geq 0} , a real solution exists only for λ ∈ [ 0 , λ c ] {\displaystyle \lambda \in [0,\lambda _{c}]} , where λ c {\displaystyle \lambda _{c}} is the critical parameter called as Frank-Kamenetskii parameter. The critical parameter is λ c = 0.8785 {\displaystyle \lambda _{c}=0.8785} for n = 1 {\displaystyle n=1} , λ c = 2 {\displaystyle \lambda _{c}=2} for n = 2 {\displaystyle n=2} and λ c = 3.32 {\displaystyle \lambda _{c}=3.32} for n = 3 {\displaystyle n=3} . For n = 1 , 2 {\displaystyle n=1,\ 2} , two solution exists and for 3 ≤ n ≤ 9 {\displaystyle 3\leq n\leq 9} infinitely many solution exists with solutions oscillating about the point λ = 2 ( n − 2 ) {\displaystyle \lambda =2(n-2)} . For n ≥ 10 {\displaystyle n\geq 10} , the solution is unique and in these cases the critical parameter is given by λ c = 2 ( n − 2 ) {\displaystyle \lambda _{c}=2(n-2)} . Multiplicity of solution for n = 3 {\displaystyle n=3} was discovered by Israel Gelfand in 1963 and in later 1973 generalized for all n {\displaystyle n} by Daniel D. Joseph and Thomas S. Lundgren. The solution for n = 1 {\displaystyle n=1} that is valid in the range λ ∈ [ 0 , 0.8785 ] {\displaystyle \lambda \in [0,0.8785]} is given by
ψ = − 2 ln [ e − ψ m / 2 cosh ( λ 2 e − ψ m / 2 r ) ] {\displaystyle \psi =-2\ln \left[e^{-\psi _{m}/2}\cosh \left({\frac {\sqrt {\lambda }}{\sqrt {2}}}e^{-\psi _{m}/2}r\right)\right]}
where ψ m = ψ ( 0 ) {\displaystyle \psi _{m}=\psi (0)} is related to λ {\displaystyle \lambda } as
e ψ m / 2 = cosh ( λ 2 e − ψ m / 2 ) . {\displaystyle e^{\psi _{m}/2}=\cosh \left({\frac {\sqrt {\lambda }}{\sqrt {2}}}e^{-\psi _{m}/2}\right).}
The solution for n = 2 {\displaystyle n=2} that is valid in the range λ ∈ [ 0 , 2 ] {\displaystyle \lambda \in [0,2]} is given by
ψ = ln [ 64 e ψ m ( λ e ψ m r 2 + 8 ) 2 ] {\displaystyle \psi =\ln \left[{\frac {64e^{\psi _{m}}}{(\lambda e^{\psi _{m}}r^{2}+8)^{2}}}\right]}
where ψ m = ψ ( 0 ) {\displaystyle \psi _{m}=\psi (0)} is related to λ {\displaystyle \lambda } as
( λ e ψ m + 8 ) 2 − 64 e ψ m = 0. {\displaystyle (\lambda e^{\psi _{m}}+8)^{2}-64e^{\psi _{m}}=0.}
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