In the application of the discrete ray method to the mixed differential-difference equation below , as $n \to \infty $:
$G'_n(x) = e^{-\frac{x^{2}}{2\,n\,(n+1)}}\,G_{n-1}(x)$, $G_{0}(x)=1$
and setting $G_{n}(x) = \epsilon^{-n}\,\phi(\epsilon\,x,\epsilon\,n)$, one obtains the following differential equation:
$\epsilon\,\frac{\partial \phi(u,v)}{\partial\,u} = e^{-\frac{u^{2}}{2\,v(v+\epsilon)}}\,\phi(u,v-\epsilon)$; $u = \epsilon\;x, v = \epsilon\;n, n \in \mathbb{N}^{+}, \epsilon\ \to 0$
At this point the following ansatz is inserted into the pde above:
$\phi(u,v)\sim \exp(\epsilon^{-1}\psi(u,v))K(u,v)$
Expanding both sides, in a Taylor series $\mathcal{O}(\epsilon^{2})$ around $\epsilon=0$, and comparing like terms of $\epsilon$, two equations will emerge. The first is discussed here.
Eikonal Equation
$F(u,v,\psi,p,q) = \ln(p)$ + $\frac{u^{2}}{2\:v^{2}}+q=0$ ; where p = $\frac{\partial\psi(u,v)}{\partial u}$, q = $\frac{\partial\psi(u,v)}{\partial v}$
The method of characteristics applied to the above -
$\{ \frac{d\,u}{d\,t}= \frac{1}{p}, \frac{d\,v}{d\,t}=1,\frac{d\,p}{d\,t}=-\frac{u^{2}}{v^{2}}, \frac{d\,q}{d\,t}=\frac{u^{2}}{v^{3}},\frac{d\,\psi}{d\,t}=q+1 \}$
with the following initial conditions
$u(s,0)=0,v(s,0)=s,p(s,0)=1,q(s,0)=0,\psi(s,0)=0$
Upon setting $\beta=erf^{-1}(a\:\frac{v-\alpha s}{s})$ and solving for $p$ first, which is an Emden-Fowler type equation, we obtain:
$u(s,t)=\sqrt{e}\:s\:\exp(-\beta^{2})+\sqrt{2}\:v\:\beta \\ v(s,t)=s+t \\ p(s,t)=\frac{\sqrt{e}\;s}{v}\exp(-\beta^{2}) \\ q(s,t)=\ln(\frac{v}{s})+\beta^{2}-\frac{u^{2}}{2\;v^{2}}-\frac{1}{2} \\ \psi(s,t)=v\;(\ln(\frac{v}{s}-\frac{1}{2}))+\frac{e\;s^{2}}{2\;v}\exp(-2\;\beta^{2}) $
With: $a=\sqrt{\frac{2}{\pi\;e}}$, $\alpha =\sqrt{\frac{\pi\;e}{2}}\;erf(\frac{1}{\sqrt{2}})+1$
(u,v) $\in \mathbb{R}^{+}\times\mathbb{R}^{+} $, $v\neq 0$
Any help or suggestions in finding t(u,v) e.g. a series expansion, a uniform expansion, etc. would be greatly appreciated.
Regards,
Andre