What is the asymptotic solution (for $s\gg 1$) of the following integral equation $$z(s)=1+\gamma\int\limits_{-\infty}^s ds_1\int\limits_{-\infty}^{s_1}ds_2 \cos{(s_1^2-s_2^2)}z(s_2)\;?$$ In fact I need to show that $$\lim_{s\to\infty} z(s)=2\exp{\left(\frac{\pi\gamma}{4}\right)}-1.$$ The integral equation is equivalent to the following third order differential equation $$sz^{\prime\prime\prime}-z^{\prime\prime}-s(\gamma-4s^2)z^{\prime}+ \gamma z=0,$$ and the initial conditions $z(-\infty)=1,\,z^\prime(-\infty)=0,\,z^{\prime\prime}(-\infty)=\gamma$.

The question arose in the context of remarkable connection between the Landau-Zener problem and the ball rolling along the Cornu spiral (that's how I do now what the $\lim_{s\to\infty} z(s)$ should be) established by Bloch and Rojo in http://ajp.aapt.org/resource/1/ajpias/v78/i10/p1014_s1