Photon soliton and fine structure due to nonlinear Compton scattering

Abstract

Induced Compton scattering of high-intensity radiation, $N(\ensuremath{\nu},t)=\frac{{c}^{2}I(\ensuremath{\nu},t)}{2h{\ensuremath{\nu}}^{3}}\ensuremath{\gg}1$, by a plasma is governed by a nonlinear integro-differential kinetic equation of the Boltzmann type. For a Maxwellian electron distribution, the transition probability kernel takes the form $K(\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}})\ensuremath{\propto}(\ensuremath{\nu}\ensuremath{-}{\ensuremath{\nu}}^{\ensuremath{'}})$ $\mathrm{exp}[\ensuremath{-}\frac{{(\ensuremath{\nu}\ensuremath{-}{\ensuremath{\nu}}^{\ensuremath{'}})}^{2}}{2{(\ensuremath{\Delta}{\overline{\ensuremath{\nu}}}_{D})}^{2}}]$, where $\ensuremath{\nu}$ and ${\ensuremath{\nu}}^{\ensuremath{'}}$ are the photon frequencies before and after scattering and $\ensuremath{\Delta}{\overline{\ensuremath{\nu}}}_{D}={[(\frac{2k{T}_{e}}{{m}_{e}{c}^{2}})(1\ensuremath{-}cos\ensuremath{\alpha})]}^{\frac{1}{2}}\ensuremath{\nu}$ is the Doppler width associated with the scattering angle $\ensuremath{\alpha}$. Only limiting cases with local approximative forms of the kernel have been treated up to now: for a broad photon spectrum ($\ensuremath{\delta}\ensuremath{\nu}\ensuremath{\gg}\ensuremath{\Delta}{\overline{\ensuremath{\nu}}}_{D}$) the kinetic equation was approximated by the Kompaneets differential equation corresponding to $K(\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}})\ensuremath{\propto}{\ensuremath{\delta}}^{\ensuremath{'}}(\ensuremath{\nu}\ensuremath{-}{\ensuremath{\nu}}^{\ensuremath{'}})$; for a narrow photon spectrum ($\ensuremath{\delta}\ensuremath{\nu}\ensuremath{\ll}\ensuremath{\Delta}{\overline{\ensuremath{\nu}}}_{D}$), the kernel was approximated by $K(\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}})\ensuremath{\propto}(\ensuremath{\nu}\ensuremath{-}{\ensuremath{\nu}}^{\ensuremath{'}})$. Qualitative red-shift evolutions of particular initial spectrum profiles were found. Here, the general integro-differential kinetic equation which governs the nonlinear and nonlocal interaction between photons and electrons is studied by a numerical treatment and asymptotic analytical approaches are exhibited. Starting from several initial conditions---narrow or broad spectrum profiles, in the presence or in the absence of a noise spectrum---general spectral evolutions for one-dimensional conservative photon systems are obtained. In the presence of a constant noise spectrum ${N}_{0}$ a universal asymptotic solution of the soliton type is obtained: Whatever the initial conditions, a narrow photon spectrum $N(\ensuremath{\nu},t)$ evolves towards a photon soliton moving downwards in the frequency axis with the form $\mathrm{ln}[\frac{N(\ensuremath{\nu},t)}{{N}_{0}}]=[\mathrm{ln}(\frac{{N}_{m}}{{N}_{0}})]\mathrm{exp}{\ensuremath{-}[\frac{{({\ensuremath{\nu}}_{S}\ensuremath{-}\ensuremath{\sigma}t\ensuremath{-}\ensuremath{\nu})}^{2}}{2{(\ensuremath{\Delta}{\overline{\ensuremath{\nu}}}_{D})}^{2}}}$, where ${N}_{m}$ is the maximum amplitude and $\ensuremath{\sigma}=\mathrm{const}({N}_{m}\ensuremath{-}{N}_{0}){[\mathrm{ln}(\frac{{N}_{m}}{{N}_{0}})]}^{\ensuremath{-}\frac{3}{2}}$ is the speed. A broad spectrum, however, is decomposed into a set of such solitons which are ordered by amplitude (proportional to their speeds) in their downward motion. In the absence of a constant noise spectrum, an initial narrow Gaussian profile moves downwards (on the $\ensuremath{\nu}$ axis) with decreasing speed, growing in amplitude and narrowing in width.