Single-spike solutions to the 1D shadow Gierer-Meinhardt problem

A fundamental example of reaction-diffusion system exhibiting Turing type pattern formation is the Gierer-Meinhardt system, which reduces to the shadow Gierer-Meinhardt problem in a suitable singular limit. Thanks to its applicability in a large range of biological applications, this singularly perturbed problem has been widely studied in the last few decades via rigorous, asymptotic, and numerical methods. However, standard matched asymptotics methods do not apply (Ni, 1998 [1]; Wei, 1998 [2]), and therefore analytical expressions for single spike solutions are generally lacking. By introducing an ansatz based on generalized hyperbolic functions, we determine exact radially symmetric solutions to the one-dimensional shadow Gierer- Meinhardt problem for any 1 < p n pi*internalconversion process in 4-thiouracil (4TU), triggered by an optical pump. The element-sensitive spectroscopic signatures are recorded by a resonant X-ray probe tuned to thesulfur, oxygen, or nitrogen K-edge. We employ high-level electronic structure methodsoptimized for core-excited electronic structure calculation combined with quantumnuclear wavepacket dynamics computed on two relevant nuclear modes, fully accountingfor their quantum nature of nuclear motions. We critically discuss the capabilities andlimitations of the resonant technique. For sulfur and nitrogen, we document a pre-edgespectral window free from ground-state background and rich with pi pi*andn pi*absorptionfeatures. The lowest sulfur K-edge shows strong absorption for both pi pi*andn pi*. In thelowest nitrogen K-edge window, we resolve a state-specificfingerprint of the pi pi*and anapproximate timing of the conical intersection via its depletion. A spectral signature of then pi*transition, not accessible by UV-vis spectroscopy, is identified. The oxygen K-edge isnot sensitive to molecular deformations and gives steady transient absorption features without spectral dynamics. The pi pi*/n pi*coherence information is masked by more intense contributions from populations. Altogether, element-specific time-resolvedresonant X-ray spectroscopy provides a detailed picture of the electronic excited-state dynamics and therefore a sensitive window into the photophysics of thiobases.