We show that, for the 1d cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for example, the dynamics of single solitons subject to external forces as in [7,8].

Background Cooking oil fumes (COFs) are composed of particulate matter, polycyclic aromatic hydrocarbons, volatile organic compounds, aldehydes, and ketones, and are currently a global health concern. Some agents in COFs are mutagenic and carcinogenic. However, only a few reports have addressed the hazardous effects of COF exposure on the female reproductive system. In this study, we explored the effects of subchronic exposure to COFs on female gonads in vivo and the possible involvement of the G-protein-coupled receptor 30 signaling pathway. Methods COFs were generated by heating commercially available canola oil in an iron pot. Adult female Wistar rats at 2 months of age were exposed to COFs at 32 mg/m 3 for 0, 0.5, 1, 2, or 4 h/day for 56 days. The estrous cycle in rats was studied twice at 7:00 a.m. and 7:00 p.m. on the 43rd treatment day until the current estrous cycle was complete. The rat body weight was measured before the experiment and at day 56 post-exposure. At the end of the experiment, rat blood was collected for gonadal hormone assay, and ovaries were collected for histology and mRNA isolation. The mRNA levels of GPR30, EGFR, STAT3, and ERK were determined by quantitative RT-PCR. Results At a concentration of 32.21 ± 5.11 mg/m3, COF exposure extended the estrous cycle in rats, and ovary coefficient decreased. COFs showed various effects on the sex hormone levels and follicles, depending on its exposure level. Exposure to COFs led to the changes in mRNA levels of the G-protein-coupled receptor 30 (GPR30), epidermal growth factor receptor (EGFR), signal transducer and activator of transcription 3 (STAT3), and extracellular signal-regulated kinase (ERK). Conclusion This study indicated that cooking oil fume exposure disrupted the estrous cycle, sex hormone patterns, and follicle development in female rats in a dose-dependent manner. These adverse effects of cooking oil fumes on female reproductive health were correlated with the G-protein-coupled receptor 30-mediated signaling pathway.

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We study the dynamics of soliton solutions to the perturbed mKdV equationThis type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain O( t 1/2 ) close to a soliton on an O( −1 ) time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale O( −1/2 ). Over this time scale, the perturbation can impart an O(1) influence on the soliton position.