Even solutions of the Toda system with prescribed asymptotic behavior (Q652078)

The author studies the Toda system \[ \begin{aligned} q''_{1} &= -\exp (q_{1}-q_{2}),\\ q''_{i} &= \exp (q_{i-1}-q_{i})-\exp (q_{i}-q_{i+1}), \;i=2,\dots ,n-1,\\ q''_{n} &= \exp (q_{n-1}-q_{n})\end{aligned} \] and proves the existence, uniqueness and the asymptotic behavior for \(t\rightarrow \infty \) of its even solutions. The behavior is asymptotically linear in agreement with the earlier \textit{J. Moser} results [``Finitely many mass points on the line under the influence of an exponential potential - an integrable system'', Dyn. Syst., Theor. Appl., Battelle Seattle 1974 Renc., Lect. Notes Phys. 38, 467--497 (1975; Zbl 0323.70012)]. Such solutions are related to the ones of the Allen-Cahn partial differential equation.