Comparison study of dynamics in one-sided and two-sided solid-combustion models
Comparing two-sided and one-sided solid-combustion models, this paper concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. Features include linear instability of basic solutions and weakly nonlinear evolution of small perturbations, as well as the complex dynamics of period doubling, quadrupling, and eventual chaotic oscillations. Both asymptotic and numerical methods are used for different solution regimes. First, multiscale weakly nonlinear analysis takes into account the cumulative effect of small nonlinearities to obtain a correct description of the evolution over long times. For a range of parameters, the asymptotic method with some dominant modes captures the formation of coherent structures. In other cases, numerical solutions reveal period-folding behaviors. In general, the oneand two-sided models agree qualitatively for all solution regimes, which is consistent with prior numerical comparisons and extends our results from [L. K. Gross and J. Yu, SIAM J. Appl. Math., 65 (2005), pp. 1708–1725].