Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hlder regularity (dimension three) of the shear flows. Remarks are made on the quadratic and linear laws of front speed expectation in the small and large root mean square regimes.

We study the stability and oscillation of traveling fronts in a three-component, advection-reaction biodegradation model. The three components are pollutant, nutrient, and bacteria concentrations. Under an explicit condition on the biomass growth and decay coefficients, we derive reduced, two-component, semilinear hyperbolic models through a relaxation procedure, during which biomass is slaved to pollutant and nutrient concentration variables. The reduced two-component models resemble the Broadwell model of the discrete velocity gas. The traveling fronts of the reduced system are explicit and are expressed in terms of hyperbolic tangent function in the nutrient-deficient regime. We perform energy estimates to prove the asymptotic stability of these fronts under explicit conditions on the coefficients in the system. In the small damping limit, we carry out Wentzel--Kramers--Brillouin (WKB) analysis on front

We consider a model semilinear reaction-diffusion system with cubic nonlinear reaction terms and small spatially decaying initial data on R 1. The model system is motivated by the thermal-diffusive system in combustion, and it reduces to a scalar reaction-diffusion equation with Zeldovich nonlinearity when the Lewis number is one and proper initial data are prescribed. For scalar equations of similar type it is well known that while a nonlinearity of degree greater than three (supercritical case) has no effect for large times a cubic nonlinearity qualitatively changes the long time behaviour. The latter case has been treated in the literature by a rescaling method under the additional assumption of smallness of the nonlinearity. Although for our system the cubic nonlinearity is also critical we establish large time behaviour when the nonlinearity is not necessarily small which essentially differs from the supercritical case. This

The sine-Gordon (SG) equation and perturbed nonlinear Schrdinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called <i>light bullets</i>) in nonlinear dispersive optical media. We begin with the (2+1) SG equation obtained as an asymptotic reduction in the two level dissipationless MaxwellBloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusingdefocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet

G-equations are well-known front propagation models in combustion and are HamiltonJacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of one-dimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.