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Register a new userCombustion waves in a model with chain branching reaction and their stability
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In this paper the travelling wave solutions in the adiabatic model with two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the properties of nonadiabatic one-step combustion waves in that there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. The difference between the nonadiabatic one-step and adiabatic two-step models is found in the behaviour of the combustion waves near the extinction condition. It is shown that the flame velocity drops down to zero and a standing combustion wave is formed as the extinction condition is reached. Prospects of further work are also discussed.
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We study asymptotic stability of solitary wave solutions in the one-dimensional Benney-Luke equation, a formally valid approximation for describing two-way water wave propagation. For this equation, as for the full water wave problem, the classic variational method for proving orbital stability of solitary waves fails dramatically due to the fact that the second variation of the energy-momentum functional is infinitely indefinite. We establish nonlinear stability in energy norm under the spectral stability hypothesis that the linearization admits no non-zero eigenvalues of non-negative real part. We also verify this hypothesis for waves of small energy.
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.
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By combining results of Mizumachi on the stability of solitons for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam (FPU) model are linearly stable and hence by the results of Friesecke and Pego that they are also nonlinearly, asymptotically stable.
We consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schrodinger models. We show that stability of bright solitons in presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane, by means of both numerical simulations and analytical estimates. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. We characterize soliton stability using two different criteria based on the evolution of the Hamiltonian of the soliton and its power. The first criterion allows us to estimate a time (or distance) over which the soliton preserves its form. The second criterion gives the life-time of the solitary wave packet in terms of its radiative power losses. We derive a simplified mean field approach which allows us to calculate the power loss analytically in the physically relevant case of weakly correlated noise, which in turn serves as a lower estimate of the life-time for correlated noise in general case.
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