Nonlinear Asymptotic Stability of the Semistrong Pulse Dynamics in a Regularized Gierer–Meinhardt Model (original) (raw)
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2005
We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the strongly localized activator pulses interact through the weakly localized inhibitor, the interaction is not tail-tail as in the weak interaction limit, and the pulses change amplitude and even stability as their separation distance evolves on algebraically slow time scales. The RG approach employed here validates the interaction laws of quasi-steady pulse patterns obtained formally in the literature, and establishes that the pulse dynamics reduce to a closed system of ordinary differential equations for the activator pulse locations. Moreover, we fully justify the reduction to the nonlocal eigenvalue problem (NLEP) showing the large difference between the quasi-steady NLEP operator and the operator arising from linearization about the pulse is controlled by the resolvent. 2...
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The peculiar intergrability of the Davey-Stewartson equation allows us to find analytically solutions describing the simultaneous formation and interaction of onedimensional and two-dimensional localized coherent structures. The predicted phenomenology allows us to address the issue of interaction of solitons of different dimensionality that may serve as a starting point for the understanding of hybridodimensional collisions recently observed in nonlinear optical media. Nonlinear wave propagation occurs in many different physical systems (e.g. water waves and optics) and leads to a myriad of different interesting and useful phenomena that are in striking contrast to linear propagation effects [1]. Among these, the emergence of localized non-dispersive coherent pulses, solitons, that do not suffer deformation and that undergo elastic-like collisions, have attracted much attention [2]. Perhaps their most peculiar physical feature is associated with their interaction dynamics. Whereas classical soliton experimental studies have been confined to lower dimensional (one-dimensional, 1+1D) systems, where diffraction occurs in one dimension only, in the last decade experiments in nonlinear optics have allowed the stable observation of both 1+1D solitons in a bulk environment, known as stripe or wall
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We consider the semistrong limit of pulse interaction in a thermally driven, parametrically forced, nonlinear Schrödinger (TDNLS) system modeling pulse interaction in an optical cavity. The TDNLS couples a parabolic equation to a hyperbolic system, and in the semistrong scaling we construct pulse solutions which experience both short-range, tail-tail interactions and long-range thermal coupling. We extend the renormalization group (RG) methods used to derive semistrong interaction laws in reaction-diffusion systems to the hyperbolic-parabolic setting of the TDNLS system. A key step is to capture the singularly perturbed structure of the semigroup through the control of the commutator of the resolvent and a re-scaling operator. The RG approach reduces the pulse dynamics to a closed system of ordinary differential equations for the pulse locations.
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Physical Review Letters, 1997
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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
In a scalar reaction–diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state’s critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction–diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have non-zero Maslov index, and hence be unstable. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.
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Nonlinear Analysis: Real World Applications, 2012
The pulse solution of the spatially discrete excitable FitzHugh-Nagumo (FHN) system is approximately constructed using matched asymptotic expansions in the limit of large time scale separation (as measured by a small dimensionless parameter ϵ). The pulse profile typically consists of slowly varying regions of the excitatory variable separated by sharp wave fronts. In the FHN system, the velocity of a pulse is decided by the interaction between its leading and trailing fronts, but the leading order approximation gives only a fair result when compared with direct numerical solutions. A higher order approximation to the wave fronts comprising the FHN pulse is found. Our approximation provides an ϵ-dependent pulse velocity that approximates much better the velocity obtained from numerical solutions. As a result, the reconstruction of the FHN pulse using the improved wave fronts is much closer to the numerically obtained pulse.