Multi-dimensional stability of waves travelling through rectangular lattices in rational directions

Hoffman, Aaron; Hupkes, Hermen Jan; Vleck, Erik S. Van

doi:10.1090/s0002-9947-2015-06392-2

Cited by 18 publications

(56 citation statements)

References 40 publications

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“…However, besides the partial results in [34], the first successful use of the comparison principle in two dimensions was actually a byproduct of the analysis [9] on which this paper is based. It is therefore not surprising that part of our work here can be seen as a companion paper to [24], in the sense that we give an alternative proof of the nonlinear stability of travelling waves to the unobstructed planar LDE (1.1) with Λ = Z 2 .…”

Section: Stability Of Wavesmentioning

confidence: 87%

“…First of all, it ensures that all diffusive effects are made visible. It is here that the link to our prior work [24] based on spectral techniques becomes apparent, since we recover a condition on the essential spectrum that ensures that the diffusion coefficient is non-zero. Furthermore, the decay rate of all remaining terms is pushed to at least t −3/2 , corresponding to third order differences in θ.…”

Section: Discrete Settingmentioning

confidence: 99%

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Entire solutions for bistable lattice differential equations with obstacles

2017

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We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by "holes") are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.(1.9) with z decreasing from z 0 to 0 and with Z increasing from 0 to Z ∞ . To better understand the crucial relationship between the asymptotic phaseshift Z ∞ and the additive perturbation z, we note that the super-solution residual J = u t − u xx − g(u) is given by(1.10)Close to the interface, the term g(Φ) − g(Φ + z) ∼ −g ′ (Φ)z is negative and must be dominated by the positive termŻΦ ′ . This requires thatŻ dominate z andż. On the other hand, close to the Post-interaction regime -convergence to waveContinuous Setting In this final step of the program, the large transients generated in the interaction regime must be controlled in a frame that moves along with the unobstructed wave. As discussed above, this analysis leads naturally to a large basin nonlinear stability result for planar travelling wave solutions to the unobstructed PDE (1.2) with Ω = R 2 . For presentation purposes, we will focus our discussion here on this unobstructed special case. Indeed, the inclusion of the obstacle merely adds technical complications that do not contribute to the understanding of the differences between the continuous and discrete frameworks. 7The main task is to construct a super-solution for (1.2) with Ω = R 2 of the form( 1.15) which adds transverse effects to the Ansatz (1.9) discussed earlier. As before, the function z decreases from z 0 to 0 while Z increases from 0 to Z ∞ . Both z and Z should be thought of as small terms. By contrast, the new function θ should be allowed to be arbitrarily large at t = 0, provided that it is localized in the sense θ(·, 0) ∈ L 2 and that it decays to zero as t → ∞ uniformly in y. This function controls deformations of the wave interface in the transverse direction. We note that any localized initial perturbation from the wave can be dominated by the initial condition in (1.15) by choosing z 0 positive and as small as we wish, at the cost of a larger value for θ(·, 0) L 2 . Assuming for the moment that Z ∞ scales with z 0 , this freedom implies that we can dominate the transients caused by such a perturbation by a family of super-solutions that have arbitrarily small asymptotic phase offsets Z ∞ . A similar argument with sub-solutions then establishes the convergence to the planar wave without any asymptotic phase shift.The difference in behaviour between one and two spatial dimensions is hence caused by the extra transverse direction, al...

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Section: Stability Of Wavesmentioning

confidence: 87%

Section: Discrete Settingmentioning

confidence: 99%

Entire solutions for bistable lattice differential equations with obstacles

2017

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“…In our case, we use point-wise Green's functions estimates to obtain sharp decay estimates of the linear part of our linearized operator. These types of estimates are reminiscent of the ones obtained by Hoffman and coworkers [11] in the study of multi-dimensional stability of planar traveling of lattice differential equations, which are discrete version of equation (1.1). In the nonlocal setting, using super-and sub-solution technique, Chen [5] has been able to prove the uniform multidimensional stability of the traveling wave ϕ of equation (1.1).…”

Section: Hypothesis (H2)mentioning

confidence: 66%

Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

Faye¹

2015

DCDS-A

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“…Our final condition requires λ z to depend quadratically on z, which means that this group velocity has to vanish. We emphasize that the inequality [∂ 2 z λ z ] z=0 > 0 was required in [24] to obtain the nonlinear stability of the planar wave (c * , Φ * ).…”

Section: )mentioning

confidence: 99%

Travelling corners for spatially discrete reaction-diffusion systems

Hupkes¹,

Morelli²

2020

Communications on Pure &Amp; Applied Analysis

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We consider reaction-diffusion equations on the planar square lattice that admit spectrally stable planar travelling wave solutions. We show that these solutions can be continued into a branch of travelling corners. As an example, we consider the monochromatic and bichromatic Nagumo lattice differential equation and show that both systems exhibit interior and exterior corners.Our result is valid in the setting where the group velocity is zero. In this case, the equations for the corner can be written as a difference equation posed on an appropriate Hilbert space. Using a non-standard global center manifold reduction, we recover a two-component difference equation that describes the behaviour of solutions that bifurcate off the planar travelling wave. The main technical complication is the lack of regularity caused by the spatial discreteness, which prevents the symmetry group from being factored out in a standard fashion.

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Multi-dimensional stability of waves travelling through rectangular lattices in rational directions

Cited by 18 publications

References 40 publications

Entire solutions for bistable lattice differential equations with obstacles

Entire solutions for bistable lattice differential equations with obstacles

Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation

Travelling corners for spatially discrete reaction-diffusion systems

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