Approximations and error bounds for traveling and standing wave solutions of the one-dimensional $$\hbox {M}^5$$M5-model for mesenchymal motion

Cruz-García, Salvador; García‐Reimbert, Catherine

doi:10.1007/s40590-019-00233-7

Cited by 1 publication

(3 citation statements)

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“…In this section, we show that unlike the standing case, in which the dimension of the eigenspace associated to the zero eigenvalue is infinity (see [3]), in the case of traveling waves, zero is an eigenvalue whose eigenspace has dimension one for all wave speed c ∈ (0, s).…”

Section: The Eigenvalue λ =mentioning

confidence: 91%

“…By Theorem 1.1, 2sq + l − (s + c) > 0 and 2sq + r − (s + c) < 0, hence, since 0 < q + r , q + l < 1, we conclude that ∂q + r /∂q + l < 0. Recently, M 5 -traveling wave solutions for system (1) have been investigated by Cruz-García et al [3]. Using Lagrange's interpolation method, the authors derived an exactly solvable approximate equation which yields analytical approximations for the standing and traveling waves.…”

Section: -Traveling Wavesmentioning

confidence: 99%

“…Using Lagrange's interpolation method, the authors derived an exactly solvable approximate equation which yields analytical approximations for the standing and traveling waves. In [3], upper and lower error bounds are computed for the approximate standing wave solutions. Furthermore, error bounds for the approximate traveling wave solutions were provided under necessary conditions on the right and left endpoints q + r and q + l .…”

Section: -Traveling Wavesmentioning

confidence: 99%

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Spectrum of the M⁵-traveling waves

Cruz-García

2020

Math. Model. Nat. Phenom.

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In this paper we study the essential spectrum of the operator obtained by linearizing at traveling waves that occur in the one-dimensional version of the M$^5$-model for mesenchymal cell movement inside a directed tissue made up of highly aligned fibers. We show that traveling waves are spectrally unstable in $L^2(\mathbb{R};\mathbb{C}^3)$ as the essential spectrum includes the imaginary axis. Tools in the proof include exponential dichotomies and Fredholm properties. We prove that a weighted space $L^2_{w}(\mathbb{R};\mathbb{C}^3)$ with the same function for the tree variables of the linearized operator is no suitable to shift the essential spectrum to the left of the imaginary axis. We find a pair of appropriate weight functions whereby on the weighted space $L^2_{w_{\alpha}}(\mathbb{R};\mathbb{C}^2)\times L^2_{w_{\varepsilon}}(\mathbb{R};\mathbb{C})$ the essential spectrum lies on $\left\{\mathfrak{R}e\hspace{.03cm}\lambda<0\right\}$, outside the imaginary axis.

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Section: The Eigenvalue λ =mentioning

confidence: 91%

Section: -Traveling Wavesmentioning

confidence: 99%

Section: -Traveling Wavesmentioning

confidence: 99%

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