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Orbital stability and instability of periodic wave solutions for the \(\phi ^4\)-model

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Abstract

In this work we find explicit periodic wave solutions for the classical \(\phi ^4\)-model, and study their corresponding orbital stability/instability in the energy space. In particular, for this model we find at least four different branches of spatially-periodic wave solutions, which can be written in terms of Jacobi elliptic functions. Two of these branches correspond to superluminal waves (speeds larger than the speed of light), the third-one corresponds to sub-luminal waves and the remaining one corresponds to stationary complex-valued waves. In this work we prove the orbital instability of real-valued sub-luminal traveling waves. Furthermore, we prove that under some additional hypothesis, complex-valued stationary waves as well as the real-valued zero-speed sub-luminal wave are all stable. This latter case is related (in some sense) to the classical Kink solution of the \(\phi ^4\)-model.

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The present article under submission does not make use of any data from any repository. The results and theorems proved in this article do not require any supporting data, but the mathematical theory developed therein.

Notes

  1. Notice that, as far as local well-posedness in \(H^1\times L^2\) is concerned, there is no fundamental difference between these cases and our periodic setting.

  2. Notice that, for the case of Theorem 1.3, all three of these symmetries preserves the oddness of the solution.

  3. We call anti-snoidal to \(-\phi _{\mathrm {sn}}\), where \(\phi _{\mathrm {sn}}\) is the snoidal wave solution. Notice that \(-\phi _{\mathrm {sn}}\) is also a solution.

  4. Notice that if \(\phi _c\) is a solution, then so is \(-\phi _c\), and hence, there is no loss of generality in this assumption.

  5. We shall rigorously prove this inequality in the proof of Proposition 3.1 below.

  6. We shall rigorously prove this inequality in the proof of Proposition 3.2 below.

  7. See notation (7.2) below.

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Acknowledgements

The author is grateful to professor Fabio Natali for pointing out a flaw in the proof of Theorem 7.1 in a previous version of this work.

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  1. Institut Denis Poisson, CNRS, Université de Tours, Université d’Orleans, Parc Grandmont, 37200, Tours, France

    José Manuel Palacios

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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

Appendix

Appendix

1.1 Proof of Proposition 3.3

In fact, first of all notice that for c fixed, \(T_{\mathrm {sb}}\) as in (3.20) regarded as a function of \(\beta _1\) satisfies \(T_{\mathrm {sb}}((1,\sqrt{2}))=(2\pi \sqrt{\omega _{\mathrm {sb}}},+\infty )\). Moreover, notice that due to condition (3.21) we have the bound \(2\pi \sqrt{\omega _{\mathrm {sb}}}<L\). Then, as an application of the Implicit Function Theorem, in order to conclude the uniqueness of \(\beta _1(c)\) it is enough to show that \(\tfrac{d}{d\beta _1}T_{\mathrm {sb}}<0\). Indeed, notice that by direct differentiation of the equation defining \(\kappa \) in (3.22) with respect to \(\beta _1\) we have

$$\begin{aligned} \dfrac{d\kappa }{d\beta _1}=-\dfrac{2}{\beta _1^2\sqrt{2-\beta _1^2}}<0 \quad \hbox {for all } \,\beta _1\in (1,\sqrt{2}). \end{aligned}$$

Therefore, recalling that \(K(\kappa )\) is an strictly increasing function, and due to the sign of \(\kappa '\) given by the latter inequality, by differentiating relation (3.20) with respect to \(\beta _1\) we conclude

$$\begin{aligned} \dfrac{d}{d\beta _1}T_{\mathrm {sb}}=-\dfrac{4\sqrt{2\omega _{sb}}}{\beta _1^2}K+\dfrac{4\sqrt{2\omega _\mathrm {sb}}}{\beta _1}K'(\kappa )\dfrac{d\kappa }{d\beta _1}<0, \end{aligned}$$
(7.10)

what finish the proof. \(\square \)

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Palacios, J.M. Orbital stability and instability of periodic wave solutions for the \(\phi ^4\)-model. Partial Differ. Equ. Appl. 3, 56 (2022). https://doi.org/10.1007/s42985-022-00185-0

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