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

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  • José Manuel Palacios

    (Université de Tours, Université d’Orleans)

Abstract

In this work we find explicit periodic wave solutions for the classical $$\phi ^4$$ ϕ 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 $$\phi ^4$$ ϕ 4 -model.

Suggested Citation

  • José Manuel Palacios, 2022. "Orbital stability and instability of periodic wave solutions for the $$\phi ^4$$ ϕ 4 -model," Partial Differential Equations and Applications, Springer, vol. 3(4), pages 1-31, August.
    • Handle: RePEc:spr:pardea:v:3:y:2022:i:4:d:10.1007_s42985-022-00185-0
    DOI: 10.1007/s42985-022-00185-0
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    Keywords

    35B35; 37K45; 35C08; 35L71; 35B10; 35C07; 35Q51;
    All these keywords.

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