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

Abstract

In this work we find explicit periodic wave solutions for the classical 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 corresponds to superluminal waves, a third-one corresponding to a sub-luminal wave and the remaining one corresponding to a stationary complex-valued wave. 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 zerospeed sub-luminal wave are all stable. This latter case is related (in some sense) to the classical Kink solution.

Periodic traveling waves in the ϕ^4 model: Instability, stability and localized structures

“…As noted above, the relevant solutions and associated existence results have been previously presented in [27]. Hence, here we briefly review the corresponding findings by means of first integral (ODE) considerations.…”

“…(2) As stated before, the work of [27] has proved the orbital instability of snoidal waves to (1) with respect to co-periodic perturbations. Yet, the stability and instability of cnoidal and dnoidal solutions of the φ 4 model have not been investigated, to the best of our knowledge.…”

“…Recently the work of [27] has proved the existence of three different forms of periodic traveling wave solutions to (1). These solutions are given in Propositions 3.1 (dnoidal waves), 3.2 (cnoidal waves) and 3.3 (snoidal waves) of [27]. The same work included a proof of the orbital instability of snoidal waves with respect to co-periodic perturbations [27].…”

“…However, there have been relatively few studies of stability of periodic traveling wave solutions to (1). Recently the work of [27] has proved the existence of three different forms of periodic traveling wave solutions to (1). These solutions are given in Propositions 3.1 (dnoidal waves), 3.2 (cnoidal waves) and 3.3 (snoidal waves) of [27].…”

“…These solutions are given in Propositions 3.1 (dnoidal waves), 3.2 (cnoidal waves) and 3.3 (snoidal waves) of [27]. The same work included a proof of the orbital instability of snoidal waves with respect to co-periodic perturbations [27]. It is important to note here that the latter waveform exists only for subluminal speeds (i.e., for c 2 < 1), while the cnoidal and dnoidal waveforms are only present for superluminal speeds (i.e., for c 2 > 1).…”