Abstract
We study the periodic cubic derivative nonlinear Schrödinger equation (DNLS) and the (focussing) quintic nonlinear Schrödinger equation (NLS). These are both \(L^2\) critical dispersive models, which exhibit threshold-type behavior, when posed on the line \({{\mathbb {R}}}\). We describe the (three-parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide criteria for stability, depending on the sign of a scalar quantity. The proof relies on an instability index count, which in turn critically depends on a detailed spectral analysis of a self-adjoint matrix Hill operator. We exhibit a region in parameter space, which produces spectrally stable waves. We also provide an explicit description of the stability of all bell-shaped traveling waves for the quintic NLS, which turns out to be a two-parameter subfamily of the one exhibited for DNLS. We give a complete description of their stability—as it turns out some are spectrally stable, while other are spectrally unstable, with respect to co-periodic perturbations.
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Notes
It is important to observe that in the gauge relation (1.5), we have that \(|q|=|u|\), so the phase function can be written either with q or u inside of it
In the work Liu et al. (2013), the authors exhibit explicit sech-type solutions for all powers \(\sigma \)
Even though, there are certainly interesting solutions for \(\omega <0\) as well
Indeed, as the resolvent operators \(({{\mathscr {J}}}{{\mathscr {L}}}-\lambda )^{-1}, \lambda \in {\mathbb {R}}, \lambda>>1\) are smoothing of order two, this guarantees that \(({{\mathscr {J}}}{{\mathscr {L}}}-\lambda )^{-1}:L^2[-T,T]\times L^2[-T,T] \rightarrow L^2[-T,T]\times L^2[-T,T] \) is compact, whence its spectrum consists entirely of eigenvalues converging toward zero. It follows that \(\sigma ({{\mathscr {J}}}{{\mathscr {L}}})\) consists of eigenvalues only.
which proved to be extremely non-trivial to obtain with the explicit waves under consideration
But note that this is not necessary. For example, one might have \(k_\mathrm{Ham}=2=2k_i^-, k_r=k_i^-=0\), which means that no instabilities are present, but there is a pair of purely imaginary eigenvalues, with a negative Krein signature.
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Communicated by Peter Miller.
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Milena Stanislavova is partially supported by NSF-DMS, # 1614734. Atanas Stefanov acknowledges partial support from NSF-DMS, # 1908626.
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Hakkaev, S., Stanislavova, M. & Stefanov, A. On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS. J Nonlinear Sci 31, 54 (2021). https://doi.org/10.1007/s00332-021-09712-6
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DOI: https://doi.org/10.1007/s00332-021-09712-6