Abstract
In this chapter we consider the stability of solitary waves.
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Notes
- 1.
It is not enough to require that the perturbed solitary wave remains close to \(\psi ^\mathrm{solitary}_{\omega }\) for \(0\le z \le Z\), where \(0<Z<\infty \), since by Corollary 5.4, all the perturbed solitary waves that exist in \([0, Z]\) satisfy this requirement.
- 2.
When the NLS is not invariant under translations in \(\mathbf{x}\), orbital stability refers to stability up to phase shifts. This is the case, e.g., with the NLS on a bounded domain (Sect. 16.5.2), or the NLS with a potential that depends on \(\mathbf{x}\).
- 3.
- 4.
When \(\Omega _\mathrm{im} = 0\), this expression reduces to (9.7a).
- 5.
I.e., all solutions of the subcritical NLS exist globally, whereas the critical and supercritical NLS admit singular solutions.
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Fibich, G. (2015). Stability of Solitary Waves. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_9
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DOI: https://doi.org/10.1007/978-3-319-12748-4_9
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