Abstract
It is a matter of experience that nonlinear waves in a dispersive medium, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics—the amplitude, the phase, the wave number, etc.—slowly vary in large space and time scales. In the 1960s, Whitham developed an asymptotic (WKB) method to study the effects of small “modulations” on nonlinear dispersive waves. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham’s formal theory. We discuss some recent advances in the mathematical understanding of the dynamics, in particular, the instability, of slowly modulated waves for equations of KdV type.
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Notes
- 1.
Note that \(\mathcal{O}(1)\) terms cancel out thanks to the definition of \((a_{0},E_{0},c_{0})\), and hence the leading order term is \(\mathcal{O}(\varepsilon )\).
- 2.
The requirement that the perturbation be integrable is not the only choice possible. Other natural candidates of classes of perturbations include periodic perturbations with fundamental period nT for some \(n = 1, 2, 3,\ldots\). While such periodic classes of perturbations are natural from a variational viewpoint, they may impose artificial constraints on the physical problem. As we will see below, the class of localized, i.e. integrable on the line, perturbations includes information about all quasi-periodic perturbations.
- 3.
Here we ignore issues of well-posedness of Eq. (4.1) to such initial data.
- 4.
In general, spectral stability does not imply linear stability, as is familiar from the ODE theory. Nevertheless, spectral instability often does imply linear (and nonlinear) instability.
- 5.
Here, we take the convention that the left hand side is zero when k = 0.
- 6.
- 7.
- 8.
The condition α > 1∕2 is an artifact of the corresponding existence theory. See [45] for details.
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Acknowledgements
JCB is supported by the National Science Foundation grant DMS-1211364. VMH is supported by the National Science Foundation grant CAREER DMS-1352597 and an Alfred P. Sloan Foundation Fellowship. MAJ is supported by the National Science Foundation grant DMS-1211183.
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Bronski, J.C., Hur, V.M., Johnson, M.A. (2016). Modulational Instability in Equations of KdV Type. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_4
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