Abstract
In this article, we provide an alternative way to construct small amplitude traveling waves for general Whitham type equations, in both periodic and whole line contexts. More specifically, Fourier analysis techniques allow us to reformulate the problem to the study of waves that are small and regular perturbations of well-understood ODE’s. In addition, rigorous spectral stability of these waves is established.
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Notes
In principle, one could compute explicitly the next terms, up to any degree of accuracy.
In this case, the only symmetry is the translation in the x variable.
Note that the precise form for \(\nu \) is primarily for convenience. We could let \(\nu = m(0) + \kappa \epsilon ^2\) for any \(\kappa >0\) and all that would really change are some nonessential coefficients.
And in fact, for the Whitham example, where \(m(k)=\sqrt{\frac{\tanh (k)}{k}}\) it is not smoothing.
The most common example one should think is \(X=H^s\), a Sobolev space, with dual \(X^*=H^{-s}\).
A much more precise result is contained in Theorem 2.3, [14], but we state this corollary, as it is enough for our purposes.
And hence \(k_0^{\le 0}({{\mathscr {L}}}_\epsilon )=1\), since the left hand side of (24) is non-negative.
And in fact it has a single negative eigenvalue.
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Stefanov is partially supported by NSF-DMS under Grant # 1614734. Wright is partially supported by NSF-DMS under Grant # 1511488.
Appendix: Assorted Proofs
Appendix: Assorted Proofs
Proof
(Lemma 3) Take \(K>k_*/\epsilon \). Then we clearly have
Since \(K>k_* /\epsilon \), we know that \(m(\epsilon K) \le m_1\) by (5). Thus we have
Here we have used that fact that \(m''(0) < 0\), which is implied by (4). This in turn implies:
The triangle inequality gives:
Now suppose that \(|K| \le k_*/\epsilon \). We have
The fact that m is even and \(C^{3,1}\) implies, by way of Taylor’s theorem, that there exists \(C>0\) such that
when \(|k| \le k_*\). This implies that
when \(|K| \le k_*/\epsilon \).
The fundamental theorem of calculus implies that
Here we used the fact that m(k) is even. Then we use (4) to see that
so long as \(|k| \le k_*\). Thus, for \(|K| \le k_*/\epsilon \) we have
Since \(m''(0)\) and \(m_2\) are both negative this implies:
when \(|K| \le k_*/\epsilon \).
Thus we can control the left hand side of (33) using (34) and (35) as:
when \(K\le k_*/\epsilon \). Since \(K^4/(1+K^2)^2 \le 1\) we have
\(\square \)
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Stefanov, A., Wright, J.D. Small Amplitude Traveling Waves in the Full-Dispersion Whitham Equation. J Dyn Diff Equat 32, 85–99 (2020). https://doi.org/10.1007/s10884-018-9713-8
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DOI: https://doi.org/10.1007/s10884-018-9713-8