Principal investigators
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Prof. Dr. Guido Schneider |
(7/2015 - 6/2019) |
Project summary
Amplitude, modulation, or envelope equations, such as the Ginzburg-Landau, the KdV, or the NLS equation play a big role for the qualitative understanding of pattern forming systems or of dispersive wave systems in spatially unbounded domains. The last decades saw a big number of approximation results for these multiple scaling problems. It has been shown that the amplitude equations make correct predictions about the dynamics of the original systems.
For very few special situations it has been known that amplitude equations can fail to make correct predictions. Therefore, it has been the topic of the research project A8 to investigate the failure of amplitude equations systematically. While working on the project we had to revise our starting hypothesis that failure of amplitude equations is a rather seldom phenomenon. It turns out that a big number of relevant systems show this phenomenon.
Moreover, it turned out that the validity question of amplitude equations in such situations is really a subtle question. Therefore, the question of failure and validity of amplitude equations cannot be separated. The amplitude equation can fail for Sobolev initial conditions, but can make correct predictions for analytic conditions. It can fail for periodic boundary conditions, but can make correct predictions on the whole real line.
Beside new examples, new rigorous proofs for the failure of amplitude equations, we were able to confirm some conjectures about the failure and validity of amplitude equations.
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The first system which came into our focus was the Klein-Gordon-Zakhavov (KGZ) system \[c^{-2}\partial_{T}^2 z = \Delta z - c^2 z \pm nz\,,\qquad \alpha^{-2}\partial_{T}^2 n = \Delta n + \Delta \vert z\vert^2 \] from project (B1). Beside a number of positive results, see project (B1), in the subsequent reference it has been shown that for periodic b.c. the Klein Gordon (KG) equation fails to make correct predictions about the KGZ system in the limit: \(c\) fixed, \(\alpha \to \infty\). The KG equation has to be modified to make correct predictions on the torus. See [DSS16].
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The limit KGZ to KG and the limit Zakharov to NLS, fall into the class of fast-slow systems of the form \[\begin{equation}\label{A8:eq1} \partial_t U = \Lambda_U U + B_U(U,V)\,, \qquad \varepsilon \partial_t V = \Lambda_V V + B_V(U,U)\,,\end{equation}\] resp. \[\begin{equation}\label{A8:eq2} \partial_t U = \Lambda_U U + B_U(U,V)\,, \qquad \partial_t V = \varepsilon^{-1} \Lambda_V V + \varepsilon^{-1} B_V(U,U)\,, \end{equation} \] where \(0 < \varepsilon \ll 1 \) is a small perturbation parameter. At a first view the validity of the limit system \[\begin{equation}\label{A8:eq3} \partial_t \psi_U = \Lambda_U \psi_U - B_U(\psi_U, \psi_V)\,, \qquad \textrm{with} \qquad \psi_V= \Lambda_V^{-1} B_V(\psi_U,\psi_U) \end{equation} \] is a hard problem due the \( \varepsilon^{-1} \) on the right hand side of \(\eqref{A8:eq2}\) which can lead to \(\mathcal{O}(\varepsilon^{-1})\) growth rates on the natural \(\mathcal{O}(1)\) time scale of the limit system \(\eqref{A8:eq3}\). At a second view it turns out that by a simple change of variables \( W = V + M(U,U) \), the dangerous term \( \varepsilon^{-1} B_V(U,U) \) can be removed, and so it can be expected that for all systems, which can be written in the form \( \eqref{A8:eq1} \), an approximation property holds. However, this is not true. At a third view the approximation problem turns out to be rather subtle. It turns out that for the KGZ-KG limit the sign of the nonlinear terms plays no role, whereas for the Zakkarov-NLS limit, in case of a 'wrong' sign in the nonlinear terms of the Zakhavov system, the NLS equation makes wrong predictions. See [BSSZ19].
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The second system which came into our focus was the complex Ginzburg-Landau equation \[ \begin{equation}\label{A8:eq4} \partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi |\Psi|^2, \end{equation} \] and modulations of periodic waves \[ \begin{equation}\label{equi1} \Psi_{per}(X,T) = \Psi_0(\zeta) e^{i(\zeta X + \Omega_0(\zeta)T)}\,, \end{equation} \] near the Eckhaus boundary. Depending on the parameters \( \alpha , \beta \), and the wave number \( \zeta \) for their description a number of different amplitude equations can be derived. In [HdRS18] we concentrated on the validity of the KdV equation. Beside some positive results it turned out that there is a parameter regime, where the KdV equation can be derived, but fails to make correct predictions, even in case of analytic initial conditions.
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Unstable quadratic resonances in dispersive wave systems can be used to prove that NLS and FWI approximations fail in case of \(2\pi/k_0\)-periodic b.c. assuming that the three resonant wave numbers \(k_1\), \(k_2\), and \(k_3\) are integer multiples of \(k_0\). For a Boussinesq equation it could be shown that the FWI system in case of unstable resonances fails to make correct predictions without imposing periodic b.c.. On the other hand the NLS approximation is valid in case of different group velocities at the resonant wave numbers. Hence the localization of the solutions in terms of \(\varepsilon\) plays a fundamental role whether an approximation is valid or not. This will be the subject of a forthcoming paper.
