Project A10 • Standing and moving pulses in periodic media (7/2019 - 6/2023)
Principal investigators
Prof. Dr. Wolfgang Reichel
(7/2019 - 6/2023)
Prof. Dr. Guido Schneider
(7/2019 - 6/2023)
Project summary
In this project we investigate the dynamics of dispersive systems in infinitely extended, time- or spatially periodic media with an emphasis on the existence and dynamics of moving and standing exact and approximate modulating pulse solutions. We are interested in multiple scaling problems where in an asymptotic limit one can analyse the more complicated original system. Important examples of such dispersive systems are water waves over a spatially periodic bottom, electromagnetic waves in photonic crystal fibers or nanotubes, and mechanical waves in polyatomic Fermi-Pasta-Ulam-Tsingou (FPUT) chains. Moving and standing modulating pulse solutions play a fundamental role in information technology. We have so far achieved results about the (non-) validity of dispersion management models, about the existence of both small and large amplitude breather solutions on necklace graphs, about the existence of moving breather solutions in a periodic FPUT model, and about the validity of the Derivative Nonlinear Schrödinger approximation for some model problems. In our analysis we use, e.g., spatial dynamics, variational methods, bifurcation theory, and Bloch wave expansions.
Dispersion management
Interchanging the role of space and time is widely used in nonlinear optics for modeling the evolution of light pulses in glass fibers, cf. Figure 1. A phenomenological model for the mathematical description of light pulses in glass fibers with a periodic structure in this set-up is the so-called dispersion management equation.
Figure 1. In geometric or nonlinear optics the space variable $ x_{phys} $ is very often taken as the evolutionary variable. The initial condition is given by the time evolution at the lower end of the fiber. Since the system is periodic in space we have chosen \eqref{A10:A1} to be periodic w.r.t. our evolutionary variable $ t = x_{phys} $.
The purpose of [FS22] was to answer the question whether the dispersion management equation or other modulation equations are more than just phenomenological models. Using Floquet theory for a time-periodic model problem \begin{equation} \label{A10:A1} \partial_t^2 u = a_1(t) \partial_x^2 u + a_2(t) -a_3(t) u^3 \quad \mbox{ with } a_j(t) = a_j(t+ 2 \pi) \end{equation} we showed that in case of comparable wave lengths of the light and of the fiber periodicity the NLS equation $$ i \partial_T A = \nu_1 \partial_X^2 A + \nu_2 A |A|^2 \quad \mbox{ with } \nu_j \in \mathbb{R} $$ and NLS-like modulation equations with constant coefficients $ \nu_ j $ can be derived as amplitude equations by an ansatz $$ u(x,t) = \varepsilon A (\varepsilon(x- ct),\varepsilon^2 t) e^{i(k_0 x - \omega_0 t)} \Phi_0(t) + c.c. , \qquad (0 < \varepsilon \ll 1), $$ with $ \Phi_0(t) = \Phi_0(t+ 2 \pi) $ the Bloch wave associated with $ \omega_0 $. The above ansatz is justified through error estimates assuming rather strong stability and non-resonance conditions. If these conditions are not satisfied we were able to prove in [FS22] that the modulation equations make wrong predictions in general. In particular, we explained that in the scaling which is necessary for the derivation of the dispersion management equation $$ i \partial_T A = \widetilde \varepsilon^{-1} \nu_1( \widetilde \varepsilon^{-1 }T ) \partial_X^2 A + \nu_2 A |A|^2, \qquad (\nu_2 \in \mathbb{R}, 0 < \widetilde \varepsilon \ll 1, \nu_1(t) = \nu_1(t+ 2\pi) ) $$ the time-periodic model problem behaves totally differently than predicted by the dispersion management equation.
The Derivative NLS approximation
The Derivative NLS equation occurs as an envelope equation for general dispersive wave systems in parameter regimes where the cubic coefficient for the associated NLS equation vanishes for the spatial wave number of the underlying slowly modulated wave packet. In spatially periodic situations such parameter regimes can be obtained by adapting the spatially periodic coefficients in the original system.
In the current funding period we started to investigate the validity of the Derivative NLS equation in the spatially homogeneous situation for a Klein-Gordon model \begin{equation} \label{nKGe} \partial_t^2 u = \partial_x^2 u - u + \varrho(\partial_x) u^3, \quad \mbox{ where } \quad \varrho(ik) = \frac{k^2-1}{k^2+1}, \end{equation} with a cubic nonlinearity. The coefficient function $ \varrho $ was chosen such that $ \varrho(i) = 0 $. For this system the (generalized) Derivative NLS equation \begin{equation}\label{DNLSintro} i \partial_T A = \nu_1 \partial_X^2 A + \nu_2 A |A|^2 +i \nu_3 |A|^2 \partial_X A + i \nu_4 A^2 \partial_X \overline{A} + \nu_5 A |A|^4, \end{equation} with $ \nu_j \in \mathbb{R} $, can be derived as an envelope equation via a multiple scaling perturbation ansatz \begin{align} \label{psiA} u(x,t) \approx \varepsilon^{1/2} \psi_A (x,t) = \varepsilon^{1/2} A(\varepsilon (x-c_gt),\varepsilon^2 t) e^{i(k_0 x - \omega_0 t)} + c.c. , \end{align} with $ 0 < \varepsilon \ll 1 $ and $ k_0 = 1 $. For the first time in the literature we proved in [HS22a, HS22b] with two different approaches, that the Derivative NLS equation makes correct predictions about the dynamics of the Klein-Gordon model \eqref{nKGe}. The proof in [HS22a] works for analytic initial conditions and is based on normal form transformations and modulational Gevrey spaces. The proof in [HS22b] works in Sobolev spaces, is based on energy estimates and normal form transformations, but requires the validity of more non-resonance conditions than in [HS22a]. New challenges arise due to the fact that not all cubic terms can be eliminated by normal form transformations. There is a total resonance and a second-order resonance, see Figure 2.
Figure 2. Graphical evaluation of the non-resonance condition $ r_{jj_1j_2j_3}(k) = 0 $. Plots of $ r_{jj_1j_2j_3}(k) = -j \omega(k) - \omega(j_1)- \omega(j_2) + j_3 \omega(k-j_1-j_2) $ for the resonances with the second order touching where $ \omega(k) = \sqrt{1+k^2}\text{sign}(k) $ and $ j , j_1,j_2,j_3\in \{-1,1\}$. The left panel shows the case $ (j,j_1,j_2,j_3)= (-1,1,1,-1) $ and the right panel shows the case $ (j,j_1,j_2,j_3)= (-1,-1,-1,1) $. In case of a total resonance we have $ r_{jj_1j_2j_3}(k) \equiv 0 $.
Moving modulating pulses in spatially periodic media
In [HdRS] we considered an FPUT system with attracting nearest neighbor interaction and repelling next-to-nearest neighbor interaction, cf. Figure 3. For this case such a lattice admits moving modulating pulse solutions. This research was motivated by the fact that this is the first rigorous construction of solitary waves for FPUT systems whose profiles are not described by a KdV equation but by an NLS equation. The proof is based on a center manifold reduction for the spatial dynamics system which is now an advance-delay system.
Figure 3. Sketch of a possible physical realization of the oscillator chain considered in [HdRS]. It consists of two lines of oscillators where in between each line the forces are repelling and across the lines the forces are attracting.
Higher dimensional waves in space in Fermi–Pasta–Ulam–Tsingou (FPUT) systems
In [PS22KP] we showed that the Kadomtsev-Petviashvili (KP)-approximation makes correct predictions about the dynamics of a scalar FPUT system \begin{eqnarray*} \partial_t^2 q_{m,n} & = & W'(q_{m+1,n}-q_{m,n})- W'(q_{m,n}-q_{m-1,n}) \\ && + W'(q_{m,n+1}-q_{m,n})- W'(q_{m,n}-q_{m,n-1}), \qquad (m,n) \in \mathbb{Z^2} \end{eqnarray*} on a square 2D lattice, cf. Figure 4. The KP equation $$ \partial_T \partial_X A + \partial_Y^2 A + \partial_X^4 A - \partial_X^2 (A^2) = 0 $$ describes unidirectional long waves of small amplitude with slowly varying transverse modulations. Existing approximation results were extended to include now arbitrary directions of the wave propagation.
Figure 4. The picture shows a physical realization of a 2D square FPUT system.
Large amplitude breathers on periodic necklace graphs
In [MRS22] we proved the existence of infinitely many localized breathers for the Klein-Gordon equation \begin{equation} \label{klein_gordon_graph} \partial_t^2 u = \partial_x^2 u - \alpha u \pm |u|^{p-1}u \mbox{ on } \Gamma\times\mathbb{R} \end{equation} for the full range of exponents \(1< p <\infty\), \(\alpha\geq 0\), and where $\Gamma$ is the periodic necklace graph equipped with Kirchhoff boundary conditions at the vertices. This result complements our previous paper [Mai20], where small solutions of size $\varepsilon$ bifurcating from $0$ were found. In the new result the breathers were derived by variational methods. They are determined as critical points of a functional $J:H\to\mathbb{R}$ on a suitable Hilbert space $H$. Since the functional is strongly indefinite, the breathers are saddle points. In contrast to previous publications embedding limitations we have now removed all embedding restrictions and proved that $H\to L^q(\Gamma)$ for all $q\in [2,\infty)$ which allowed us to treat \eqref{klein_gordon_graph} for all exponents $p\in(1,\infty)$.
