Project A1 • Large signals in nonlinear fiber optics

Principal investigators

Project summary

This project is motivated by signal transmission in long glass fibre cables. The propagation of signals in such cables is effectively described by a one-dimensional cubic non-linear Schrödinger equation (NLS) \begin{equation}\label{A1:NLS} i \partial_t u-\partial_x^2u=c|u|^2u,\quad u(t,\cdot)=u_0, \end{equation} where \(t\) denotes position along the cable and \(x\) is retarded time. So the initial value \(u_0\) is the signal seen at the position \(t=0\), and very long cables correspond ideally to no limit on \(t\). Thus we aim at global (in \(t\)) existence of solutions for \(x\in\mathbb{R}\).

Signals transporting information are neither decaying in \(x\) nor are they periodic in \(x\). The primary goal of the project is to establish global well-posedness for such signals. This means that we have to leave the well-studied setting of the usual theory for the NLS \eqref{A1:NLS} in the scale of \(L^2\)-based Sobolev spaces \(H^s(\mathbb{R})\) (decaying) on the real line or \(H^s(\mathbb{T})\) (periodic) on the torus. In these spaces the free Schrödinger group, which governs the evolution of the linear case \(c=0\) in \eqref{A1:NLS}, is bounded. Roughly speaking, local wellposedness in \(H^s\) for large \(s\) is established by direct fixed point arguments, Strichartz estimates can be used to lower the regularity index \(s\), and global wellposedness is shown via conserved quantities such as mass or the Hamiltonian, which are directly related to the norms in \(L^2\) and \(H^1\).

Modulation spaces

An important part of the project concentrated on the study of the NLS in modulation spaces \(M_{p, q}^s(\mathbb{R}^d)\), which is a scale of spaces that comprises the usual Sobolev spaces \(H^s(\mathbb{R}^d)\) as well as spaces containing non-decaying and non-periodic signals. Moreover the free Schrödinger group acts boundedly in \(M^s_{p,q}(\mathbb{R}^d)\) for all \(s\in\mathbb{R}\) and \(p,q\in[1,\infty]\). This is in sharp contrast to the more common Bessel potential spaces \(H^s_p(\mathbb{R}^d)\) for \(p\neq 2\). Modulation spaces are defined in terms of the short-time Fourier transform \begin{equation*} (V_g f)(x, k) = \mathcal{F} (f g(\cdot - x))(k),\qquad x,k\in\mathbb{R}^d.\end{equation*} Here, \(f\) is the function under study, \(g\) a smooth function used for localization (window function), \(x\) the position at which \(f\) has been localized, \(\mathcal{F}\) is the (usual) Fourier transform on \(\mathbb{R}^d\) with variable \(k\). For \(p,q\in[1,\infty]\) and a regularity index \(s\in\mathbb{R}\), the modulation space \(M_{p,q}^s(\mathbb{R}^d)\) is the set of all \(f \in\mathcal{S}'(\mathbb{R}^d)\) that satisfy \begin{equation*} \left\Vert u\right\Vert_{M_{p, q}^s} = \left\Vert k \mapsto \left(1 + \left|k\right|^2\right)^{\frac{s}{2}} \left\Vert(V_gf)(\cdot, k)\right\Vert_{L^p(\mathbb{R}^d)}\right\Vert_{L^q(\mathbb{R}^d)} < \infty. \end{equation*} Here, \(\mathcal{S}'(\mathbb{R}^d)\) is the ambient space of tempered distributions on \(\mathbb{R}^d\) and \(\left\Vert\cdot\right\Vert_{L^p}\) and \(\left\Vert\cdot\right\Vert_{L^q}\) are the usual norms on the Lebesgue spaces \(L^p(\mathbb{R}^d)\) and \(L^q(\mathbb{R}^d)\) respectively. One has \(M^s_{2,2}(\mathbb{R}^d)=H^s(\mathbb{R}^d)\) for any \(s\in\mathbb{R}\), the spaces are inceasing with \(p\) and \(q\) and decreasing with \(s\), and non-decaying signals \(f\) correspond to having \(p = \infty\).

Progress within the project

In the first funding period we have achieved the following results.

• The NLS with algebraic non-linearity is locally wellposed in intersections modulation spaces \(M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\) for all \(s\ge0\), \(p\in[1,\infty]\), and \(q\in[1,\infty)\). These spaces are algebras for pointwise multiplication. We also obtained a new Hölder type inequality for modulation spaces (see [CHKP16]).

• We obtained global existence for the NLS \eqref{A1:NLS} for initial values with large norm in modulation spaces \(M_{r, r'}(\mathbb{R})\) with \(r>2\) sufficiently close to \(2\) (see [CHKP17]). This relies on a splitting method going back to Bourgain and has been inspired by a paper of Hyakuna and Tsutsumi (see [HTT11]). The results has been extended in several ways in [Cha18]. These are the first known global existence results for large norms in modulation spaces outside the \(H^s\)-scale.

• We proved an abstract version of the boundedness of the free Schrödinger group in modulation spaces (see [Kun18]). It allows for more flexibility in the construction of modulation type function spaces. Special emphasis is put on the limit cases \(p=\infty\) and/or \(q=\infty\) and the continuity and denseness issues that arise in these cases. This closes a gap in the existing literature, where these problems do not seem to have been studied in this generality.

During our work on the project we realized that the normal form reduction method via differentiation by parts that has been successfully applied to the NLS \eqref{A1:NLS} on the torus \(\mathbb{T}\), i.e. in periodic Sobolev spaces \(H^s(\mathbb{T})\) (see [GKO13]), can be adapted to work in modulation spaces and in spaces containing certain non-decaying signals. Technically this means that Fouier coefficients have to be replaced by localized Fourier transforms and multiplication by complex numbers has to be replaced by suitable multilinear operators. The advantage of the method is that the nonlinearity can be studied without the use of Strichartz estimates and this can lead to results on unconditional uniqueness. This notion of uniqueness means that given an initial value \(u_0\in X\), the solution \(u\) is unique in the space of continuous functions on a time interval with values in \(X\) without intersecting with another function space (which arises, e.g., by the use of Strichartz estimates). Our adaptation of the method lead to the following:

• We obtained local wellposedness for the NLS \eqref{A1:NLS} in modulation spaces \(M^s_{p, q}(\mathbb{R})\) for certain ranges of \(p, q\) and \(s\ge0\) which are not algebras (see [Pat18], [CHKP18b]). For certain (smaller) ranges of \(p\), \(q\), and \(s\) we could also prove unconditional uniqueness results. This transforms the results from [GKO13] to the real line and extends them to modulation spaces.

• We obtained local wellposedness and unconditional uniqueness in low regularity for a hybrid "tooth problem" where a periodic signal (an infinite sequence of "teeth") is perturbed by a local signal (e.g. some of these teeth are missing), i.e. the NLS \eqref{A1:NLS} is studied in spaces \(H^s(\mathbb{T})+H^s(\mathbb{R})\), \(s\ge0\) (see [CHKP18a]). These are the first results for such a sort of problem in the literature. Here, we also profited from our experience with Bourgain's decomposition method (see above).

Next steps include the extension of the last mentioned results to other types of non-decaying signals and the investigation of suitable conserved quantities to obtain global existence in these problems.

Publications

1. R. Schippa. On smoothing estimates in modulation spaces and the NLS with slowly decaying initial data. J. Funct. Anal., 282(15):109352, March 2022. URL https://doi.org/10.1016/j.jfa.2021.109352. [preprint] [bibtex]

2. F. Klaus, P. Kunstmann, and N. Pattakos. Unconditional uniqueness of higher order nonlinear Schrödinger equations. Czechoslovak Math. J., 71(3):709–742, October 2021. URL https://doi.org/10.21136/CMJ.2021.0078-20. [preprint] [bibtex]

3. L. Chaichenets and N. Pattakos. On the global wellposedness of the Klein–Gordon equation for initial data in modulation spaces. Proc. Amer. Math. Soc., 149(9):3849–3861, September 2021. URL https://doi.org/10.1090/proc/15497. [preprint] [bibtex]

4. L. Chaichenets, D. Hundertmark, P. Kunstmann, and N. Pattakos. On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R})+H^1(\mathbb{T})$. NoDEA Nonlinear Differential Equations Appl., 28(2):11–28, January 2021. URL https://doi.org/10.1007/s00030-020-00670-8. [preprint] [bibtex]

5. L. Chaichenets, D. Hundertmark, P. C. Kunstmann, and N. Pattakos. Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces $M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)$. In W. Dörfler, M. Hochbruck, D. Hundertmark, W. Reichel, A. Rieder, R. Schnaubelt, and B. Schörkhuber, editors, Mathematics of Wave Phenomena, Trends in Mathematics, pages 89–107, October 2020. Birkhäuser Basel. [bibtex]

6. L. Chaichenets and N. Pattakos. The global Cauchy problem for the NLS with higher order anisotropic dispersion. Glasgow Math. J., 1–9, December 2019. URL https://doi.org/10.1017/S0017089519000491. Online first. [preprint] [bibtex]

7. L. Chaichenets, D. Hundertmark, P. Kunstmann, and N. Pattakos. Knocking out teeth in one-dimensional periodic nonlinear Schrödinger equation. SIAM J. Math. Anal., 51(5):3714–3749, September 2019. URL https://doi.org/10.1137/19M1249679. [preprint] [bibtex]

8. L. Chaichenets, D. Hundertmark, P. Kunstmann, and N. Pattakos. Nonlinear Schrödinger equation, differentiation by parts and modulation spaces. J. Evol. Equ., 19(3):803–843, September 2019. URL https://doi.org/10.1007/s00028-019-00501-z. [preprint] [bibtex]

9. N. Pattakos. NLS in the modulation space $M_{2,q}(\mathbb{R})$. J. Fourier Anal. Appl., 25(4):1447–1486, August 2019. URL https://doi.org/10.1007/s00041-018-09655-9. [preprint] [bibtex]

10. P. C. Kunstmann. Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations. Semigroup Forum, 98(3):645–668, June 2019. URL https://doi.org/10.1007/s00233-019-10016-1. [preprint] [bibtex]

11. L. Chaichenets, D. Hundertmark, P. Kunstmann, and N. Pattakos. On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M_{p,q}(\Bbb R)$. J. Differ. Equ., 263(8):4429–4441, October 2017. URL https://doi.org/10.1016/j.jde.2017.04.020. [preprint] [bibtex]

Preprints

12. F. Klaus. Wellposedness of NLS in modulation spaces. CRC 1173 Preprint 2022/21, Karlsruhe Institute of Technology, April 2022. [bibtex]

13. R. Schippa. Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces. CRC 1173 Preprint 2022/20, Karlsruhe Institute of Technology, March 2022. [bibtex]

14. F. Klaus and P. Kunstmann. Global wellposedness of NLS in $H^1(\mathbb{R})+ H^s(\mathbb{T})$. CRC 1173 Preprint 2021/39, Karlsruhe Institute of Technology, September 2021. [bibtex]

15. F. Klaus and R. Schippa. A priori estimates for the derivative nonlinear Schrödinger equation. CRC 1173 Preprint 2020/21, Karlsruhe Institute of Technology, August 2020. Revised version from April 2021. To be published in Funkcialaj Ekvacioj. [bibtex]

Theses

16. L. Chaichenets. Modulation spaces and nonlinear Schrödinger equationscs. PhD thesis, Karlsruhe Institute of Technology (KIT), September 2018. [bibtex]

Other references

17. Z. Guo, S. Kwon, and T. Oh. Poincaré–Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS. Commun. Math. Phys., 322(1):19–48, August 2013. URL https://doi.org/10.1007/s00220-013-1755-5. [bibtex]

18. R. Hyakuna, T. Tanaka, and M. Tsutsumi. On the global well-posedness for the linear Schrödinger equations with large initial data of infinite $L^2$ norm. Nonlinear Anal., 74(4):1304–1319, 2011. URL https://doi.org/10.1016/j.na.2010.10.003. [bibtex]