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 nonlinear 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 wellposedness 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 \(p,q\in[1,\infty]\), which 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 and \(\mathcal{F}\) 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. We write $M_{p,q}(\mathbb{R}^d)≔M^0_{p,q}(\mathbb{R}^d)$. One has \(M^s_{2,2}(\mathbb{R}^d)=H^s(\mathbb{R}^d)\) for any \(s\in\mathbb{R}\), the spaces are increasing 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 two funding periods we have achieved the following results.

• The NLS with algebraic nonlinearity 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 were 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 Fourier 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 [Pat19], [CHKP19a]). 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 [CHKP19b]). These were 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 we aimed to extend the last mentioned results both to other types of non-decaying signals and to lower regularity, and to investigate suitable conserved quantities to obtain global existence in these problems. One of the major challenges for new local wellposedness results in modulation spaces \(M^s_{p,q}(\mathbb{R})\) is the interplay of low regularity (small \(s\), large \(q\)) and slow decay (large \(p\)) which makes it hard to estimate the nonlinearity. The problem in the prospect of global wellposedness is that even though the equation \eqref{A1:NLS} is completely integrable, none of the infinitely many conserved quantities is well-defined without a fixed asymptotic behavior of the function \(u\) when \(x\to\pm\infty\). In the hybrid "tooth problem" case one can suitably renormalize the conserved quantities, e.g. when \(u = w + v \in H^1(\mathbb{T}) + L^2(\mathbb{R})\) then the renormalized mass \begin{equation} \int_{\mathbb{R}}|u|^2-|w|^2\,dx = \int|v|^2+2\text{Re}(\bar{v}w)\,dx \end{equation} is well-defined and formally conserved. Unfortunately due to the emergence of \(L^1\) contributions and the lack of coercivity it is not clear whether these quantities can be used effectively. Both of these issues were addressed and we were able to achieve the following results.

• In [CHKP21], we were able to show global wellposedness for the "tooth problem" of a quadratic NLS with nonlinearity \(\pm|u|u\) with the aid of a Gronwall argument for the mass of the local part \(v \in L^2(\mathbb{R})\).

• In [Sch22], we obtained local wellposedness results for \(M_{4,2}(\mathbb{R})\) and \(M^{\varepsilon}_{6,2}(\mathbb{R})\), \(\varepsilon > 0\), and – in the defocusing case – global results for initial data in \(M^s_{4,2}(\mathbb{R})\) and \(M^s_{6,2}(\mathbb{R})\) for \(s > 3/2\). We combined the ansatz of using finite Picard iterates ([DSS21]) with local smoothing estimates obtained by decoupling techniques from [BD15].

• In [KK22] we obtained global wellposedness for the "tooth problem" in \(H^1(\mathbb{R})+ H^s(\mathbb{T})\) for \(s > 3/2\) for the cubic NLS \eqref{A1:NLS} in the defocusing case. By modifying arguments from [DSS21] we were able to use a combination of mass and energy for the local part \(v\) (both quantities are non-conserved). To be more precise, the study of the Hamiltonian corresponding to the modified NLS equation for the local part \(v\) leads to an exponential type bound on the solution.

• The local and global wellposedness results in our work [Kla23] (and see also [Kla22]) cover wellposedness results for a large new range of modulation spaces (see Figure 1 for an overview of the results if \(s = 0\) and \(s = 1\). Previously, the works [Guo17] and [OW20] showed global wellposedness for $L^2$-based modulation spaces \(M_{2,q}(\mathbb{R})\), \(q \in (2, \infty)\) by making use of the conserved quantities from [KVZ18]. In [Kla23] it is shown that these conserved quantities can be adapted to the spaces \(M_{2,q}(\mathbb{R})\) with \(q \in [1, 2)\). In the special case \(q = 1\) this solved a long-standing open problem (see [RSW12, Question 7.2]). Additional local wellposedness results are obtained via multilinear interpolation, as are some new global wellposedness results via persistance of regularity. Finally in the defocusing case for slow decay \(2 < p < \infty\) global existence is established in modulation spaces of sufficient regularity.

  

Apart of the result on NLS \eqref{A1:NLS} we note that the techniques we developed yielded also new results on other dispersive models:

• In [CP21a] we showed global wellposedness results and scattering for an NLS with higher order anisotropic dispersion

• The work [CP21b] proves global wellposedness of the Klein–Gordon equation for initial data in modulation spaces \(M^1_{p,p'}(\mathbb{R})\times M_{p,p'}\) with \(p\) sufficiently close to \(2\).

• In [KKP21] we prove unconditional uniqueness of higher order nonlinear Schrödinger equations using an infinite normal form reduction method (compare to [Pat19], [CHKP19a]).

• In [KS22] new a priori estimates in low regularity for the completely integrable derivative NLS are derived by making use of the formalism developed in [KVZ18].

• In the work [Sch23] we prove the existence of infinite energy solutions for energy critical NLS by resorting to modulation spaces and proving new bilinear estimates.

• We show low regularity wellposedness for KP-I equations in the work [SaS22].

Next steps include a possible extension of our results to the focusing case of NLS \eqref{A1:NLS} and to other types of non-decaying signals. Moreover we aim to investigate qualitative properties of the solutions constructed (stability, scattering vs blow-up, etc).

Publications

1. R. Schippa. Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces. J. Math. Anal. Appl., 519(1):Paper No. 126748, March 2023. URL https://doi.org/10.1016/j.jmaa.2022.126748. [preprint] [bibtex]

2. F. Klaus. Wellposedness of NLS in modulation spaces. J. Fourier Anal. Appl., 29:9, February 2023. URL https://doi.org/10.1007/s00041-022-09985-9. [preprint] [bibtex]

3. F. Klaus and R. Schippa. A priori estimates for the derivative nonlinear Schrödinger equation. Funkc. Ekvacioj, 65(3):329–346, December 2022. URL https://doi.org/10.1619/fesi.65.329. [preprint] [bibtex]

4. F. Klaus and P. Kunstmann. Global wellposedness of NLS in $H^1(\Bbb R)+H^s(\Bbb T)$. J. Math. Anal. Appl., 514(2):Paper No. 126359, 14, October 2022. URL https://doi.org/10.1016/j.jmaa.2022.126359. [preprint] [bibtex]

5. 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]

6. 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]

7. 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]

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

9. L. Chaichenets and N. Pattakos. The global Cauchy problem for the NLS with higher order anisotropic dispersion. Glasgow Math. J., 63(1):45–53, January 2021. URL https://doi.org/10.1017/S0017089519000491. [preprint] [bibtex]

10. 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]

11. 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]

12. 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]

13. 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]

14. 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]

15. 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. Differential Equations, 263(8):4429–4441, October 2017. URL https://doi.org/10.1016/j.jde.2017.04.020. [preprint] [bibtex]

Preprints

16. F. Klaus, H. Koch, and B. Liu. Well-posedness for the KdV hierarchy. CRC 1173 Preprint 2023/23, Karlsruhe Institute of Technology, November 2023. [bibtex]

17. F. Klaus. A Strichartz estimate for quasiperiodic functions. CRC 1173 Preprint 2023/15, Karlsruhe Institute of Technology, June 2023. [bibtex]

18. F. Klaus. A priori estimates for a quadratic dNLS. CRC 1173 Preprint 2023/14, Karlsruhe Institute of Technology, May 2023. [bibtex]

19. A. Sanwal and R. Schippa. Low regularity well-posedness for KP-I equations: the dispersion-generalized case. CRC 1173 Preprint 2022/44, Karlsruhe Institute of Technology, September 2022. [bibtex]

Theses

20. F. Klaus. Nonlinear Schrödinger equations with rough data. PhD thesis, Karlsruhe Institute of Technology (KIT), November 2022. [bibtex]

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

Other references

22. B. Dodson, A. Soffer, and T. Spencer. Global well-posedness for the cubic nonlinear Schrödinger equation with initial data lying in $L^p$-based Sobolev spaces. J. Math. Phys., 62(7):Paper No. 071507, 13, 2021. URL https://doi.org/10.1063/5.0042321. [bibtex]

23. T. Oh and Y. Wang. Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces. J. Differential Equations, 269(1):612–640, 2020. URL https://doi.org/10.1016/j.jde.2019.12.017. [bibtex]

24. R. Killip, M. Vişan, and X. Zhang. Low regularity conservation laws for integrable PDE. Geom. Funct. Anal., 28(4):1062–1090, 2018. URL https://doi.org/10.1007/s00039-018-0444-0. [bibtex]

25. S. Guo. On the 1D cubic nonlinear Schrödinger equation in an almost critical space. J. Fourier Anal. Appl., 23(1):91–124, 2017. URL https://doi.org/10.1007/s00041-016-9464-z. [bibtex]

26. J. Bourgain and C. Demeter. The proof of the $l^2$ decoupling conjecture. Ann. of Math. (2), 182(1):351–389, 2015. URL https://doi.org/10.4007/annals.2015.182.1.9. [bibtex]

27. 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]

28. M. Ruzhansky, M. Sugimoto, and B. Wang. Modulation spaces and nonlinear evolution equations. In Evolution equations of hyperbolic and Schrödinger type, volume 301 of Progr. Math., pages 267–283. Birkhäuser/Springer Basel AG, Basel, 2012. [bibtex]

29. 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]