Skip to main content
Log in

On Selection of Standing Wave at Small Energy in the 1D Cubic Schrödinger Equation with a Trapping Potential

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we show how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associated to a repulsive potential by a series of Darboux transformations. Even though, by its non translation invariance, our model avoids some of the difficulties related to the effect that translation has on virial inequalities of the kink stability problem for wave equations, it still is a classical model and it retains some of the main difficulties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alammari, M., Snelson, S.: On asymptotic stability for near-constant solutions of variable–coefficient scalar field equations. arXiv:2104.13909

  2. Alammari, M., Snelson, S.: Linear and orbital stability analysis for solitary-wave solutions of variable-coefficient scalar field equations. J. Hyperbolic Differ. Equ. 19, 175–201 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bambusi, D., Cuccagna, S.: On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential. Am. J. Math. 133(5), 1421–1468 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buslaev, V., Perelman, G.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Uraltseva, N.N. (ed.) Nonlinear Evolution Equations. Translations: Series 2, vol. 164, pp. 75–98. American Mathematical Society, Providence (1995)

    Google Scholar 

  5. Cazenave, T., Hareaux, A.: An Introduction to Semilinear Equations. Claredon Press, Oxford (1998)

    Google Scholar 

  6. Chang, S.M., Gustafson, S., Nakanishi, K., Tsai, T.P.: Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal. 39, 1070–1111 (2007/08)

  7. Chen, G.: Long-time dynamics of small solutions to 1d cubic nonlinear Schrödinger equations with a trapping potential, arXiv:1907.07115

  8. Chen, G., Pusateri, F.: The 1d nonlinear Schrödinger equation with a weighted \(L^1 \) potential. arXiv:1912.10949

  9. Cuccagna, S., Maeda, M.: A note on small data soliton selection for nonlinear Schrödinger equations with potential. arXiv:2107.13878

  10. Cuccagna, S., Maeda, M.: Coordinates at small energy and refined profiles for the nonlinear Schrödinger equation. Ann. PDE 7 ,no. 2, Paper No. 16, 34 pp (2021)

  11. Cuccagna, S., Maeda, M.: On small energy stabilization in the NLS with a trapping potential. Anal. PDE 8(6), 1289–1349 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cuccagna, S., Maeda, M.: On stability of small solitons of the 1-D NLS with a trapping delta potential. SIAM J. Math. Anal. 51(6), 4311–4331 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cuccagna, S., Maeda, M., V. Phan, T.: On small energy stabilization in the NLKG with a trapping potential. Nonlinear Anal. 146, 32–58 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cuccagna, S., Tarulli, M.: On asymptotic stability of standing waves of discrete Schrödinger equation in \({\mathbb{Z} }\). SIAM J. Math. Anal. 41, 861–885 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cuccagna, S., Visciglia, N., Georgiev, V.: Decay and scattering of small solutions of pure power NLS in R with \(p > 3\) and with a potential. Commun. Pure Appl. Math. 67, 957–981 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math. 32, 121–251 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert Problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deift, P., Zhou, X.: Perturbation theory for infinite-dimensional integrable systems on the line. A case study. Acta Math. 188, 163–262 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Delort, J.-M.: Modified scattering for odd solutions of cubic nonlinear Schrodinger equations with potential in dimension one. preprint hal–01396705

  20. Delort, J.-M., Masmoudi, N.: Long time Dispersive Estimates for perturbations of a kink solution of one dimensional wave equations, preprint hal–02862414

  21. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  22. Zhou, G., Sigal, I.M.: Relaxation of solitons in nonlinear Schrödinger equations with potential. Adv. Math. 216, 443–490 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gravejat, P., Smets, D.: Asymptotic stability of the black soliton for the Gross–Pitaevskii equation. Proc. Lond. Math. Soc. 111, 305–353 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Germain, P., Pusateri, F.: Quadratic Klein-Gordon equations with a potential in one dimension. arXiv:2006.15688

  25. Germain, P., Pusateri, F., Rousset, F.: The Nonlinear Schrödinger equation with a potential in dimension 1. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 1477–1530 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Gustafson, S., Nakanishi, K., Tsai, T.P.: Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves. Int. Math. Res. Not. 2004(66), 3559–3584 (2004)

    Article  MATH  Google Scholar 

  27. Kowalczyk, M., Martel, Y., Muñoz, C.: Kink dynamics in the \(\phi ^4\) model: asymptotic stability for odd perturbations in the energy space. J. Am. Math. Soc. 30, 769–798 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kowalczyk, M., Martel, Y., Muñoz, C.: Nonexistence of small, odd breathers for a class of nonlinear wave equations. Lett. Math. Phys. 107, 921–931 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Kowalczyk, M., Martel, Y., Muñoz, C.: Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes. arXiv:1903.12460 (to appear in Jour. Eur. Math. Soc.)

  30. Kowalczyk, M., Martel, Y., Muñoz, C., Van Den Bosch, H.: A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models. Ann. PDE 7, no. 1, Paper No. 10, 98 pp (2021)

  31. Li, Z.: Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case. Front. Math. China 15, 923–957 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lindblad, H., Luhrmann, J., Schlag, W., Soffer, A.: On modified scattering for 1D quadratic Klein–Gordon equations with non-generic potentials. arXiv:2012.15191

  33. Lindblad, H., Lührmann, J., Soffer, A.: Decay and asymptotics for the one-dimensional Klein–Gordon equation with variable coefficient cubic nonlinearities. SIAM J. Math. Anal. 52, 6379–6411 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lindblad, H., Soffer, A.: A remark on asymptotic completeness for the critical nonlinear Klein–Gordon equation. Lett. Math. Phys. 73, 249–258 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Lindblad, H., Soffer, A.: A remark on long range scattering for the nonlinear Klein–Gordon equation. J. Hyperbolic Differ. Equ. 2, 77–89 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lindblad, H., Soffer, A.: Scattering and small data completeness for the critical nonlinear Schrödinger equation. Nonlinearity 19, 345–353 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lindblad, H., Soffer, A.: Scattering for the Klein–Gordon equation with quadratic and variable coefficient cubic nonlinearities. Trans. Am. Math. Soc. 367, 8861–8909 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lührmann, J., Schlag, W.: Asymptotic stability of the sine-Gordon kink under odd perturbations. arXiv:2106.09605

  39. Martinez, M.: Decay of small odd solutions for long range Schrödinger and Hartree equations in one dimension. Nonlinearity 33, 1156–1182 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Martinez, M.: On the decay problem for the Zakharov and Klein–Gordon–Zakharov systems in one dimension. J. Evol. Equ. 21, 3733–3763 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  41. Masaki, S., Murphy, J., Segata, J.: Modified scattering for the one-dimensional cubic NLS with a repulsive delta potential. Int. Math. Res. Not. 24, 7577–7603 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. 161, 157–222 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Merle, F., Raphael, P.: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13, 591–642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Merle, F., Raphael, P.: On universality and blow-up profile for \(L^2\)- critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Merle, F., Raphael, P.: On a sharp lower bound on the blow-up rate for the \(L^2\) critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19, 37–90 (2006)

    Article  MATH  Google Scholar 

  46. Mizumachi, T.: Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 48(3), 471–497 (2008)

    MathSciNet  MATH  Google Scholar 

  47. Nakanishi, K., Phan, T.V., Tsai, T.P.: Small solutions of nonlinear Schrödinger equations near first excited states. J. Funct. Anal. 263, 703–781 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Naumkin, I.P.: Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential. J. Math. Phys. 57, 051501 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Schlag, W.: Dispersive estimates for Schrödinger operators: a survey. In: Mathematical Aspects of Nonlinear Dispersive Equations, Annals of Mathematical Studies, 163. Princeton University Press, Princeton, pp. 255–285 (2007)

  50. Snelson, S.: Asymptotic stability for odd perturbations of the stationary kink in the variable-speed \(\phi ^4\) model. Trans. Am. Math. Soc. 370, 7437–7460 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  51. Soffer, A., Weinstein, M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9–74 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Soffer, A., Weinstein, M.I.: Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16(8), 977–1071 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sogge, C.: Fourier Integrals in Classical Analysis. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  54. Sterbenz, J.: Dispersive decay for the 1D Klein–Gordon equation with variable coefficient nonlinearities. Trans. Am. Math. Soc. 368, 2081–2113 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  55. Tsai, T.P., Yau, H.T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and radiation dominated solutions. Commun. Pure Appl. Math. 55, 153–216 (2002)

    Article  MATH  Google Scholar 

  56. Tsai, T.P., Yau, H.T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31, 1629–1673 (2002)

    Article  MATH  Google Scholar 

  57. Tsai, T.P., Yau, H.T.: Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data. Adv. Theor. Math. Phys. 6, 107–139 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tsai, T.P., Yau, H.T.: Stable directions for excited states of nonlinear Schrödinger equations. Commun. PDE 27, 2363–2402 (2002)

    Article  MATH  Google Scholar 

  59. Taylor, M.: Pseudo Differential Operators. Princeton University Press, Princeton (1981)

    Google Scholar 

  60. Weder, R.: \(L^p-L ^{p^{\prime }}\) estimates for the Schrödinger equation on the line and Inverse Scattering for the Nonlinear Schrödinger equation with a potential. J. Funct. Anal. 170, 37–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

C. was supported by the Prin 2020 project Hamiltonian and Dispersive PDEs N. 2020XB3EFL. M. was supported by the JSPS KAKENHI Grant Number 19K03579, G19KK0066A and JP17H02853.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Scipio Cuccagna.

Ethics declarations

Conflict of Interest

The authors declare that there was no conflict of interest.

Additional information

Communicated by K. Nakanishi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A Appendix: Proof of Lemma 9.3

A Appendix: Proof of Lemma 9.3

It is equivalent to show that there is a constant \(C>0\) such that for all v

$$\begin{aligned}&\left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) \prod _{j=1}^{N}R _{H}( \omega _j) P_c \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v\right\| _{L^2({\mathbb {R}})} \le C \left\| {\mathrm {sech}}\left( \frac{a x}{20} \right) v \right\| _{L^2({\mathbb {R}})} . \end{aligned}$$
(A.1)

By (8.1), for \(x<y\) we have the formula

$$\begin{aligned} R _{H}(z^2) (x,y)&= \frac{T(z)}{2\mathrm{i}z} f_- (x, z) f_+ (y, z) \nonumber \\&\quad = \frac{1}{z^2+\omega _j} \frac{ f_- (x, \mathrm{i}\sqrt{|\omega _j|}) f_+ (y, \mathrm{i}\sqrt{|\omega _j|}) }{ \int _{{\mathbb {R}}} f_-(x',\mathrm{i}\sqrt{|\omega _j|}) f_+(x',\mathrm{i}\sqrt{|\omega _j|}) dx'} + \widetilde{R} _{H}(z^2) (x,y) , \end{aligned}$$
(A.2)

where \( \frac{T(z)}{2\mathrm{i}z}= \frac{1}{[ f_+ (x, z) , f_- (x, z)]}\), where in the denominator in the r.h.s. we have the Wronskian, where \( \widetilde{R} _{H}(z^2) (x,y)\) is not singular in \(z=\mathrm{i}\sqrt{|\omega _j|}\). On the other hand,

$$\begin{aligned}&T(z) = \frac{\text {Res}(T,\mathrm{i}\sqrt{|\omega _j|}) }{z-\mathrm{i}\sqrt{|\omega _j|} }+ \widetilde{T}(z) , \end{aligned}$$

with \( \widetilde{T}(z)\) non singular and with residue, see p. 146 [16],

$$\begin{aligned}&\text {Res}(T,\mathrm{i}\sqrt{|\omega _j|}) = \mathrm{i}\left( \int _{{\mathbb {R}}} f_-(x',\mathrm{i}\sqrt{|\omega _j|}) f_+(x',\mathrm{i}\sqrt{|\omega _j|}) dx' \right) ^{-1} . \end{aligned}$$

It is elementary to conclude, comparing the terms in (A.2), that

$$\begin{aligned}&\widetilde{R} _{H}( \omega _j) (x,y) = K_j(x,y) +C(\omega _j) \phi _j (x) \phi _j (y) \text { with}\nonumber \\ {}&K_j(x,y)= \frac{1}{2\mathrm{i}\sqrt{|\omega _j|}} \frac{ \left. \partial _{z}\left( f_-(x,z) f_+(y,z) \right) \right| _{z=\mathrm{i}\sqrt{|\omega _j|}} }{ \int _{{\mathbb {R}}} f_-(x',\mathrm{i}\sqrt{|\omega _j|}) f_+(x',\mathrm{i}\sqrt{|\omega _j|}) dx'} . \end{aligned}$$
(A.3)

for some constant \( C(\omega _j)\). For \(x>y\) we obtain the same formula, interchanging x and y. Denoting by \(K_j\) the operator with the kernel (A.3) for \(x<y\) and the formula obtained from (A.3) interchanging x and y if \(x>y\), we notice that

$$\begin{aligned}&\prod _{j=1}^{N}R _{H}( \omega _j) P_c = K_1...K_N . \end{aligned}$$

It is also easy to check, following the discussion in p. 134 [16], that there is a fixed \(C>0\) s.t. \(|K_j(x,y) |\le C\left\langle x-y\right\rangle e^{-\sqrt{|\omega _j|} |x-y|} \). Then, for any value \(a\in [ 0 , \sqrt{|\omega _N|}]\) we have

$$\begin{aligned}&\Vert {\mathrm {sech}}\left( \frac{a x}{10} \right) \prod _{j=1}^{N}R _{H}( \omega _j) P_c \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v \Vert _{L^2}\\ {}&\quad \lesssim \Vert \prod _{j=1}^{N}R _{H}( \omega _j) P_c {\mathrm {sech}}\left( \frac{a x}{10} \right) \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v \Vert _{L^2}. \end{aligned}$$

We have

$$\begin{aligned}&{\mathrm {sech}}\left( \frac{a x}{10} \right) \mathcal {A} =P_{N}(x, \mathrm{i}\partial _x ) {\mathrm {sech}}\left( \frac{a x}{10} \right) , \end{aligned}$$

for an N–th order differential operator with smooth and bounded coefficients.

Next, we write

$$\begin{aligned}&{\mathrm {sech}}\left( \frac{a x}{10} \right) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} = \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} {\mathrm {sech}}\left( \frac{a x}{10} \right) + \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ -N} \left[ {\mathrm {sech}}\left( \frac{a x}{10 } \right) , \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right] ,\end{aligned}$$

so that

$$\begin{aligned}&\left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) \prod _{j=1}^{N}R _{H}( \omega _j) P_c \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v\right\| _{L^2({\mathbb {R}})} \\ {}&\quad \lesssim \left\| \prod _{j=1}^{N}R _{H}( \omega _j) P_c P_{N}(x, \mathrm{i}\partial _x ) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} {\mathrm {sech}}\left( \frac{a x}{10} \right) v \right\| _{L^2({\mathbb {R}})} \\ {}&\qquad + \left\| \prod _{j=1}^{N}R _{H}( \omega _j) P_c P_{N}(x, \mathrm{i}\partial _x ) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ -N} \left[ {\mathrm {sech}}\left( \frac{a x}{10} \right) , \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right] v \right\| _{L^2({\mathbb {R}})} \\ {}&\quad =:I+II . \end{aligned}$$

We have

$$\begin{aligned} I&\le \left\| \prod _{j=1}^{N}R _{H}( \omega _j) P_c P_{N}(x, \mathrm{i}\partial _x ) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right\| _{L^2\rightarrow L^2} \left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) v \right\| _{L^2({\mathbb {R}})} \\&\quad \le C \left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) v \right\| _{L^2({\mathbb {R}})} \end{aligned}$$

with a fixed constant C independent from \(\varepsilon \in (0,1)\). Next, we have

$$\begin{aligned}&II \le \left\| \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ -N} \left[ {\mathrm {sech}}\left( \frac{a x}{10} \right) , \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right] v \right\| _{L^2({\mathbb {R}})} \le C \left\| {\mathrm {sech}}\left( \frac{a x}{20} \right) v \right\| _{L^2({\mathbb {R}})} \end{aligned}$$

by Lemma 5.5, because \(\int e^{-\mathrm{i}kx} {\mathrm {sech}}(x) dx = \pi \ {\mathrm {sech}}\left( \frac{\pi }{2} k \right) \) (which can be proved by an elementary application of the Residue Theorem) so that in the strip \(k=k_1+ \mathrm{i}k_2\) with \(|k_2|\le \mathbf {b}:=a/20\), then \({\mathrm {sech}}\left( \frac{\pi }{2} \ \frac{10}{a} k \right) \) satisfies the estimates required on \(\widehat{\mathcal {V}}\) in (5.11). This completes the proof of (A.1).

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cuccagna, S., Maeda, M. On Selection of Standing Wave at Small Energy in the 1D Cubic Schrödinger Equation with a Trapping Potential. Commun. Math. Phys. 396, 1135–1186 (2022). https://doi.org/10.1007/s00220-022-04487-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-022-04487-7

Navigation