Abstract
In this paper, we investigate dynamical properties of two-dimensional vortex solitons in optical materials with a nonlocal focusing nonlinearity. We establish the existence of the vortex solitons via variational method and some new tricks. Our results open up an intriguing perspective for research of stable vortex solitons in other nonlocal nonlinear systems including Bose–Einstein condensates with pronounced long-range interparticle interaction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosetti, A., Ruiz, D.: Multiple bound stats for the Schrödinger–Poisson equation. Commun. Contemp. Math. 10, 1–14 (2008)
Akhmanov, S., Krindach, D., Migulin, A., Sukhorukov, A., Khokhlov, R.: Thermal self-actions of laser beams. IEEE J. Quantum Electron. 4, 568–575 (1968)
Assanto, G., Peccianti, M.: Spatial solitons in nematic liquid crystals. IEEE J. Quantum Electron. 39, 13–21 (2003)
Azzollini, A., d’Avenia, P., Pomponio, A.: On the Schrödinger–Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincare Anal. Non Lineaire 27, 779–791 (2010)
Bertolotti, M., Li Voti, R., Marchetti, S., Sibili, C.: Interaction of soliton-like beam in a diffusive nonlinear planar waveguide. Opt. Commun. 133, 578–586 (1997)
Biloshytskyi, V.M., Oliynyk, A.O., Kruglenko, P.M., Desyatnikov, A.S., Yakimenko, A.I.: Solitons with rings and vortex rings on solitons in nonlocal nonlinear media. arXiv: 1702.04494v1
Briedis, D., Petersen, D.E., Edmundson, D., Krolikowski, W., Bang, O.: Ring vortex solitons in nonlocal nonlinear media. EQEC’05 European 13, 435–443 (2005)
Cerami, G., Vaira, G.: Positive solutions of some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 248, 521–543 (2010)
Chen, S., Tang, X.: On the planar Schrdöinger–Poisson system with the axially symmetric potential. J. Differ. Equ. 268, 945–976 (2020)
Chen, S., Tang, X.: Axially symmetric solutions for the planar Schrödinger–Poisson system with critical exponential growth. J. Differ. Equ. 269, 9144–9174 (2020)
Cingolani, S., Weth, T.: On the planar Schrödinger–Poisson system. Ann. Inst. Henri Poincare Anal. Non Lineaire 33, 169–197 (2016)
Davydova, A., Fishchuk, A.I.: Upper hybrid nonlinear wave structures. Ukr. J. Phys. 40, 487–494 (1995)
D’Aprile, T., Mugnai, D.: Non-existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)
Du, M., Weth, T.: Ground states and high energy solutions of the planar Schrödinger–Poisson system. Nonlinearity 30, 3492–3515 (2017)
Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28, 1633–1659 (1997)
Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on \(R^N\). Proc. R. Soc. Edinb. A 129, 787–809 (1999)
Królikowski, W., Bang, O., Nikolov, N.I., Neshev, D., Wyller, J., Rasmussen, J.J., Edmundson, D.: Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media. J. Opt. B 6, S288–S294 (2004)
Królikowski, W., Bang, O., Wyller, J., Rasmussen, J.J.: Optical beams in nonlocal nonlinear media. Acta Phys. Pol. A 103, 133–147 (2003)
Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)
Lieb, E.: Sharp constants in the Hardy–Littlewood–Sobolev inequality and related inequalities. Ann. Math. 118, 349–374 (1983)
Litvak, A.G., Mironov, V.A., Fraiman, G.M., Yunakovskii, A.D.: Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity. Sov. J. Plasma Phys. 1, 31–37 (1975)
McLaughlin, D.W., Muraki, D.J., Shelley, M.J., Xiao, W.: A paraxial model for optical self-focussing in a nematic liquid crystal. Physica D 88, 55–81 (1995)
Parola, A., Salasnich, L., Reatto, L.: Structure and stability of bosonic clouds: alkali-metal atoms with negative scattering length. Phys. Rev. A 57, R3180–R3183 (1998)
Pecseli, H.L., Rasmussen, J.J.: Nonlinear electron waves in strongly magnetized plasmas. Plasma Phys. 22, 421–438 (1980)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. III. Scattering Theory, Academic Press, New York (1979)
Ruiz, D.: Semiclassical states for coupled Schrödinger–Maxwell equations concentration around a sphere. Math. Models Methods Appl. Sci. 15, 141–164 (2005)
Snyder, W., Mitchell, J.: Accessible solitons. Science 276, 1538–1541 (1997)
Sun, J., Ma, S.: Ground state solutions for some Schrödinger–Poisson systems with periodic potentials. J. Differ. Equ. 260, 2119–2149 (2016)
Stubbe, J.: Bound states of two-dimensional Schrödinger–Newton equations. arXiv: 0807.4059v1
Suter, D., Blasberg, T.: Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium. Phys. Rev. A 48, 4583–4587 (1993)
Willem, M.: Minimax Theorems. Birkhäser, Boston (1996)
Acknowledgements
This work was supported in part by the National Natural Science Foundation of He’ nan Province of China (Grant No. 222300420416) and the National Natural Science Foundation of China under Grants 11471099 and 11971148.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Han, H., Zhang, R. Existence of vortex solitons in nonlocal nonlinear media. Z. Angew. Math. Phys. 73, 158 (2022). https://doi.org/10.1007/s00033-022-01794-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-022-01794-w
Keywords
- Vortex soliton solution
- Schrödinger–Poisson system
- Variational methods
- Logarithmic convolution potential
- Axially symmetric function