Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T23:10:21.352Z Has data issue: false hasContentIssue false
  • English
  • Français

Standing waves of modified Schrödinger equations coupled with the Chern–Simons gauge theory

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  12 March 2019

Pietro d'Avenia ,
Alessio Pomponio  and
Tatsuya Watanabe
Pietro d'Avenia
Affiliation:
Dipartimento di Meccanica, Matematica e Management Politecnico di Bari Via Orabona 4, 70125Bari, Italy ([email protected]; [email protected])
Alessio Pomponio
Affiliation:
Dipartimento di Meccanica, Matematica e Management Politecnico di Bari Via Orabona 4, 70125Bari, Italy ([email protected]; [email protected])
Tatsuya Watanabe
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City603-8555, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We are interested in standing waves of a modified Schrödinger equation coupled with the Chern–Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the nonexistence for nontrivial solutions.

Keywords

ground state solutiongauged Schrödinger equationvariational method

MSC classification

Primary: 35J62: Quasilinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35Q60: PDEs in connection with optics and electromagnetic theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1915 - 1936
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adachi, S. and Watanabe, T.. Uniqueness of the ground state solutions of quasilinear Schrödinger equations. Nonlinear Anal. 75 (2012), 819–833.CrossRefGoogle Scholar
2Bergé, L., de Bouard, A. and Saut, J. C.. Blowing up time-dependent solutions of the planar Chern–Simons gauged nonlinear Schrödinger equation. Nonlinearity 8 (1995), 235–253.CrossRefGoogle Scholar
3Brihaye, Y., Hartmann, B. and Zakrzewski, W.. Spinning solitons of a modified nonlinear Schrödinger equation. Phys. Rev. D 69 (2004), 087701.CrossRefGoogle Scholar
4Brizhik, L., Eremko, A., Piette, B. and Zahkrzewski, W. J.. Static solutions of a D-dimensional modified nonlinear Schrödinger equation. Nonlinearity 16 (2003), 1481–1497.CrossRefGoogle Scholar
5Brüll, L. and Lange, H.. Solitary waves for quasilinear Schrödinger equations. Expo. Math. 4 (1986), 279–288.Google Scholar
6Byeon, J., Huh, H. and Seok, J.. Standing waves of nonlinear Schrödinger equations with the gauge field. J. Funct. Anal. 263 (2012), 1575–1608.CrossRefGoogle Scholar
7Byeon, J., Huh, H. and Seok, J.. On standing waves with a vortex point of order N for the nonlinear Chern–Simons–Schrödinger equations. J. Diff. Eqns. 261 (2016), 1285–1316.CrossRefGoogle Scholar
8Colin, M. and Jeanjean, L.. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 56 (2004), 213–226.CrossRefGoogle Scholar
9Colin, M., Jeanjean, L. and Squassina, M.. Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity 23 (2010), 1353–1385.CrossRefGoogle Scholar
10Cunha, P. L., d'Avenia, P., Pomponio, A. and Siciliano, G.. A multiplicity result for Chern–Simons–Schrödinger equation with a general nonlinearity. Nonlinear Differ. Equ. Appl. 22 (2015), 1831–1850.CrossRefGoogle Scholar
11Di Benedetto, E. and Trudinger, N. S.. Harnack inequalities for quasi-minima of variational integrals. AIHP Anal. Nonlinéaire. 1 (1984), 295–308.Google Scholar
12Felsager, B.. Geometry, particles and fields (New York: Springer-Verlag, 1998).CrossRefGoogle Scholar
13Huh, H.. Standing waves of the Schrödinger equation coupled with the Chern–Simons gauge field. J. Math. Phys. 53 (2012), 063702.CrossRefGoogle Scholar
14Huh, H.. Energy Solution to the Chern–Simons–Schrödinger equations. J. Abstr. Appl. Anal. 2013 (2013), Article ID 590653, 7 pp.Google Scholar
15Jackiw, R. and Pi, S. Y.. Soliton solutions to the gauged nonlinear Schrödinger equations on the plane. Phys. Rev. Lett. 64 (1990), 2969–2972.CrossRefGoogle Scholar
16Jackiw, R. and Pi, S. Y.. Self-dual Chern–Simons solitons. Progr. Theoret. Phys. Suppl. 107 (1992), 1–40.CrossRefGoogle Scholar
17Jiang, Y., Pomponio, A. and Ruiz, D.. Standing waves for a gauged nonlinear Schrödinger equation with a vortex point. Commun. Contemp. Math. 18 (2016), 1550074, 20 pp.CrossRefGoogle Scholar
18Krolikowski, W., Bang, O., Rasmussen, J. J. and Wyller, J.. Modulational instability in nonlocal nonlinear Kerr media. Phys. Rev. E 64 (2001), 016612.CrossRefGoogle ScholarPubMed
19Kurihara, S.. Large-amplitude quasi-solitons in superfluid films. J. Phys. Soc. Japan 50 (1981), 3262–3267.CrossRefGoogle Scholar
20Ladyzhenskaya, O. A. and Uraltseva, N. N.. Linear and quasilinear elliptic equations (New York: Academic Press, 1968).Google Scholar
21Liu, B. and Smith, P.. Global wellposedness of the equivariant Chern–Simons–Schrödinger equation. Rev. Mat. Iberoam. 32 (2016), 751–794.CrossRefGoogle Scholar
22Liu, J., Wang, Y. and Wang, Z. Q.. Solutions for quasilinear Schrödinger equations via the Nehari method. Commun. Partial Diff. Eqns. 29 (2004), 879–901.CrossRefGoogle Scholar
23Liu, B., Smith, P. and Tataru, D.. Local wellposedness of Chern–Simons–Schrödinger. Int. Math. Res. Not. IMRN 2014 (2014), 6341–6398.CrossRefGoogle Scholar
24Pomponio, A.. Some results on the Chern–Simons–Schrödinger equation. Lect. Notes Semin. Interdiscip. Mat. 13 (2016), 67–93.Google Scholar
25Pomponio, A. and Ruiz, D.. A variational analysis of a gauged nonlinear Schrödinger equation. J. Eur. Math. Soc. 17 (2015), 1463–1486.CrossRefGoogle Scholar
26Pomponio, A. and Ruiz, D.. Boundary concentration of a gauged nonlinear Schrödinger equation. Calc. Var. PDE 53 (2015), 289–316.CrossRefGoogle Scholar
27Rabier, P. and Stuart, C. A.. Exponential decay of the solutions of quasilinear second-order equations and Pohozaev identities. J. Diff. Eqns. 165 (2000), 199–234.CrossRefGoogle Scholar
28Ruiz, D. and Siciliano, G.. Existence of ground states for a modified nonlinear Schrödinger equation. Nonlinearity 23 (2010), 1221–1233.CrossRefGoogle Scholar
29Shen, Y. and Wang, Y.. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal. 80 (2013), 194–201.CrossRefGoogle Scholar
30Strauss, W. A.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
31Tolksdorf, P.. Regularity for a more general class of quasilinear elliptic equations. J. Diff. Eqns. 51 (1984), 126–150.CrossRefGoogle Scholar
32Trudinger, N. S.. On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967), 721–747.CrossRefGoogle Scholar
33Wan, Y. and Tan, J.. The existence of nontrivial solutions to Chern–Simons–Schrödinger systems. Disc. Cont. Dyn. Syst. 37 (2017), 2765–2786.CrossRefGoogle Scholar
34Yuan, J.. Multiple normalized solutions of Chern–Simons–Schrödinger system. Nonlinear Differ. Equ. Appl. 22 (2015), 1801–1816.CrossRefGoogle Scholar