On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

Journal of Differential Equations

Volumn 321, Issue , 2022, Pages 258-295

  Fukaya, Noriyoshi ^a   Georgiev, Vladimir ^b,c,d   Ikeda, Masahiro ^e,f  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

^b UNIVERSITY OF PISA   (Italy)

^c WASEDA UNIVERSITY   (Japan)

^d INSTITUTE OF MATHEMATICS AND INFORMATICS   (Bulgaria)

^e Disaster Resilience Science Team   (Japan)

^f KEIO UNIVERSITY   (Japan)

Author keywords

Instability; Nonlinear Schr dinger equation; Point interaction; Stability; Standing wave

Indexed keywords

EID: 85126563092     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2022.03.008     Document Type: Article

References (64)

1. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., McClain, M.A., (eds.), 2021 NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/ Release 1.1.2 of 2021-06-15.
2. Adami, R., Boni, F., Carlone, R., Tentarelli, L., Ground states for the planar NLSE with a point defect as minimizers of the constrained energy. arXiv:2109.09482.
3. Adami, R., Carlone, R., Correggi, M., Tentarelli, L., Blow-up for the pointwise NLS in dimension two: absence of critical power. J. Differ. Equ. 269 (2020), 1–37.
4. Adami, R., Carlone, R., Correggi, M., Tentarelli, L., Stability of the standing waves of the concentrated NLSE in dimension two. Math. Eng., 3, 2021, 15 Paper No. 011.
5. 2 supercritical point NLS. Trans. Am. Math. Soc. 374 (2021), 35–60.
6. Adami, R., Noja, D., Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect. J. Phys. A, 42, 2009, 495302 19.
7. Adami, R., Noja, D., Ortoleva, C., Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three. J. Math. Phys., 54, 2013, 013501 33.
8. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., The low energy expansion in nonrelativistic scattering theory. Ann. Inst. H. Poincaré Sect. A (N. S.) 37 (1982), 1–28.
9. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H., Point interactions in two dimensions: basic properties, approximations and applications to solid state physics. J. Reine Angew. Math. 380 (1987), 87–107.
10. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H., Solvable Models in Quantum Mechanics. second ed., 2005, AMS Chelsea Publishing, Providence, RI with an appendix by Pavel Exner.
11. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Kirsch, W., On point interactions in one dimension. J. Oper. Theory 12 (1984), 101–126.
12. Albeverio, S., Høegh-Krohn, R., Point interactions as limits of short range interactions. J. Oper. Theory 6 (1981), 313–339.
13. Almgren, F.J. Jr., Lieb, E.H., Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 2 (1989), 683–773.
14. Berestycki, H., Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82 (1983), 313–345.
15. Berestycki, H., Cazenave, T., Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci., Sér. 1 Math. 293 (1981), 489–492.
16. Berezin, F.A., Faddeev, L.D., Remark on the Schrödinger equation with singular potential. Dokl. Akad. Nauk SSSR 137 (1961), 1011–1014.
17. Brézis, H., Lieb, E., A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88 (1983), 486–490.
18. Brothers, J.E., Ziemer, W.P., Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384 (1988), 153–179.
19. Buslaev, V.S., Komech, A.I., Kopylova, E.A., Stuart, D., On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator. Commun. Partial Differ. Equ. 33 (2008), 669–705.
20. Cacciapuoti, C., Finco, D., Noja, D., Well posedness of the nonlinear Schrödinger equation with isolated singularities. J. Differ. Equ. 305 (2021), 288–318.
21. Caspers, W., Clément, P., A different approach to singular solutions. Differ. Integral Equ. 7 (1994), 1227–1240.
22. Cazenave, T., Lions, P.-L., Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85 (1982), 549–561.
23. Cazenave, T., Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10, 2003, New York University, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence, RI.
24. Cazenave, T., An Introduction to Semilinear Elliptic Equations. second ed., 2018, Editora do IM-UFRJ, Rio de Janeiro.
25. p-boundedness of wave operators. Rev. Math. Phys., 31, 2019, 1950012 32.
26. Cuccagna, S., Maeda, M., On stability of small solitons of the 1-D NLS with a trapping delta potential. SIAM J. Math. Anal. 51 (2019), 4311–4331.
27. D'Ancona, P., Pierfelice, V., Teta, A., Dispersive estimate for the Schrödinger equation with point interactions. Math. Methods Appl. Sci. 29 (2006), 309–323.
28. p-boundedness of wave operators for the three-dimensional multi-centre point interaction. Ann. Henri Poincaré 19 (2018), 283–322.
29. Duchêne, V., Marzuola, J.L., Weinstein, M.I., Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications. J. Math. Phys., 52, 2011, 013505 17.
30. 2-critical nonlinear Schrödinger equations with attractive inverse-power potential. RIMS Kôkyûroku Bessatsu, 2022 in press.
31. Fukaya, N., Hayashi, M., Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities. Transl. Am. Math. Soc. 374 (2021), 1421–1447.
32. Fukaya, N., Ohta, M., Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential. Osaka J. Math. 56 (2019), 713–726.
33. Fukuizumi, R., Stability of standing waves for nonlinear Schrödinger equations with critical power nonlinearity and potentials. Adv. Differ. Equ. 10 (2005), 259–276.
34. Fukuizumi, R., Jeanjean, L., Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. Discrete Contin. Dyn. Syst. 21 (2008), 121–136.
35. Fukuizumi, R., Ohta, M., Instability of standing waves for nonlinear Schrödinger equations with potentials. Differ. Integral Equ. 16 (2003), 691–706.
36. Fukuizumi, R., Ohta, M., Stability of standing waves for nonlinear Schrödinger equations with potentials. Differ. Integral Equ. 16 (2003), 111–128.
37. Fukuizumi, R., Ohta, M., Ozawa, T., Nonlinear Schrödinger equation with a point defect. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25 (2008), 837–845.
38. Georgiev, V., Michelangeli, A., Scandone, R., On fractional powers of singular perturbations of the Laplacian. J. Funct. Anal. 275 (2018), 1551–1602.
39. Goodman, R.H., Holmes, P.J., Weinstein, M.I., Strong NLS soliton-defect interactions. Physica D 192 (2004), 215–248.
40. Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), 160–197.
41. 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, 3559–3584.
42. Holmer, J., Marzuola, J., Zworski, M., Fast soliton scattering by delta impurities. Commun. Math. Phys. 274 (2007), 187–216.
43. Ikeda, M., Inui, T., Global dynamics below the standing waves for the focusing semilinear Schrödinger equation with a repulsive Dirac delta potential. Anal. PDE 10 (2017), 481–512.
44. Kaminaga, M., Ohta, M., Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity. Saitama Math. J. 26:2009 (2010), 39–48.
45. Komech, A., Kopylova, E., Stuart, D., On asymptotic stability of solitons in a nonlinear Schrödinger equation. Commun. Pure Appl. Anal. 11 (2012), 1063–1079.
46. n. Arch. Ration. Mech. Anal. 105 (1989), 243–266.
47. Le Coz, S., Standing Waves in Nonlinear Schrödinger Equations, Analytical and Numerical Aspects of Partial Differential Equations. 2009, Walter de Gruyter, Berlin, 151–192.
48. Le Coz, S., Fukuizumi, R., Fibich, G., Ksherim, B., Sivan, Y., Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Physica D 237 (2008), 1103–1128.
49. Masaki, S., Murphy, J., Segata, J.-i., Asymptotic stability of solitary waves for the 1d NLS with an attractive delta potential. arXiv:2008.11645, August 2020.
50. Masaki, S., Murphy, J., Segata, J.-i., Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential. Anal. PDE 13 (2020), 1099–1128.
51. Michelangeli, A., Ottolini, A., Scandone, R., Fractional powers and singular perturbations of quantum differential Hamiltonians. J. Math. Phys., 59, 2018, 072106 27.
52. Michelangeli, A., Olgiati, A., Scandone, R., Singular Hartree equation in fractional perturbed Sobolev spaces. J. Nonlinear Math. Phys. 25 (2018), 558–588.
53. Michelangeli, A., Scandone, R., Point-like perturbed fractional Laplacians through shrinking potentials of finite range. Complex Anal. Oper. Theory 13 (2019), 3717–3752.
54. Ohta, M., Instability of solitary waves for nonlinear Schrödinger equations of derivative type. SUT J. Math. 50 (2014), 399–415.
55. Ohta, M., Yamaguchi, T., Strong instability of standing waves for nonlinear Schrödinger equations with a delta potential. Harmonic Analysis and Nonlinear Partial Differential Equations RIMS Kôkyûroku Bessatsu, B56, Res. Inst. Math. Sci., 2016, RIMS, Kyoto, 79–92.
56. Okazawa, N., Suzuki, T., Yokota, T., Energy methods for abstract nonlinear Schrödinger equations. Evol. Equ. Control Theory 1 (2012), 337–354.
57. Rose, H.A., Weinstein, M.I., On the bound states of the nonlinear Schrödinger equation with a linear potential. Physica D 30 (1988), 207–218.
58. Shatah, J., Stable standing waves of nonlinear Klein-Gordon equations. Commun. Math. Phys. 91 (1983), 313–327.
59. Shatah, J., Strauss, W., Instability of nonlinear bound states. Commun. Math. Phys. 100 (1985), 173–190.
60. Suzuki, T., Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Discrete Contin. Dyn. Syst., 2015, 1019–1024.
61. Vakhitov, N.G., Kolokolov, A.A., Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16 (1973), 783–789.
62. Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87 (1982/83), 567–576.
63. Weinstein, M.I., Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39 (1986), 51–67.
64. p-boundedness of wave operators for 2D Schrödinger operators with point interactions. Ann. Henri Poincaré 22 (2021), 2065–2101.