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Evolution Equations and Control Theory (EECT)
 

Stability of ground states for logarithmic Schrödinger equation with a $\delta^{'}$-interaction
Pages: 155 - 175, Issue 2, June 2017

doi:10.3934/eect.2017009      Abstract        References        Full text (510.6K)           Related Articles

Alex H. Ardila - Department of Mathematics, IME-USP, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil (email)

1 R. Adami and D. Noja, Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect, J. Phys. A Math. Theor., 42 (2009), 495302, 19pp.       
2 R. Adami and D. Noja, Nonlinearity-defect interaction: Symmetry breaking bifurcation in a NLS with $\delta^{'}$ impurity, Nanosystems, 2 (2011), 5-19.
3 R. Adami and D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a $\delta^{'}$ interaction, Comm. Math. Phys., 318 (2013), 247-289.       
4 R. Adami and D. Noja, Exactly solvable models and bifurcations: The case of the cubic NLS with a $\delta $ or a $\delta'$ interaction in dimension one, Math. Model. Nat. Phenom., 9 (2014), 1-16.       
5 R. Adami, D. Noja and N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1155-1188.       
6 S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New York, 1988.       
7 J. Angulo and A. H. Ardila, Stability of standing waves for logarithmic Schrödinger equation with attractive delta potential, Indiana Univ. Math. J., to appear.
8 A. H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, 335 (2016), 1-9.
9 I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys, 100 (1976), 62-93.       
10 H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.       
11 T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal., T.M.A., 7 (1983), 1127-1140.       
12 T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003.       
13 T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.       
14 R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136.       
15 R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schrödinger equation with a point defect, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 837-845.       
16 R. Fukuizumi and A. Sacchetti, Bifurcation and stability for nonlinear Schrödinger equations with double well potential in the semiclassical limit, J. Stat. Phys., 145 (2011), 1546-1594.       
17 A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions, vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981.       
18 E. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev, 32 (1985), 1201-1204.
19 R. K. Jackson and M. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Stat. Phys., 116 (2004), 881-905.       
20 M. Kaminaga and M. Ohta, Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48.       
21 C. M. Khalique and A. Biswas, Gaussian soliton solution to nonlinear Schrödinger's equation with log law nonlinearity, International Journal of Physical Sciences, 5 (2010), 280-282.
22 E. W. Kirr, P. Kevrekidis and D. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.       
23 A. Kostenko and M. Malamud, Spectral theory of semibounded Schrödinger operators with $\delta^{'}$-interactions, Ann. Henri Poincaré, 15 (2014), 501-541.       
24 S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a dirac potential, Phys. D, 237 (2008), 1103-1128.       
25 E. Lieb and M. Loss, Analysis, 2nd edition, vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.       
26 A. Sacchetti, Universal critical power for nonlinear Schrödinger equations with symmetric double well potential, Phys. Rev. Lett., 103 (2009), 194101.
27 K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht, 2012.       
28 J. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, 12 (1984), 191-202.       
29 K. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297.       

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