# American Institute of Mathematical Sciences

June  2017, 6(2): 155-175. doi: 10.3934/eect.2017009

## Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction

 Department of Mathematics, IME-USP, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil

Received  July 2016 Revised  February 2017 Published  April 2017

In this paper we study the one-dimensional logarithmic Schrödin-\break ger equation perturbed by an attractive
 $δ^{\prime}$
-interaction
 $i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log|}}u|2 = 0,(x,t) \in \mathbb{R} \times \mathbb{R} ,$
where $γ>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $δ^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0 < γ≤ 2$, then there is a single ground state and it is an odd function; if $γ>2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.
Citation: Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
##### References:
 [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. doi: 10.1088/1751-8113/42/49/495302. Google Scholar [2] R. Adami and D. Noja, Nonlinearity-defect interaction: Symmetry breaking bifurcation in a NLS with δ' impurity, Nanosystems, 2 (2011), 5-19. Google Scholar [3] R. Adami and D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ' interaction, Comm. Math. Phys., 318 (2013), 247-289. doi: 10.1007/s00220-012-1597-6. Google Scholar [4] R. Adami and D. Noja, Exactly solvable models and bifurcations: The case of the cubic NLS with a δ or a δ' interaction in dimension one, Math. Model. Nat. Phenom., 9 (2014), 1-16. doi: 10.1051/mmnp/20149501. Google Scholar [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. doi: 10.3934/dcdsb.2013.18.1155. Google Scholar [6] S. Albeverio, F. Gesztesy, R. H∅egh-Krohn and H. Holden, Solvable Models in Quantum Mechanics Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-88201-2. Google Scholar [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.Google Scholar [8] A.H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, 335 (2016), 1-9. Google Scholar [9] I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys, 100 (1976), 62-93. doi: 10.1016/0003-4916(76)90057-9. Google Scholar [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. doi: 10.2307/2044999. Google Scholar [11] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal., T.M.A., 7 (1983), 1127-1140. doi: 10.1016/0362-546X(83)90022-6. Google Scholar [12] T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010. Google Scholar [13] T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504. Google Scholar [14] R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive {D}irac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136. doi: 10.3934/dcds.2008.21.121. Google Scholar [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. doi: 10.1016/j.anihpc.2007.03.004. Google Scholar [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. doi: 10.1007/s10955-011-0356-y. Google Scholar [17] A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981. Google Scholar [18] E. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev, 32 (1985), 1201-1204. doi: 10.1103/PhysRevA.32.1201. Google Scholar [19] R.K. Jackson and M. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a {G}ross-{P}itaevskii equation, J. Stat. Phys., 116 (2004), 881-905. doi: 10.1023/B:JOSS.0000037238.94034.75. Google Scholar [20] M. Kaminaga and M. Ohta, Stability of standing waves for nonlinear {S}chrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00220-011-1361-3. Google Scholar [23] A. Kostenko and M. Malamud, Spectral theory of semibounded Schrödinger operators with $δ^{\prime}$-interactions, Ann. Henri Poincaré, 15 (2014), 501-541. doi: 10.1007/s00023-013-0245-9. Google Scholar [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. doi: 10.1016/j.physd.2007.12.004. Google Scholar [25] E. Lieb and M. Loss, Analysis 2nd edition, vol. ~14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar [26] A. ~Sacchetti, Universal critical power for nonlinear Schrödinger equations with symmetric double well potential Phys. Rev. Lett. 103 (2009), 194101. doi: 10.1103/PhysRevLett.103.194101. Google Scholar [27] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1. Google Scholar [28] J. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, 12 (1984), 191-202. doi: 10.1007/BF01449041. Google Scholar [29] K. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: {O}rigin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297. doi: 10.1134/S0202289310040067. Google Scholar

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##### References:
 [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. doi: 10.1088/1751-8113/42/49/495302. Google Scholar [2] R. Adami and D. Noja, Nonlinearity-defect interaction: Symmetry breaking bifurcation in a NLS with δ' impurity, Nanosystems, 2 (2011), 5-19. Google Scholar [3] R. Adami and D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ' interaction, Comm. Math. Phys., 318 (2013), 247-289. doi: 10.1007/s00220-012-1597-6. Google Scholar [4] R. Adami and D. Noja, Exactly solvable models and bifurcations: The case of the cubic NLS with a δ or a δ' interaction in dimension one, Math. Model. Nat. Phenom., 9 (2014), 1-16. doi: 10.1051/mmnp/20149501. Google Scholar [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. doi: 10.3934/dcdsb.2013.18.1155. Google Scholar [6] S. Albeverio, F. Gesztesy, R. H∅egh-Krohn and H. Holden, Solvable Models in Quantum Mechanics Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-88201-2. Google Scholar [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.Google Scholar [8] A.H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, 335 (2016), 1-9. Google Scholar [9] I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys, 100 (1976), 62-93. doi: 10.1016/0003-4916(76)90057-9. Google Scholar [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. doi: 10.2307/2044999. Google Scholar [11] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal., T.M.A., 7 (1983), 1127-1140. doi: 10.1016/0362-546X(83)90022-6. Google Scholar [12] T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010. Google Scholar [13] T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504. Google Scholar [14] R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive {D}irac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136. doi: 10.3934/dcds.2008.21.121. Google Scholar [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. doi: 10.1016/j.anihpc.2007.03.004. Google Scholar [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. doi: 10.1007/s10955-011-0356-y. Google Scholar [17] A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981. Google Scholar [18] E. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev, 32 (1985), 1201-1204. doi: 10.1103/PhysRevA.32.1201. Google Scholar [19] R.K. Jackson and M. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a {G}ross-{P}itaevskii equation, J. Stat. Phys., 116 (2004), 881-905. doi: 10.1023/B:JOSS.0000037238.94034.75. Google Scholar [20] M. Kaminaga and M. Ohta, Stability of standing waves for nonlinear {S}chrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00220-011-1361-3. Google Scholar [23] A. Kostenko and M. Malamud, Spectral theory of semibounded Schrödinger operators with $δ^{\prime}$-interactions, Ann. Henri Poincaré, 15 (2014), 501-541. doi: 10.1007/s00023-013-0245-9. Google Scholar [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. doi: 10.1016/j.physd.2007.12.004. Google Scholar [25] E. Lieb and M. Loss, Analysis 2nd edition, vol. ~14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar [26] A. ~Sacchetti, Universal critical power for nonlinear Schrödinger equations with symmetric double well potential Phys. Rev. Lett. 103 (2009), 194101. doi: 10.1103/PhysRevLett.103.194101. Google Scholar [27] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1. Google Scholar [28] J. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, 12 (1984), 191-202. doi: 10.1007/BF01449041. Google Scholar [29] K. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: {O}rigin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297. doi: 10.1134/S0202289310040067. Google Scholar
The graph of the curve $\mathcal{I}_{1}\cup \mathcal{I}_{2}$.
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