American Institute of Mathematical Sciences

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July  2019, 18(4): 2093-2116. doi: 10.3934/cpaa.2019094

On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction

Jaime Angulo Pava 1,, and  César A. Hernández Melo 2,

1. 

Department of Mathematics, IME-USP, Rua do Matão 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil
2. 

Department of Mathematics, DMA-UEM, Av. Colombo, 5790 Jd. Universitário, CEP 87020-900, Maringá, PR, Brazil

* Corresponding author

Received  May 2018 Revised  August 2018 Published  January 2019

Fund Project: The first author is supported by Grant CNPq/Brazil

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schródinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index.

Keywords: Nonlinear stability, standing wave solutions, nonlinear Schródinger equation with point interaction, analytic perturbation theory, self-adjoint extensions.
Mathematics Subject Classification: Primary: 35Q55, 37K40, 37K45; Secondary: 47E05, 35J61.
Citation: Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2093-2116. doi: 10.3934/cpaa.2019094
References:
[1]

R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415. doi: 10.1016/j.jde.2016.01.029. Google Scholar

[2]

R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777. doi: 10.1016/j.jde.2014.07.008. Google Scholar

[3]

R. Adami, D. Noja and N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discr. Cont. Dyn. Syst.- B, 18 (2013), 155-1188. doi: 10.3934/dcdsb.2013.18.1155. Google Scholar

[4]

G. Agrawal, Nonlinear Fiber Optics, Academic Press, 4th edition, 2007.Google Scholar

[5]

S. Albeverio, Z. Brzezniak and L. Dabrowski, Fundamental solution of the heat and Schródinger equations with point interaction, J. Funct. Anal., 130 (1995), 220-254. doi: 10.1006/jfan.1995.1068. Google Scholar

[6]

S. Albeverio, F. Gesztesy, R. Krohn and H. Holden, Solvable Models in Quantum Mechanics, AMS Chelsea publishing, 2004., Google Scholar

[7] S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators, London Mathematical Society Lecture Note Series 271, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511758904. Google Scholar
[8]

J. Angulo, Instability of cnoidal-peak for the NLS-$\delta$ equation, Math. Nachr., 285 (2012), 1572-1602. doi: 10.1002/mana.201100209. Google Scholar

[9]

J. Angulo and A. Hernandez, Stability of standing waves for logarithmic Schródinger equation with attractive delta potential, IUMJ, 67 (2018), 471-494. doi: 10.1512/iumj.2018.67.7273. Google Scholar

[10]

J. Angulo and N. Goloshchapova, On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discr. Cont. Dyn. Syst.- A, 38 (2018), 5039-5066. doi: 10.3934/dcds.2018221. Google Scholar

[11]

J. Angulo and N. Goloshchapova, On the standing waves of the NLS-log equation with point interaction on a star graph, preprint, arXiv: 1803.07194.Google Scholar

[12]

J. Angulo and N. Goloshchapova, Extension theory approach in the stability of the standing waves for NLS equation with point interactions on a star graph, Advances in Differential Equations, 23 (2018), 793-846. Google Scholar

[13]

J. Angulo and N. Goloshchapova, Stability of standing waves for NLS-log equation with $\delta$-interaction, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 27. doi: 10.1007/s00030-017-0451-0. Google Scholar

[14]

J. Angulo, O. Lopes and A. Neves, Instability of travelling waves for weakly coupled KdV systems, Nonlinear Anal., 69 (2008), 1870-1887. doi: 10.1016/j.na.2007.07.039. Google Scholar

[15]

J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874. Google Scholar

[16]

J. Angulo and G. Ponce, The nonlinear Schródinger equation with a periodic $\delta$-interaction, Bull. Braz. Math. Soc., New Series, 44 (2013), 497-551. doi: 10.1007/s00574-013-0024-8. Google Scholar

[17]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun, 219 (2003), 427-433. Google Scholar

[18]

V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Modern Physics Letter - B, 18 (2004), 627-651. Google Scholar

[19]

F. A. Berezin and M. A. Shubin, The Schródinger Equation, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4. Google Scholar

[20]

V. Caudrelier, M. Mintchev and E. Ragoucy, Solving the quantum non-linear Schródinger equation with $\delta$-type impurity, J. Math. Phys., 46 (2005), 042703-1-24. doi: 10.1063/1.1842353. Google Scholar

[21]

T. Cazenave, Semilinear Schródinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003. doi: 10.1090/cln/010. Google Scholar

[22]

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schródinger equation with a Dirac Potential, Phys. D, 237 (2008), 1103-1128. doi: 10.1016/j.physd.2007.12.004. Google Scholar

[23]

K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. PDE., 34 (2009), 1074-1173. doi: 10.1080/03605300903076831. Google Scholar

[24]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in gas of sodium atoms, Phys. Rev. Lett., 74 (1995), 3969-3973. Google Scholar

[25]

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond and V. Skarka, Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett., 110 (2013), 013901.Google Scholar

[26]

E. L. Falcão-Filho, C. B. de Araújo and J. J. Rodrigues Jr., High-order nonlinearities of aqueous colloids containing silver nanoparticles, J. Opt. Soc. Am. - B, 24 (2007), 2948-2956. Google Scholar

[27]

R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schródinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136. doi: 10.3934/dcds.2008.21.121. Google Scholar

[28]

R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schródinger 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

[29]

B. Gaveau and L. S. Schulman, Explicit time-dependent Schródinger propagators, J. Physics A: Math. Gen., 19 (1986), 1833-1846. Google Scholar

[30]

F. Genoud, F. B. Malomed and R. Weishäupl, Stable NLS solitons in a cubic-quintic medium with a delta-function potential, Nonlinear Anal., 133 (2016), 28-50. doi: 10.1016/j.na.2015.11.016. Google Scholar

[31]

B. V. Gisin, R. Driben and B. A. Malomed, Bistable guided solitons in the cubic-quintic medium, J. Optics B: Quantum and Semiclassical Optics, 6 (2004), S259–S264.Google Scholar

[32]

R. H. Goodman, J. Holmes and M. Weinstein, Strong NLS soliton-defect interactions, Phys. D, 192 (2004), 215-248. doi: 10.1016/j.physd.2004.01.021. Google Scholar

[33]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, Ⅰ, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. Google Scholar

[34]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, Ⅱ, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E. Google Scholar

[35]

D. Henry, J. Perez and W. Wreszinski, Stability theory for solitary-wave solutions of scalar field equation, Comm. Math. Phys., 85 (1982), 351-361. Google Scholar

[36]

J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys., 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z. Google Scholar

[37]

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. Google Scholar

[38]

T. Kato, Perturbation Theory for Linear Operators, 2$^{nd}$ edition, Springer-Verlang, New York, 1976., Google Scholar

[39]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2. Google Scholar

[40]

S. Le Coz, Y. Martel and P. Raphael, Minimal mass blow up solutions for a double power nonlinear Schródinger equation, preprint, arXiv: 1406.6002. doi: 10.4171/RMI/899. Google Scholar

[41]

M. Maeda, Stability and instability of standing waves for 1-dimensional nonlinear Schródinger equation with multiple-power nonlinearity, Kodai Math. J., 31 (2008), 263-271. doi: 10.2996/kmj/1214442798. Google Scholar

[42]

C. R. Menyuk, Soliton robustness in optical fibers, J. Opt. Soc. Am. - B, 10 (1993), 1585-1591Google Scholar

[43]

J. Moloney and A. Newell, Nonlinear Optics, CRC, Taylor & Francis Group, Boca Raton. FL. USA, 2018.,Google Scholar

[44]

M. A. Naimark, Linear Differential Operators, F. Ungar Pub. Co., New York, 1967., Google Scholar

[45]

M. Ohta, Stability and Instability of standing waves for one dimensional nonlinear Schródinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74. doi: 10.2996/kmj/1138043354. Google Scholar

[46]

M. Ohta, Instability of bound states for abstract nonlinear Schródinger equations, J. Funct. Anal., 261 (2011), 90-110. doi: 10.1016/j.jfa.2011.03.010. Google Scholar

[47]

P. Papagiannis, Y. Kominis and K. Hizanidis, Power-and momentum-dependent soliton dynamics in lattices with longitudinal modulation, Phys. Rev. A, 84 (2011), 013820Google Scholar

[48]

S. Reed and B. Simon, Methods of Modern Mathematical Physics: Analysis of Operators, Academic Press, vol. Ⅳ, 1978. Google Scholar

[49]

H. Sakaguchi and M. Tamura, Scattering and trapping of nonlinear Schródinger solitons in external potentials, J. Phys. Soc. Japan, 73 (2004), 503-506. doi: 10.1143/JPSJ.73.503. Google Scholar

[50]

B. T. Seaman, L. D. Car and M. J. Holland, Effect of a potential step or impurity on the Bose-Einstein condensate mean field, Phys. Rev. A, 71 (2005), 033609.Google Scholar

show all references

References:
[1]

R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415. doi: 10.1016/j.jde.2016.01.029. Google Scholar

[2]

R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777. doi: 10.1016/j.jde.2014.07.008. Google Scholar

[3]

R. Adami, D. Noja and N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discr. Cont. Dyn. Syst.- B, 18 (2013), 155-1188. doi: 10.3934/dcdsb.2013.18.1155. Google Scholar

[4]

G. Agrawal, Nonlinear Fiber Optics, Academic Press, 4th edition, 2007.Google Scholar

[5]

S. Albeverio, Z. Brzezniak and L. Dabrowski, Fundamental solution of the heat and Schródinger equations with point interaction, J. Funct. Anal., 130 (1995), 220-254. doi: 10.1006/jfan.1995.1068. Google Scholar

[6]

S. Albeverio, F. Gesztesy, R. Krohn and H. Holden, Solvable Models in Quantum Mechanics, AMS Chelsea publishing, 2004., Google Scholar

[7] S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators, London Mathematical Society Lecture Note Series 271, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511758904. Google Scholar
[8]

J. Angulo, Instability of cnoidal-peak for the NLS-$\delta$ equation, Math. Nachr., 285 (2012), 1572-1602. doi: 10.1002/mana.201100209. Google Scholar

[9]

J. Angulo and A. Hernandez, Stability of standing waves for logarithmic Schródinger equation with attractive delta potential, IUMJ, 67 (2018), 471-494. doi: 10.1512/iumj.2018.67.7273. Google Scholar

[10]

J. Angulo and N. Goloshchapova, On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discr. Cont. Dyn. Syst.- A, 38 (2018), 5039-5066. doi: 10.3934/dcds.2018221. Google Scholar

[11]

J. Angulo and N. Goloshchapova, On the standing waves of the NLS-log equation with point interaction on a star graph, preprint, arXiv: 1803.07194.Google Scholar

[12]

J. Angulo and N. Goloshchapova, Extension theory approach in the stability of the standing waves for NLS equation with point interactions on a star graph, Advances in Differential Equations, 23 (2018), 793-846. Google Scholar

[13]

J. Angulo and N. Goloshchapova, Stability of standing waves for NLS-log equation with $\delta$-interaction, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 27. doi: 10.1007/s00030-017-0451-0. Google Scholar

[14]

J. Angulo, O. Lopes and A. Neves, Instability of travelling waves for weakly coupled KdV systems, Nonlinear Anal., 69 (2008), 1870-1887. doi: 10.1016/j.na.2007.07.039. Google Scholar

[15]

J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874. Google Scholar

[16]

J. Angulo and G. Ponce, The nonlinear Schródinger equation with a periodic $\delta$-interaction, Bull. Braz. Math. Soc., New Series, 44 (2013), 497-551. doi: 10.1007/s00574-013-0024-8. Google Scholar

[17]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun, 219 (2003), 427-433. Google Scholar

[18]

V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Modern Physics Letter - B, 18 (2004), 627-651. Google Scholar

[19]

F. A. Berezin and M. A. Shubin, The Schródinger Equation, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4. Google Scholar

[20]

V. Caudrelier, M. Mintchev and E. Ragoucy, Solving the quantum non-linear Schródinger equation with $\delta$-type impurity, J. Math. Phys., 46 (2005), 042703-1-24. doi: 10.1063/1.1842353. Google Scholar

[21]

T. Cazenave, Semilinear Schródinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003. doi: 10.1090/cln/010. Google Scholar

[22]

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schródinger equation with a Dirac Potential, Phys. D, 237 (2008), 1103-1128. doi: 10.1016/j.physd.2007.12.004. Google Scholar

[23]

K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. PDE., 34 (2009), 1074-1173. doi: 10.1080/03605300903076831. Google Scholar

[24]

K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in gas of sodium atoms, Phys. Rev. Lett., 74 (1995), 3969-3973. Google Scholar

[25]

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond and V. Skarka, Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett., 110 (2013), 013901.Google Scholar

[26]

E. L. Falcão-Filho, C. B. de Araújo and J. J. Rodrigues Jr., High-order nonlinearities of aqueous colloids containing silver nanoparticles, J. Opt. Soc. Am. - B, 24 (2007), 2948-2956. Google Scholar

[27]

R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schródinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136. doi: 10.3934/dcds.2008.21.121. Google Scholar

[28]

R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schródinger 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

[29]

B. Gaveau and L. S. Schulman, Explicit time-dependent Schródinger propagators, J. Physics A: Math. Gen., 19 (1986), 1833-1846. Google Scholar

[30]

F. Genoud, F. B. Malomed and R. Weishäupl, Stable NLS solitons in a cubic-quintic medium with a delta-function potential, Nonlinear Anal., 133 (2016), 28-50. doi: 10.1016/j.na.2015.11.016. Google Scholar

[31]

B. V. Gisin, R. Driben and B. A. Malomed, Bistable guided solitons in the cubic-quintic medium, J. Optics B: Quantum and Semiclassical Optics, 6 (2004), S259–S264.Google Scholar

[32]

R. H. Goodman, J. Holmes and M. Weinstein, Strong NLS soliton-defect interactions, Phys. D, 192 (2004), 215-248. doi: 10.1016/j.physd.2004.01.021. Google Scholar

[33]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, Ⅰ, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. Google Scholar

[34]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, Ⅱ, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E. Google Scholar

[35]

D. Henry, J. Perez and W. Wreszinski, Stability theory for solitary-wave solutions of scalar field equation, Comm. Math. Phys., 85 (1982), 351-361. Google Scholar

[36]

J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys., 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z. Google Scholar

[37]

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. Google Scholar

[38]

T. Kato, Perturbation Theory for Linear Operators, 2$^{nd}$ edition, Springer-Verlang, New York, 1976., Google Scholar

[39]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2. Google Scholar

[40]

S. Le Coz, Y. Martel and P. Raphael, Minimal mass blow up solutions for a double power nonlinear Schródinger equation, preprint, arXiv: 1406.6002. doi: 10.4171/RMI/899. Google Scholar

[41]

M. Maeda, Stability and instability of standing waves for 1-dimensional nonlinear Schródinger equation with multiple-power nonlinearity, Kodai Math. J., 31 (2008), 263-271. doi: 10.2996/kmj/1214442798. Google Scholar

[42]

C. R. Menyuk, Soliton robustness in optical fibers, J. Opt. Soc. Am. - B, 10 (1993), 1585-1591Google Scholar

[43]

J. Moloney and A. Newell, Nonlinear Optics, CRC, Taylor & Francis Group, Boca Raton. FL. USA, 2018.,Google Scholar

[44]

M. A. Naimark, Linear Differential Operators, F. Ungar Pub. Co., New York, 1967., Google Scholar

[45]

M. Ohta, Stability and Instability of standing waves for one dimensional nonlinear Schródinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74. doi: 10.2996/kmj/1138043354. Google Scholar

[46]

M. Ohta, Instability of bound states for abstract nonlinear Schródinger equations, J. Funct. Anal., 261 (2011), 90-110. doi: 10.1016/j.jfa.2011.03.010. Google Scholar

[47]

P. Papagiannis, Y. Kominis and K. Hizanidis, Power-and momentum-dependent soliton dynamics in lattices with longitudinal modulation, Phys. Rev. A, 84 (2011), 013820Google Scholar

[48]

S. Reed and B. Simon, Methods of Modern Mathematical Physics: Analysis of Operators, Academic Press, vol. Ⅳ, 1978. Google Scholar

[49]

H. Sakaguchi and M. Tamura, Scattering and trapping of nonlinear Schródinger solitons in external potentials, J. Phys. Soc. Japan, 73 (2004), 503-506. doi: 10.1143/JPSJ.73.503. Google Scholar

[50]

B. T. Seaman, L. D. Car and M. J. Holland, Effect of a potential step or impurity on the Bose-Einstein condensate mean field, Phys. Rev. A, 71 (2005), 033609.Google Scholar

Figure 1.  The peak solutions $ \phi_{\omega, Z} $ for $ Z<0 $ and $ Z>0 $
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Figure 2.  Function $ \omega\to-||\phi_{\omega, Z}||^2 $ for specific values of $ Z $. Left picture, $ \omega\in(-6, -1) $, $ Z = -0.86 $. Right picture, $ \omega\in(-6, -1) $, $ Z = -0.9 $
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Figure 3.  Function $ (\omega, Z)\to-\partial_\omega||\phi_{\omega, Z}||^2 $, for $ \omega\in(-50, -2) $. Left picture for $ Z\in(-0.9, -0.8) $. Right picture for $ Z\in(-0.8, -0.7) $
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Figure 4.  Function $ \omega\to-\partial_\omega||\phi_{\omega, Z}||^2 $ for specific values of $ Z $. Left picture, $ Z = -0.86602 $ and $ \omega\in(-10, -0.8) $. Right picture, $ Z = -0.86603 $ and $ \omega\in(-5, -1) $
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Figure 5.  The function $ \omega\to-||\phi_{\omega, Z}||^2 $, for $ \lambda_1 = 2 $, $ \lambda_2 = -1 $ and $ \omega\in (-\frac34, -\frac14) $. Left picture, $ Z = -1 $. Right picture, $ Z = 1 $
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Figure 6.  The function $ \omega\to-||\phi_{\omega, Z}||^2 $, for $ \lambda_1 = 4 $, $ \lambda_2 = -2 $ and $ \omega\in (-\frac32, -\frac14) $. Left picture, $ Z = -1 $. Right picture, $ Z = 1 $
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