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Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation
1. | Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), Recife, Pernambuco (PE) 50740-545, Brazil |
2. | Instituto de Matemática, Universidade Federal de Alagoas (UFAL), Maceió, Alagoas (AL), Brazil |
3. | Departamento de Matematicas y Estadística, Universidad Nacional de Colombia (UNAL), Cra 27 No. 64-60, 170003, Manizales, Colombia |
In a recent article [
References:
[1] |
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. |
[2] |
R. Adami, C. Cacciapuoti, D. Finco and D. Noja,
Stable standing waves for a NLS graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.
doi: 10.1016/j.jde.2016.01.029. |
[3] |
K. Ammari and E. Crépeau,
Feedback stabilization and boundary controllability of the Korteweg–de Vries equation on a star-shaped network, SIAM J. Control Optim., 56 (2018), 1620-1639.
doi: 10.1137/17M113959X. |
[4] |
K. Ammari and E. Crépeau,
Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 127 (2019), 39-43.
doi: 10.1016/j.sysconle.2019.03.005. |
[5] |
J. Angulo Pava and N. Goloshchapova,
On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discrete Contin. Dyn. Syst. A., 38 (2018), 5039-5066.
doi: 10.3934/dcds.2018221. |
[6] |
J. Angulo Pava and N. Goloshchapova,
Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, Adv. Differential Equations, 23 (2018), 793-846.
|
[7] |
L. Baudouin and M. Yamamoto,
Inverse problem on a tree-shaped network: Unified approach for uniqueness, Applicable Analysis, 94 (2015), 2370-2395.
doi: 10.1080/00036811.2014.985214. |
[8] |
M. Ben-Artzi, H. Koch and J.-C Saut,
Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math, 330 (2000), 87-92.
doi: 10.1016/S0764-4442(00)00120-8. |
[9] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013.
doi: 10.1090/surv/186. |
[10] |
J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008. |
[11] |
J. L. Bona and R. C. Cascaval,
Nonlinear dispersive waves on trees, Can. Appl. Math. Q., 16 (2008), 1-18.
|
[12] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[13] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part II: The KDV-equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[14] |
R. Burioni, D. Cassi, M. Rasetti, P. Sodano and A. Vezzani,
Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34 (2001), 4697-4710.
|
[15] |
R. A. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic Schrödinger equation, Appl. Math. Optim., (2019).
doi: 10.1007/s00245-019-09640-8. |
[16] |
R. de A. Capistrano-Filho, M. Cavalcante and F. A. Gallego,
Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, Pacific Journal of Mathematics, 309 (2020), 35-70.
doi: 10.2140/pjm.2020.309.35. |
[17] |
M. Cavalcante,
The initial-boundary-value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential and Integral Equations, 30 (2017), 521-554.
|
[18] |
M. Cavalcante, The Korteweg–de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69 (2018), Paper No. 124, 22 pp.
doi: 10.1007/s00033-018-1018-6. |
[19] |
M. Cavalcante and A. J. Corcho, The initial boundary value problem for the Schrödinger–Korteweg–de Vries system on the half-line, Communications in Contemporary Mathematics, 21 (2019), 1850066, 47 pp.
doi: 10.1142/S0219199718500669. |
[20] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, in Oxford Science Publications, Oxford University Press, 1998.
![]() ![]() |
[21] |
E. Cerpa, E. Crépeau and C. Moreno,
On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network, IMA Journal of Mathematical Control and Information, 37 (2020), 226-240.
doi: 10.1093/imamci/dny047. |
[22] |
E. Cerpa, E. Crépeau and J. Valein,
Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network, Evol. Equ. Control Theory, 9 (2020), 673-692.
doi: 10.3934/eect.2020028. |
[23] |
J. E. Colliander and C. E. Kenig,
The generalized Korteweg-de Vries equation on the half-line, Comm. Partial Differential Equations, 27 (2002), 2187-2266.
doi: 10.1081/PDE-120016157. |
[24] |
S. Cui and C. Guo,
Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(R^n)$ and applications, Nonlinear Analysis, 67 (2007), 687-707.
doi: 10.1016/j.na.2006.06.020. |
[25] |
G. Fibich, B. Ilan and G. Papanicolaou,
Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.
doi: 10.1137/S0036139901387241. |
[26] |
M. Freedman, L. Lovász and A. Schrijver,
Reflection positivity, rank connectivity, and homomorphism of graphs, Journal of the American Mathematical Society, 20 (2007), 37-51.
doi: 10.1090/S0894-0347-06-00529-7. |
[27] |
J. Ginibre, Y. Tsutsumi and G. Velo,
On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.
doi: 10.1006/jfan.1997.3148. |
[28] |
F. Gregorio and D. Mugnolo,
Bi-Laplacians on graphs and networks, J. Evol. Equ., 20 (2020), 191-232.
doi: 10.1007/s00028-019-00523-7. |
[29] |
J. Holmer,
The initial-boundary value problem for the 1d nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668.
|
[30] |
J. Holmer,
The initial-boundary value problem for the Korteweg-de Vries equation, Comm. in Partial Differential Equations, 31 (2006), 1151-1190.
doi: 10.1080/03605300600718503. |
[31] |
L. I. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the Schrödinger equation on a tree, Inverse Problems, 28 (2012), 015011, 30 pp.
doi: 10.1088/0266-5611/28/1/015011. |
[32] |
V. I. Karpman,
Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1103/PhysRevE.53.R1336. |
[33] |
V. I. Karpman and A. G. Shagalov,
Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[34] |
P. Kuchment,
Quantum graphs, I. Some basic structures, Waves Random Media, 14 (2004), 107-128.
doi: 10.1088/0959-7174/14/1/014. |
[35] |
D. Mugnolo, Mathematical Technology of Networks, Springer Proceedings in Mathematics & Statistics, Bielefeld, 128, 2015.
doi: 10.1007/978-3-319-16619-3. |
[36] |
D. Mugnolo, D. Noja and C. Seifert,
Airy-type evolution equations on star graphs, Analysis & PDE, 11 (2018), 1625-1652.
doi: 10.2140/apde.2018.11.1625. |
[37] |
D. Mugnolo and J.-F. Rault,
Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 415-436.
doi: 10.36045/bbms/1407765881. |
[38] |
B. Pausader,
Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.
doi: 10.4310/DPDE.2007.v4.n3.a1. |
[39] |
B. Pausader,
The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.
doi: 10.1016/j.jfa.2008.11.009. |
[40] |
T. Tao, Nonlinear Dispersive Equations : Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol. 106, 2006.
doi: 10.1090/cbms/106. |
[41] |
T. Tsutsumi,
Strichartz estimates for Schrödinger equation of fourth order with periodic boundary condition, Kyoto University, 1994 (2014), 104-113.
|
[42] |
O. Tadahiro and N. Tzvetkov,
Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Relat. Fields, 169 (2016), 1121-1168.
doi: 10.1007/s00440-016-0748-7. |
show all references
References:
[1] |
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. |
[2] |
R. Adami, C. Cacciapuoti, D. Finco and D. Noja,
Stable standing waves for a NLS graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.
doi: 10.1016/j.jde.2016.01.029. |
[3] |
K. Ammari and E. Crépeau,
Feedback stabilization and boundary controllability of the Korteweg–de Vries equation on a star-shaped network, SIAM J. Control Optim., 56 (2018), 1620-1639.
doi: 10.1137/17M113959X. |
[4] |
K. Ammari and E. Crépeau,
Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 127 (2019), 39-43.
doi: 10.1016/j.sysconle.2019.03.005. |
[5] |
J. Angulo Pava and N. Goloshchapova,
On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discrete Contin. Dyn. Syst. A., 38 (2018), 5039-5066.
doi: 10.3934/dcds.2018221. |
[6] |
J. Angulo Pava and N. Goloshchapova,
Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, Adv. Differential Equations, 23 (2018), 793-846.
|
[7] |
L. Baudouin and M. Yamamoto,
Inverse problem on a tree-shaped network: Unified approach for uniqueness, Applicable Analysis, 94 (2015), 2370-2395.
doi: 10.1080/00036811.2014.985214. |
[8] |
M. Ben-Artzi, H. Koch and J.-C Saut,
Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math, 330 (2000), 87-92.
doi: 10.1016/S0764-4442(00)00120-8. |
[9] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013.
doi: 10.1090/surv/186. |
[10] |
J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008. |
[11] |
J. L. Bona and R. C. Cascaval,
Nonlinear dispersive waves on trees, Can. Appl. Math. Q., 16 (2008), 1-18.
|
[12] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
doi: 10.1007/BF01896020. |
[13] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part II: The KDV-equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[14] |
R. Burioni, D. Cassi, M. Rasetti, P. Sodano and A. Vezzani,
Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34 (2001), 4697-4710.
|
[15] |
R. A. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic Schrödinger equation, Appl. Math. Optim., (2019).
doi: 10.1007/s00245-019-09640-8. |
[16] |
R. de A. Capistrano-Filho, M. Cavalcante and F. A. Gallego,
Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, Pacific Journal of Mathematics, 309 (2020), 35-70.
doi: 10.2140/pjm.2020.309.35. |
[17] |
M. Cavalcante,
The initial-boundary-value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential and Integral Equations, 30 (2017), 521-554.
|
[18] |
M. Cavalcante, The Korteweg–de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69 (2018), Paper No. 124, 22 pp.
doi: 10.1007/s00033-018-1018-6. |
[19] |
M. Cavalcante and A. J. Corcho, The initial boundary value problem for the Schrödinger–Korteweg–de Vries system on the half-line, Communications in Contemporary Mathematics, 21 (2019), 1850066, 47 pp.
doi: 10.1142/S0219199718500669. |
[20] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, in Oxford Science Publications, Oxford University Press, 1998.
![]() ![]() |
[21] |
E. Cerpa, E. Crépeau and C. Moreno,
On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network, IMA Journal of Mathematical Control and Information, 37 (2020), 226-240.
doi: 10.1093/imamci/dny047. |
[22] |
E. Cerpa, E. Crépeau and J. Valein,
Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network, Evol. Equ. Control Theory, 9 (2020), 673-692.
doi: 10.3934/eect.2020028. |
[23] |
J. E. Colliander and C. E. Kenig,
The generalized Korteweg-de Vries equation on the half-line, Comm. Partial Differential Equations, 27 (2002), 2187-2266.
doi: 10.1081/PDE-120016157. |
[24] |
S. Cui and C. Guo,
Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(R^n)$ and applications, Nonlinear Analysis, 67 (2007), 687-707.
doi: 10.1016/j.na.2006.06.020. |
[25] |
G. Fibich, B. Ilan and G. Papanicolaou,
Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.
doi: 10.1137/S0036139901387241. |
[26] |
M. Freedman, L. Lovász and A. Schrijver,
Reflection positivity, rank connectivity, and homomorphism of graphs, Journal of the American Mathematical Society, 20 (2007), 37-51.
doi: 10.1090/S0894-0347-06-00529-7. |
[27] |
J. Ginibre, Y. Tsutsumi and G. Velo,
On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.
doi: 10.1006/jfan.1997.3148. |
[28] |
F. Gregorio and D. Mugnolo,
Bi-Laplacians on graphs and networks, J. Evol. Equ., 20 (2020), 191-232.
doi: 10.1007/s00028-019-00523-7. |
[29] |
J. Holmer,
The initial-boundary value problem for the 1d nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668.
|
[30] |
J. Holmer,
The initial-boundary value problem for the Korteweg-de Vries equation, Comm. in Partial Differential Equations, 31 (2006), 1151-1190.
doi: 10.1080/03605300600718503. |
[31] |
L. I. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the Schrödinger equation on a tree, Inverse Problems, 28 (2012), 015011, 30 pp.
doi: 10.1088/0266-5611/28/1/015011. |
[32] |
V. I. Karpman,
Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1103/PhysRevE.53.R1336. |
[33] |
V. I. Karpman and A. G. Shagalov,
Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[34] |
P. Kuchment,
Quantum graphs, I. Some basic structures, Waves Random Media, 14 (2004), 107-128.
doi: 10.1088/0959-7174/14/1/014. |
[35] |
D. Mugnolo, Mathematical Technology of Networks, Springer Proceedings in Mathematics & Statistics, Bielefeld, 128, 2015.
doi: 10.1007/978-3-319-16619-3. |
[36] |
D. Mugnolo, D. Noja and C. Seifert,
Airy-type evolution equations on star graphs, Analysis & PDE, 11 (2018), 1625-1652.
doi: 10.2140/apde.2018.11.1625. |
[37] |
D. Mugnolo and J.-F. Rault,
Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 415-436.
doi: 10.36045/bbms/1407765881. |
[38] |
B. Pausader,
Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.
doi: 10.4310/DPDE.2007.v4.n3.a1. |
[39] |
B. Pausader,
The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.
doi: 10.1016/j.jfa.2008.11.009. |
[40] |
T. Tao, Nonlinear Dispersive Equations : Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol. 106, 2006.
doi: 10.1090/cbms/106. |
[41] |
T. Tsutsumi,
Strichartz estimates for Schrödinger equation of fourth order with periodic boundary condition, Kyoto University, 1994 (2014), 104-113.
|
[42] |
O. Tadahiro and N. Tzvetkov,
Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Relat. Fields, 169 (2016), 1121-1168.
doi: 10.1007/s00440-016-0748-7. |

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