doi: 10.3934/dcdsb.2021190
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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

* Corresponding author

Received  October 2020 Revised  July 2021 Early access July 2021

In a recent article [16], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [16] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure $ \mathcal{G} $. Precisely, consider $ \mathcal{G} $ composed by $ N $ edges parameterized by half-lines $ (0,+\infty) $ attached with a common vertex $ \nu $. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.

Citation: Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021190
References:
[1]

R. AdamiC. CacciapuotiD. 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

[2]

R. AdamiC. CacciapuotiD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

M. Ben-ArtziH. 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.  Google Scholar

[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.  Google Scholar

[10]

J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.  Google Scholar

[11]

J. L. Bona and R. C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q., 16 (2008), 1-18.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. BurioniD. CassiM. RasettiP. Sodano and A. Vezzani, Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34 (2001), 4697-4710.   Google Scholar

[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.  Google Scholar

[16]

R. de A. Capistrano-FilhoM. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, in Oxford Science Publications, Oxford University Press, 1998.   Google Scholar
[21]

E. CerpaE. 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.  Google Scholar

[22]

E. CerpaE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[26]

M. FreedmanL. 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.  Google Scholar

[27]

J. GinibreY. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

P. Kuchment, Quantum graphs, I. Some basic structures, Waves Random Media, 14 (2004), 107-128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar

[35]

D. Mugnolo, Mathematical Technology of Networks, Springer Proceedings in Mathematics & Statistics, Bielefeld, 128, 2015. doi: 10.1007/978-3-319-16619-3.  Google Scholar

[36]

D. MugnoloD. Noja and C. Seifert, Airy-type evolution equations on star graphs, Analysis & PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[40]

T. Tao, Nonlinear Dispersive Equations : Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol. 106, 2006. doi: 10.1090/cbms/106.  Google Scholar

[41]

T. Tsutsumi, Strichartz estimates for Schrödinger equation of fourth order with periodic boundary condition, Kyoto University, 1994 (2014), 104-113.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. AdamiC. CacciapuotiD. 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

[2]

R. AdamiC. CacciapuotiD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

M. Ben-ArtziH. 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.  Google Scholar

[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.  Google Scholar

[10]

J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.  Google Scholar

[11]

J. L. Bona and R. C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q., 16 (2008), 1-18.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. BurioniD. CassiM. RasettiP. Sodano and A. Vezzani, Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34 (2001), 4697-4710.   Google Scholar

[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.  Google Scholar

[16]

R. de A. Capistrano-FilhoM. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, in Oxford Science Publications, Oxford University Press, 1998.   Google Scholar
[21]

E. CerpaE. 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.  Google Scholar

[22]

E. CerpaE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.  Google Scholar

[26]

M. FreedmanL. 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.  Google Scholar

[27]

J. GinibreY. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

P. Kuchment, Quantum graphs, I. Some basic structures, Waves Random Media, 14 (2004), 107-128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar

[35]

D. Mugnolo, Mathematical Technology of Networks, Springer Proceedings in Mathematics & Statistics, Bielefeld, 128, 2015. doi: 10.1007/978-3-319-16619-3.  Google Scholar

[36]

D. MugnoloD. Noja and C. Seifert, Airy-type evolution equations on star graphs, Analysis & PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[40]

T. Tao, Nonlinear Dispersive Equations : Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol. 106, 2006. doi: 10.1090/cbms/106.  Google Scholar

[41]

T. Tsutsumi, Strichartz estimates for Schrödinger equation of fourth order with periodic boundary condition, Kyoto University, 1994 (2014), 104-113.   Google Scholar

[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.  Google Scholar

Figure 1.  Star graph with $ 5 $ edges
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