Advanced Search
Article Contents
Article Contents

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

  • * Corresponding author

    * Corresponding author
Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35R02, 35Q55, 35C15, 81Q35, 35G30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Star graph with $ 5 $ edges

  • [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.
    [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.
    [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-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.
    [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. 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. 
    [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-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.
    [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. HarauxAn Introduction to Semilinear Evolution Equations, in Oxford Science Publications, Oxford University Press, 1998. 
    [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.
    [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.
    [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. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.
    [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.
    [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.
    [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. 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.
    [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.
  • 加载中



Article Metrics

HTML views(934) PDF downloads(357) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint