June  2022, 27(6): 3399-3434. doi: 10.3934/dcdsb.2021190

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 Published  June 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (6) : 3399-3434. 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.

[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. Haraux, An 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.

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.

[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. Haraux, An 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.

Figure 1.  Star graph with $ 5 $ edges
[1]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205

[2]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037

[3]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control and Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177

[4]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028

[5]

Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149

[6]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[7]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[8]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284

[9]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843

[10]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275

[11]

Editorial Office. Retraction: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3785-3785. doi: 10.3934/cpaa.2020167

[12]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174

[13]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093

[14]

Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122

[15]

Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156

[16]

Lu Yang, Guangsheng Wei, Vyacheslav Pivovarchik. Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex. Inverse Problems and Imaging, 2021, 15 (2) : 257-270. doi: 10.3934/ipi.2020063

[17]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564

[18]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113

[19]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791

[20]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (324)
  • HTML views (548)
  • Cited by (0)

[Back to Top]