American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020407

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

Van Duong Dinh 1,2, and  Sahbi Keraani 1,,

1. 

Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2. 

Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

* Corresponding author: Sahbi Keraani

Received  January 2020 Revised  June 2020 Published  July 2020

Based on recent works of Dodson-Murphy [12] and Arora-Dodson-Murphy [3], we give a unified approach for the energy scattering with radially symmetric initial data for nonlinear Schrödinger equations and nonlinear Choquard equations in any dimensions $ N\geq 2 $. We also discuss its applications for other Schrödinger-type equations.

Keywords: Nonlinear Schrödinger equation, Nonlinear Choquard equation, Ground state, Scattering, Radial Sobolev embedding.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35P25.
Citation: Van Duong Dinh, Sahbi Keraani. The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020407
References:
[1]

T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.  doi: 10.1215/21562261-2265914.  Google Scholar

[2]

A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Cont. Dyn. Syst., 39 (2019), 6643-6668.  doi: 10.3934/dcds.2019289.  Google Scholar

[3]

A. K. Arora, B. Dodson and J. Murphy, Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663.  doi: 10.1090/proc/14824.  Google Scholar

[4]

A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, preprint, arXiv: 1904.05339. Google Scholar

[5]

L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663. Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

T. Cazenave, D. Fang and J. Xie, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9.  Google Scholar

[8]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.  Google Scholar

[9]

C. V. Coffman, Uniqueness of the ground state solution for $\Delta u - u + u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684.  Google Scholar

[10]

V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions, preprint, arXiv: 1908.02987. Google Scholar

[11]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022. Google Scholar

[12]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867.  doi: 10.1090/proc/13678.  Google Scholar

[13]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.  doi: 10.4310/MRL.2018.v25.n6.a5.  Google Scholar

[14]

T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.  Google Scholar

[15]

L. G. Farah and C. Guzman, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231.  doi: 10.1016/j.jde.2017.01.013.  Google Scholar

[16]

L. G. Farah and C. Guzman, Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc., 51 (2020), 449-512.  doi: 10.1007/s00574-019-00160-1.  Google Scholar

[17]

D. Foschi, Inhomogeneous strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.  doi: 10.1142/S0219891605000361.  Google Scholar

[18]

C. Guevara, Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express., 2014 (2014), 177-243.   Google Scholar

[19]

Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Comm. Partial Differential Equations, 41 (2016), 185-207.  doi: 10.1080/03605302.2015.1116556.  Google Scholar

[20]

Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. Google Scholar

[21]

M. Hamano and M. Ikeda, Global dynamics below the ground state for the focusing Schrödinger equation with a potential, J. Evol. Equ., 2019 (in press). Google Scholar

[22]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.  Google Scholar

[23]

Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., 15 (2016), 1571-1601.  doi: 10.3934/cpaa.2016003.  Google Scholar

[24]

M. Keel and T. Tao, Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[25]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[26]

J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205.  doi: 10.1007/s00023-009-0010-2.  Google Scholar

[27]

M. K. Kwong, Uniqueness of positive solution of $\Delta u - u + u^p = 0$ in $ \mathbb R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[28]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[29]

C. Miao, G. Xu and L. Zhao, The cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44.   Google Scholar

[30]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[31]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[32]

C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228.  doi: 10.3934/dcds.2018091.  Google Scholar

[33]

E. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993.  Google Scholar

[34]

J. Stubbe, Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D, 48 (1991), 259-272.  doi: 10.1016/0167-2789(91)90087-P.  Google Scholar

[35]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[36]

T. Tao, On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48.  doi: 10.4310/DPDE.2004.v1.n1.a1.  Google Scholar

[37]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[38]

M. C. Vilela, Inhomogeneous strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.  doi: 10.1090/S0002-9947-06-04099-2.  Google Scholar

[39]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.   Google Scholar

[40]

C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp. doi: 10.1007/s00526-016-1068-6.  Google Scholar

[41]

C. Xu and T. Zhao, A remark on the scattering theory for the 2D radial focusing INLS, preprint, arXiv: 1908.00743. Google Scholar

show all references

References:
[1]

T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.  doi: 10.1215/21562261-2265914.  Google Scholar

[2]

A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Cont. Dyn. Syst., 39 (2019), 6643-6668.  doi: 10.3934/dcds.2019289.  Google Scholar

[3]

A. K. Arora, B. Dodson and J. Murphy, Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663.  doi: 10.1090/proc/14824.  Google Scholar

[4]

A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, preprint, arXiv: 1904.05339. Google Scholar

[5]

L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663. Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003. doi: 10.1090/cln/010.  Google Scholar

[7]

T. Cazenave, D. Fang and J. Xie, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9.  Google Scholar

[8]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.  Google Scholar

[9]

C. V. Coffman, Uniqueness of the ground state solution for $\Delta u - u + u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684.  Google Scholar

[10]

V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions, preprint, arXiv: 1908.02987. Google Scholar

[11]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022. Google Scholar

[12]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867.  doi: 10.1090/proc/13678.  Google Scholar

[13]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.  doi: 10.4310/MRL.2018.v25.n6.a5.  Google Scholar

[14]

T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.  Google Scholar

[15]

L. G. Farah and C. Guzman, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231.  doi: 10.1016/j.jde.2017.01.013.  Google Scholar

[16]

L. G. Farah and C. Guzman, Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc., 51 (2020), 449-512.  doi: 10.1007/s00574-019-00160-1.  Google Scholar

[17]

D. Foschi, Inhomogeneous strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.  doi: 10.1142/S0219891605000361.  Google Scholar

[18]

C. Guevara, Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express., 2014 (2014), 177-243.   Google Scholar

[19]

Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Comm. Partial Differential Equations, 41 (2016), 185-207.  doi: 10.1080/03605302.2015.1116556.  Google Scholar

[20]

Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. Google Scholar

[21]

M. Hamano and M. Ikeda, Global dynamics below the ground state for the focusing Schrödinger equation with a potential, J. Evol. Equ., 2019 (in press). Google Scholar

[22]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.  Google Scholar

[23]

Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., 15 (2016), 1571-1601.  doi: 10.3934/cpaa.2016003.  Google Scholar

[24]

M. Keel and T. Tao, Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[25]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[26]

J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205.  doi: 10.1007/s00023-009-0010-2.  Google Scholar

[27]

M. K. Kwong, Uniqueness of positive solution of $\Delta u - u + u^p = 0$ in $ \mathbb R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[28]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[29]

C. Miao, G. Xu and L. Zhao, The cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44.   Google Scholar

[30]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[31]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[32]

C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228.  doi: 10.3934/dcds.2018091.  Google Scholar

[33]

E. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993.  Google Scholar

[34]

J. Stubbe, Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D, 48 (1991), 259-272.  doi: 10.1016/0167-2789(91)90087-P.  Google Scholar

[35]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[36]

T. Tao, On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48.  doi: 10.4310/DPDE.2004.v1.n1.a1.  Google Scholar

[37]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[38]

M. C. Vilela, Inhomogeneous strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.  doi: 10.1090/S0002-9947-06-04099-2.  Google Scholar

[39]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.   Google Scholar

[40]

C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp. doi: 10.1007/s00526-016-1068-6.  Google Scholar

[41]

C. Xu and T. Zhao, A remark on the scattering theory for the 2D radial focusing INLS, preprint, arXiv: 1908.00743. Google Scholar

[1]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091
[2]

Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235
[3]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[4]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[5]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[6]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
[7]

Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597
[8]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803
[9]

Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020095
[10]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]

Dengfeng Lü. Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1781-1795. doi: 10.3934/cpaa.2016014
[12]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[13]

Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[14]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[15]

Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183
[16]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481
[17]

F. Catoire, W. M. Wang. Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Communications on Pure & Applied Analysis, 2010, 9 (2) : 483-491. doi: 10.3934/cpaa.2010.9.483
[18]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905
[19]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
[20]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

2019 Impact Factor: 1.233

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences