American Institute of Mathematical Sciences

Advanced
December  2015, 35(12): 6085-6112. doi: 10.3934/dcds.2015.35.6085

Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

Benedetta Noris 1, , Hugo Tavares 2, and  Gianmaria Verzini 3,

1. 

INdAM-COFUND Marie Curie Fellow, Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des Etats-Unis, 78035 Versailles Cédex, France
2. 

Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
3. 

Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano

Received  May 2014 Published  May 2015

For the cubic Schrödinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.
Keywords: Gross-Pitaevskii systems, constrained critical points, Cooperative and competitive elliptic systems, orbital stability., Ambrosetti-Prodi type problem.
Mathematics Subject Classification: Primary: 35C08, 35Q55, 35J50; Secondary: 35B4.
Citation: Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6085-6112. doi: 10.3934/dcds.2015.35.6085
References:
[1]

A. Aftalion, B. Noris and C. Sourdis, Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation,, Communications in Mathematical Physics, 336 (2015), 509.  doi: 10.1007/s00220-014-2281-9.  Google Scholar

[2]

S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model,, J. Funct. Anal., 256 (2009), 1118.  doi: 10.1016/j.jfa.2008.10.021.  Google Scholar

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces,, Ann. Mat. Pura Appl. (4), 93 (1972), 231.  doi: 10.1007/BF02412022.  Google Scholar

[4]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Soc. (2), 75 (2007), 67.  doi: 10.1112/jlms/jdl020.  Google Scholar

[5]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis,, Cambridge Studies in Advanced Mathematics, (1993).   Google Scholar

[6]

T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[8]

H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.  doi: 10.1007/BF01449045.  Google Scholar

[9]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[10]

S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates,, Phys. D, 196 (2004), 341.  doi: 10.1016/j.physd.2004.06.002.  Google Scholar

[11]

Z. Chen, C.-S. Lin and W. Zou, Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations,, J. Differential Equations, 255 (2013), 4289.  doi: 10.1016/j.jde.2013.08.009.  Google Scholar

[12]

E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[13]

, DispersiveWiki project,, URL , ().   Google Scholar

[14]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[15]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II,, J. Funct. Anal., 94 (1990), 308.  doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[16]

N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions,, Adv. Nonlinear Stud., 14 (2014), 115.   Google Scholar

[17]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation,, Michigan Math. J., 41 (1994), 151.  doi: 10.1307/mmj/1029004922.  Google Scholar

[18]

T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, Comm. Math. Phys., 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[19]

Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[20]

L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system,, J. Differential Equations, 229 (2006), 743.  doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[21]

L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations,, Adv. Nonlinear Stud., 10 (2010), 681.   Google Scholar

[22]

N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system,, Commun. Math. Sci., 9 (2011), 997.  doi: 10.4310/CMS.2011.v9.n4.a3.  Google Scholar

[23]

N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system,, Adv. Differential Equations, 16 (2011), 977.   Google Scholar

[24]

B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system,, Proc. Amer. Math. Soc., 138 (2010), 1681.  doi: 10.1090/S0002-9939-10-10231-7.  Google Scholar

[25]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems,, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245.  doi: 10.4171/JEMS/332.  Google Scholar

[26]

B. Noris, H. Tavares and G. Verzini, Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains,, , ().   Google Scholar

[27]

B. Noris and G. Verzini, A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems,, J. Differential Equations, 254 (2013), 1529.  doi: 10.1016/j.jde.2012.11.003.  Google Scholar

[28]

M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations,, Nonlinear Anal., 26 (1996), 933.  doi: 10.1016/0362-546X(94)00340-8.  Google Scholar

[29]

J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap,, Calc. Var. Partial Differential Equations, 49 (2014), 103.  doi: 10.1007/s00526-012-0571-7.  Google Scholar

[30]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations,, Commun. Math. Phys., 91 (1983), 313.  doi: 10.1007/BF01208779.  Google Scholar

[31]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$,, Comm. Math. Phys., 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

N. Soave, On existence and phase separation of positive solutions to nonlinear elliptic systems modelling simultaneous cooperation and competition,, , ().   Google Scholar

[33]

H. Tavares and T. Weth, Existence and symmetry results for competing variational systems,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715.  doi: 10.1007/s00030-012-0176-z.  Google Scholar

[34]

H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 279.  doi: 10.1016/j.anihpc.2011.10.006.  Google Scholar

[35]

S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates,, Arch. Ration. Mech. Anal., 194 (2009), 717.  doi: 10.1007/s00205-008-0172-y.  Google Scholar

[36]

R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems,, Topol. Methods Nonlinear Anal., 37 (2011), 203.   Google Scholar

[37]

W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Mathematics, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

A. Aftalion, B. Noris and C. Sourdis, Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation,, Communications in Mathematical Physics, 336 (2015), 509.  doi: 10.1007/s00220-014-2281-9.  Google Scholar

[2]

S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model,, J. Funct. Anal., 256 (2009), 1118.  doi: 10.1016/j.jfa.2008.10.021.  Google Scholar

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces,, Ann. Mat. Pura Appl. (4), 93 (1972), 231.  doi: 10.1007/BF02412022.  Google Scholar

[4]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Soc. (2), 75 (2007), 67.  doi: 10.1112/jlms/jdl020.  Google Scholar

[5]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis,, Cambridge Studies in Advanced Mathematics, (1993).   Google Scholar

[6]

T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[8]

H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.  doi: 10.1007/BF01449045.  Google Scholar

[9]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[10]

S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates,, Phys. D, 196 (2004), 341.  doi: 10.1016/j.physd.2004.06.002.  Google Scholar

[11]

Z. Chen, C.-S. Lin and W. Zou, Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations,, J. Differential Equations, 255 (2013), 4289.  doi: 10.1016/j.jde.2013.08.009.  Google Scholar

[12]

E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[13]

, DispersiveWiki project,, URL , ().   Google Scholar

[14]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[15]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II,, J. Funct. Anal., 94 (1990), 308.  doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[16]

N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions,, Adv. Nonlinear Stud., 14 (2014), 115.   Google Scholar

[17]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation,, Michigan Math. J., 41 (1994), 151.  doi: 10.1307/mmj/1029004922.  Google Scholar

[18]

T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, Comm. Math. Phys., 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[19]

Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[20]

L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system,, J. Differential Equations, 229 (2006), 743.  doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[21]

L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations,, Adv. Nonlinear Stud., 10 (2010), 681.   Google Scholar

[22]

N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system,, Commun. Math. Sci., 9 (2011), 997.  doi: 10.4310/CMS.2011.v9.n4.a3.  Google Scholar

[23]

N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system,, Adv. Differential Equations, 16 (2011), 977.   Google Scholar

[24]

B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system,, Proc. Amer. Math. Soc., 138 (2010), 1681.  doi: 10.1090/S0002-9939-10-10231-7.  Google Scholar

[25]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems,, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245.  doi: 10.4171/JEMS/332.  Google Scholar

[26]

B. Noris, H. Tavares and G. Verzini, Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains,, , ().   Google Scholar

[27]

B. Noris and G. Verzini, A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems,, J. Differential Equations, 254 (2013), 1529.  doi: 10.1016/j.jde.2012.11.003.  Google Scholar

[28]

M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations,, Nonlinear Anal., 26 (1996), 933.  doi: 10.1016/0362-546X(94)00340-8.  Google Scholar

[29]

J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap,, Calc. Var. Partial Differential Equations, 49 (2014), 103.  doi: 10.1007/s00526-012-0571-7.  Google Scholar

[30]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations,, Commun. Math. Phys., 91 (1983), 313.  doi: 10.1007/BF01208779.  Google Scholar

[31]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$,, Comm. Math. Phys., 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

N. Soave, On existence and phase separation of positive solutions to nonlinear elliptic systems modelling simultaneous cooperation and competition,, , ().   Google Scholar

[33]

H. Tavares and T. Weth, Existence and symmetry results for competing variational systems,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715.  doi: 10.1007/s00030-012-0176-z.  Google Scholar

[34]

H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 279.  doi: 10.1016/j.anihpc.2011.10.006.  Google Scholar

[35]

S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates,, Arch. Ration. Mech. Anal., 194 (2009), 717.  doi: 10.1007/s00205-008-0172-y.  Google Scholar

[36]

R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems,, Topol. Methods Nonlinear Anal., 37 (2011), 203.   Google Scholar

[37]

W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Mathematics, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

[1]

F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355
[2]

Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019
[3]

Imene Bendahou, Zied Khemiri, Fethi Mahmoudi. On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2367-2391. doi: 10.3934/dcds.2020118
[4]

Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481
[5]

Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715
[6]

Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184
[7]

Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905
[8]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
[9]

E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120
[10]

Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic & Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048
[11]

Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008
[12]

Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022
[13]

Mats Gyllenberg, Yi Wang. Periodic tridiagonal systems modeling competitive-cooperative ecological interactions. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 289-298. doi: 10.3934/dcdsb.2005.5.289
[14]

Yi Wang, Dun Zhou. Transversality for time-periodic competitive-cooperative tridiagonal systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1821-1830. doi: 10.3934/dcdsb.2015.20.1821
[15]

Lin Niu, Yi Wang. Non-oscillation principle for eventually competitive and cooperative systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6481-6494. doi: 10.3934/dcdsb.2019148
[16]

Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059
[17]

Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225
[18]

Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159
[19]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[20]

Shuai Li, Jingjing Yan, Xincai Zhu. Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 65-81. doi: 10.3934/cpaa.2019005

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences