    June  2018, 38(6): 2795-2808. doi: 10.3934/dcds.2018118

## Synchronization of positive solutions for coupled Schrödinger equations

 1 School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Luo-Yu Road 152, Wuhan 430079, China 2 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China 3 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

* Corresponding author: Zhi-Qiang Wang

Received  May 2017 Published  April 2018

In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation
 $\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$
where
 $2< p<\frac{n}{n-2},$
if
 $n\ge 3$
and
 $2< p<+∞$
, if
 $n = 1, 2,$
and
 $μ_1, μ_2, β>0$
are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case
 $n = 1$
by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case
 $p = 2$
for which uniqueness of positive solutions was known ().
Citation: Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118
##### References:
  A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.  Google Scholar  T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207. Google Scholar  T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar  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), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar  D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  doi: 10.1002/sapm1967461133.  Google Scholar  S. Correia, Characterization of ground-states for a system of M coupled semilinear Schrödinger equations and applications, J.Differential Equations, 260 (2016), 3302-3326.  doi: 10.1016/j.jde.2015.10.032.  Google Scholar  D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149-161.  doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar  A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅰ. Anormalous dispersion, Appl. Phys. Lett., 23 (1973), 142-144.   Google Scholar  A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-173.  doi: 10.1063/1.1654847. Google Scholar  T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n\le 3$, Communications in Mathematical Physics, 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar  H. Liu, Z. Liu and J. Chang, Existence and uniquiness of positive solutions of nonlinear Schrödinger systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 365-390.  doi: 10.1017/S0308210513000711.  Google Scholar  R. Mandel, Minimal energy solutions for repulsive nonlinear Schrödinger systems, J. Differential Equations, 257 (2014), 450-468.  doi: 10.1016/j.jde.2014.04.006.  Google Scholar  M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882.   Google Scholar  G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.  doi: 10.1002/sapm1976553231. Google Scholar  C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar  B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $R^n$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar  W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, Journal of Differential Equations, 42 (1981), 400-413.  doi: 10.1016/0022-0396(81)90113-3.  Google Scholar  Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathématique, 122 (2014), 69-85.  doi: 10.1007/s11854-014-0003-z.  Google Scholar  J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar  J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.  doi: 10.1111/1467-9590.00073.  Google Scholar  V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.  doi: 10.1007/BF00913182. Google Scholar  V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.   Google Scholar

show all references

##### References:
  A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.  Google Scholar  T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207. Google Scholar  T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar  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), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar  D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  doi: 10.1002/sapm1967461133.  Google Scholar  S. Correia, Characterization of ground-states for a system of M coupled semilinear Schrödinger equations and applications, J.Differential Equations, 260 (2016), 3302-3326.  doi: 10.1016/j.jde.2015.10.032.  Google Scholar  D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149-161.  doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar  A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅰ. Anormalous dispersion, Appl. Phys. Lett., 23 (1973), 142-144.   Google Scholar  A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-173.  doi: 10.1063/1.1654847. Google Scholar  T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n\le 3$, Communications in Mathematical Physics, 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar  H. Liu, Z. Liu and J. Chang, Existence and uniquiness of positive solutions of nonlinear Schrödinger systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 365-390.  doi: 10.1017/S0308210513000711.  Google Scholar  R. Mandel, Minimal energy solutions for repulsive nonlinear Schrödinger systems, J. Differential Equations, 257 (2014), 450-468.  doi: 10.1016/j.jde.2014.04.006.  Google Scholar  M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882.   Google Scholar  G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.  doi: 10.1002/sapm1976553231. Google Scholar  C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar  B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $R^n$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar  W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, Journal of Differential Equations, 42 (1981), 400-413.  doi: 10.1016/0022-0396(81)90113-3.  Google Scholar  Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathématique, 122 (2014), 69-85.  doi: 10.1007/s11854-014-0003-z.  Google Scholar  J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar  J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.  doi: 10.1111/1467-9590.00073.  Google Scholar  V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.  doi: 10.1007/BF00913182. Google Scholar  V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.   Google Scholar The graphs of function $\beta = \frac{r^4-2}{r^3-r}$ (left) and $\beta = \frac{r^4-1}{r^3-r}$(right)
  Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005  Chih-Wen Shih, Jui-Pin Tseng. From approximate synchronization to identical synchronization in coupled systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3677-3714. doi: 10.3934/dcdsb.2020086  Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079  V. Afraimovich, J.-R. Chazottes, A. Cordonet. Synchronization in directionally coupled systems: Some rigorous results. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 421-442. doi: 10.3934/dcdsb.2001.1.421  Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316  Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248  Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193  Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867  Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061  Shuang Liu, Wenxue Li. Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 837-859. doi: 10.3934/dcdsb.2018045  Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105  Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87  Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450  Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298  Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136  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  Iurii Posukhovskyi, Atanas G. Stefanov. On the normalized ground states for the Kawahara equation and a fourth order NLS. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4131-4162. doi: 10.3934/dcds.2020175  Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155  Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214  Antonio Iannizzotto, Kanishka Perera, Marco Squassina. Ground states for scalar field equations with anisotropic nonlocal nonlinearities. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5963-5976. doi: 10.3934/dcds.2015.35.5963

2020 Impact Factor: 1.392