# American Institute of Mathematical Sciences

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 ([19]).
Citation: Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118
##### References:
 [1] 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. [2] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207. [3] 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. [4] 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. [5] D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  doi: 10.1002/sapm1967461133. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882. [14] G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.  doi: 10.1002/sapm1976553231. [15] C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. [16] 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. [17] 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. [18] 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. [19] 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. [20] J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.  doi: 10.1111/1467-9590.00073. [21] 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. [22] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.

show all references

##### References:
 [1] 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. [2] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207. [3] 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. [4] 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. [5] D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  doi: 10.1002/sapm1967461133. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882. [14] G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.  doi: 10.1002/sapm1976553231. [15] C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. [16] 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. [17] 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. [18] 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. [19] 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. [20] J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.  doi: 10.1111/1467-9590.00073. [21] 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. [22] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.
The graphs of function $\beta = \frac{r^4-2}{r^3-r}$ (left) and $\beta = \frac{r^4-1}{r^3-r}$(right)
The graphs of function $f$ in various cases
 [1] Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005 [2] Chih-Wen Shih, Jui-Pin Tseng. From approximate synchronization to identical synchronization in coupled systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3677-3714. doi: 10.3934/dcdsb.2020086 [3] V. Afraimovich, J.-R. Chazottes, A. Cordonet. Synchronization in directionally coupled systems: Some rigorous results. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 421-442. doi: 10.3934/dcdsb.2001.1.421 [4] Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1029-1054. doi: 10.3934/dcdsb.2021079 [5] Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 [6] Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248 [7] Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193 [8] Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure and Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867 [9] Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061 [10] Shuang Liu, Wenxue Li. Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 837-859. doi: 10.3934/dcdsb.2018045 [11] Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105 [12] Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial and Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87 [13] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [14] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure and Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 [15] Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136 [16] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [17] Iurii Posukhovskyi, Atanas G. Stefanov. On the normalized ground states for the Kawahara equation and a fourth order NLS. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4131-4162. doi: 10.3934/dcds.2020175 [18] Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155 [19] Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214 [20] Antonio Iannizzotto, Kanishka Perera, Marco Squassina. Ground states for scalar field equations with anisotropic nonlocal nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5963-5976. doi: 10.3934/dcds.2015.35.5963

2020 Impact Factor: 1.392