American Institute of Mathematical Sciences

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

Synchronization of positive solutions for coupled Schrödinger equations

Chuangye Liu 1, and  Zhi-Qiang Wang 2,3,,

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]).
Keywords: Ground-states, coupled systems, synchronization.
Mathematics Subject Classification: Primary: 35J20, 35J47; Secondary: 35J50.
Citation: Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882. Google Scholar

[14]

G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238. doi: 10.1002/sapm1976553231. Google Scholar

[15]

C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152. doi: 10.1111/1467-9590.00073. Google Scholar

[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. Google Scholar

[22]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914. Google Scholar

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. Google Scholar

[2]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882. Google Scholar

[14]

G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238. doi: 10.1002/sapm1976553231. Google Scholar

[15]

C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152. doi: 10.1111/1467-9590.00073. Google Scholar

[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. Google Scholar

[22]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914. Google Scholar

Figure 1.  The graphs of function $\beta = \frac{r^4-2}{r^3-r}$ (left) and $\beta = \frac{r^4-1}{r^3-r}$(right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The graphs of function $f$ in various cases
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105
[8]

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
[9]

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
[10]

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
[11]

Antonio Iannizzotto, Kanishka Perera, Marco Squassina. Ground states for scalar field equations with anisotropic nonlocal nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5963-5976. doi: 10.3934/dcds.2015.35.5963
[12]

Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124
[13]

Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054
[14]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
[15]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[16]

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
[17]

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
[18]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[19]

R. Yamapi, R.S. MacKay. Stability of synchronization in a shift-invariant ring of mutually coupled oscillators. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 973-996. doi: 10.3934/dcdsb.2008.10.973
[20]

Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (141)
  • HTML views (206)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences