American Institute of Mathematical Sciences

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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 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. 

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