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January  2017, 16(1): 115-130. doi: 10.3934/cpaa.2017005

A complete classification of ground-states for a coupled nonlinear Schrödinger system

1. 

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P.R. China

2. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R.China

3. 

Department of Mathematics and Statistics, Utah State University, Logan UT 84322, USA

E-mail address: zhi-qiang.wang@usu.edu

Received  June 2016 Revised  August 2016 Published  November 2016

In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system
$-\Delta u_j+ u_j=\sum\limits_{i=1}^mb_{ij}u_i^2u_j, \quad\text{in}\ \mathbb{R}^n,\\ u_j(x)\to 0\ \text{as}\ |x|\ \to \infty, \quad j=1,2,\cdots, m,$
where $n=1, 2, 3, m\geq 2$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}.$ By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For $m=3$, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix $B = (b_{ij})$. In particular, there is a nontrivial ground-state solution provided that all coupling constants $b_{ij}, i\neq j$ are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when $b_{ij}, i\neq j$ are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix $B=(b_{ij})$ is positive semi-definite.
Citation: 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
References:
[1]

J. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Advances in Differential Equations, 18 (2013), 1129-1164.

[2]

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.

[3]

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

[4]

D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.

[5]

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.

[6]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-172.

[7]

T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[8]

T. C. Lin and J. Wei, Ground State of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 277 (2008), 573-576. doi: 10.1007/s00220-007-0365-5.

[9]

H. LiuZ. 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.

[10]

Z. Liu and Z. -Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.

[11]

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

[12]

N. V. Nguyen and Z. -Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Diff. Eqns., 16 (2011), 977-1000.

[13]

G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.

[14]

H. CRüeggN. Cavadini and A. Furrer, et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.

[15]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.

[16]

Z. -Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathematique, 122 (2014), 69-85. doi: 10.1007/s11854-014-0003-z.

[17]

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

[18]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 4 (1968), 190-194.

[19]

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

show all references

References:
[1]

J. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Advances in Differential Equations, 18 (2013), 1129-1164.

[2]

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.

[3]

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

[4]

D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.

[5]

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.

[6]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-172.

[7]

T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[8]

T. C. Lin and J. Wei, Ground State of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 277 (2008), 573-576. doi: 10.1007/s00220-007-0365-5.

[9]

H. LiuZ. 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.

[10]

Z. Liu and Z. -Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.

[11]

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

[12]

N. V. Nguyen and Z. -Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Diff. Eqns., 16 (2011), 977-1000.

[13]

G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.

[14]

H. CRüeggN. Cavadini and A. Furrer, et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.

[15]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.

[16]

Z. -Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathematique, 122 (2014), 69-85. doi: 10.1007/s11854-014-0003-z.

[17]

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

[18]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 4 (1968), 190-194.

[19]

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

Table 1.  The number of non-zero components of ground-state solutions
case condition 1 condition 2 type
1 $det(B) > 0$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $ p= 3 $
2 $det(B) > 0$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
3 $det(B) < 0$ $p=1, 2$
4 $rank(B)=1$ $p=1, 2, 3$
5 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=2, 3$
6 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=1, 2$
7 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p=1, 2, 3$
8 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
case condition 1 condition 2 type
1 $det(B) > 0$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $ p= 3 $
2 $det(B) > 0$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
3 $det(B) < 0$ $p=1, 2$
4 $rank(B)=1$ $p=1, 2, 3$
5 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=2, 3$
6 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=1, 2$
7 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p=1, 2, 3$
8 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
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