February  2016, 36(2): 1005-1021. doi: 10.3934/dcds.2016.36.1005

Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system

1. 

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

2. 

Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

Received  June 2014 Published  August 2015

In this paper, the existence and stability results for a two-parameter family of vector solitary-wave solutions (i.e both components are nonzero) of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{matrix} iu_t+ u_{xx} + (a |u|^2 + b |v|^2) u=0,\\ iv_t+ v_{xx} + (b |u|^2 + c |v|^2) v=0,\\ \end{matrix} \right. \end{equation*} where $u,v$ are complex-valued functions of $(x,t)\in \mathbb R^2$, and $a,b,c \in \mathbb R$ are established. The results extend our earlier ones as well as those of Ohta, Cipolatti and Zumpichiatti and de Figueiredo and Lopes. As opposed to other methods used before to establish existence and stability where the two constraints of the minimization problems are related to each other, our approach here characterizes solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. The set of minimizers is shown to be stable; and depending on the interplay between the parameters $a,b$ and $c$, further information about the structures of this set are given.
Citation: Nghiem V. Nguyen, Zhi-Qiang Wang. Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1005-1021. doi: 10.3934/dcds.2016.36.1005
References:
[1]

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

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 453.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[3]

________, Standing waves of some coupled nonlinear Schrödinger equations,, J. London Math. Soc., 75 (2007), 67.   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,, Cal. of Var. and PDEs, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

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T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems,, Journ. Part. Diff. Eqns., 19 (2006), 200.   Google Scholar

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T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[7]

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

[8]

J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems,, Jour. Diff. Eqns., 163 (2000), 429.  doi: 10.1006/jdeq.1999.3737.  Google Scholar

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T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de Métodos Matemáticos, (1989).   Google Scholar

[10]

_________, Semilinear Schrödinger equations,, AMS-Courant Lecture Notes, 10 (2003).   Google Scholar

[11]

R. Cipolatti and W. Zumpichiatti, Orbitally stable standing waves for a system of coupled nonlinear Schrödinger equations,, Nonlinear Anal., 42 (2000), 445.  doi: 10.1016/S0362-546X(98)00357-5.  Google Scholar

[12]

E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincare Anal. Non Linearaire, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[13]

D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149.  doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar

[14]

D. Garrisi, On the orbital stability of standing-waves solutions to a coupled non-linear Klein-Gordon equation,, Adv. Nonlinear Stud., 12 (2012), 639.   Google Scholar

[15]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anomalous dispersion,, Appl. Phys. Lett., 23 (1973).  doi: 10.1063/1.1654836.  Google Scholar

[16]

________, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers II. Normal dispersion,, Appl. Phys. Lett., 23 (1973).   Google Scholar

[17]

I. Ianni and S. Le Coz, Multi-speed solitary wave solutions for nonlinear Schrödinger systems,, J. London Math. Soc. (2), 89 (2014), 623.  doi: 10.1112/jlms/jdt083.  Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, Second edition,, Graduate studies in mathematics, (2001).  doi: 10.1090/gsm/014.  Google Scholar

[19]

T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, Comm. Math. Phys., 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109.   Google Scholar

[21]

_________, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223.   Google Scholar

[22]

Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[23]

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

[24]

N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves of a $3-$coupled nonlinear Schrödinger system,, Non. Anal. A: Theory, 90 (2013), 1.  doi: 10.1016/j.na.2013.05.027.  Google Scholar

[25]

M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations,, Nonlinear Anal.: Theory, 26 (1996), 933.  doi: 10.1016/0362-546X(94)00340-8.  Google Scholar

[26]

G. J. Roskes, Some nonlinear multiphase interactions,, Stud. Appl. Math., 55 (1976).   Google Scholar

[27]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbf {R^n}$,, Comm. Math. Phys., 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[28]

X. Song, Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities,, Jour. Math. Anal. Appl., 366 (2010), 345.  doi: 10.1016/j.jmaa.2009.12.011.  Google Scholar

[29]

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

[30]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Sov. Phys. Jour. Appl. Mech. Tech. Phys., 9 (1968), 190.  doi: 10.1007/BF00913182.  Google Scholar

[31]

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

[32]

A. K. Zvezdin and A. F. Popkov, Contribution to the nonlinear theory of magnetostatic spin waves,, Sov. Phys. JETP, 2 (1983).   Google Scholar

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,, Adv. Differential Equations, 18 (2013), 1129.   Google Scholar

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, C. R. Math. Acad. Sci. Paris, 342 (2006), 453.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[3]

________, Standing waves of some coupled nonlinear Schrödinger equations,, J. London Math. Soc., 75 (2007), 67.   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,, Cal. of Var. and PDEs, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[5]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems,, Journ. Part. Diff. Eqns., 19 (2006), 200.   Google Scholar

[6]

T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[7]

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

[8]

J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems,, Jour. Diff. Eqns., 163 (2000), 429.  doi: 10.1006/jdeq.1999.3737.  Google Scholar

[9]

T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de Métodos Matemáticos, (1989).   Google Scholar

[10]

_________, Semilinear Schrödinger equations,, AMS-Courant Lecture Notes, 10 (2003).   Google Scholar

[11]

R. Cipolatti and W. Zumpichiatti, Orbitally stable standing waves for a system of coupled nonlinear Schrödinger equations,, Nonlinear Anal., 42 (2000), 445.  doi: 10.1016/S0362-546X(98)00357-5.  Google Scholar

[12]

E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincare Anal. Non Linearaire, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[13]

D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149.  doi: 10.1016/j.anihpc.2006.11.006.  Google Scholar

[14]

D. Garrisi, On the orbital stability of standing-waves solutions to a coupled non-linear Klein-Gordon equation,, Adv. Nonlinear Stud., 12 (2012), 639.   Google Scholar

[15]

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anomalous dispersion,, Appl. Phys. Lett., 23 (1973).  doi: 10.1063/1.1654836.  Google Scholar

[16]

________, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers II. Normal dispersion,, Appl. Phys. Lett., 23 (1973).   Google Scholar

[17]

I. Ianni and S. Le Coz, Multi-speed solitary wave solutions for nonlinear Schrödinger systems,, J. London Math. Soc. (2), 89 (2014), 623.  doi: 10.1112/jlms/jdt083.  Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, Second edition,, Graduate studies in mathematics, (2001).  doi: 10.1090/gsm/014.  Google Scholar

[19]

T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, Comm. Math. Phys., 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109.   Google Scholar

[21]

_________, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223.   Google Scholar

[22]

Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[23]

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

[24]

N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves of a $3-$coupled nonlinear Schrödinger system,, Non. Anal. A: Theory, 90 (2013), 1.  doi: 10.1016/j.na.2013.05.027.  Google Scholar

[25]

M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations,, Nonlinear Anal.: Theory, 26 (1996), 933.  doi: 10.1016/0362-546X(94)00340-8.  Google Scholar

[26]

G. J. Roskes, Some nonlinear multiphase interactions,, Stud. Appl. Math., 55 (1976).   Google Scholar

[27]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbf {R^n}$,, Comm. Math. Phys., 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[28]

X. Song, Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities,, Jour. Math. Anal. Appl., 366 (2010), 345.  doi: 10.1016/j.jmaa.2009.12.011.  Google Scholar

[29]

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

[30]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Sov. Phys. Jour. Appl. Mech. Tech. Phys., 9 (1968), 190.  doi: 10.1007/BF00913182.  Google Scholar

[31]

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

[32]

A. K. Zvezdin and A. F. Popkov, Contribution to the nonlinear theory of magnetostatic spin waves,, Sov. Phys. JETP, 2 (1983).   Google Scholar

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