April  2016, 36(4): 1789-1811. doi: 10.3934/dcds.2016.36.1789

Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

1. 

Trocaire College, Mathematics Department, 360 Choate Ave, Buffalo, NY 14220, United States

Received  July 2014 Revised  July 2015 Published  September 2015

In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system \begin{equation*} i\partial_t u_{j}+ \partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj} |u_k|^{p}\right)|u_j|^{p-2}u_j = 0, \ j=1,2,3, \end{equation*} where $u_j$ are complex-valued functions of $(x,t)\in \mathbb{R}^{2}$ and $a_{kj}$ are positive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case). Our approach improves many of the previously known results. In all variational methods used previously to study the stability of solitary waves, which we are aware of, the constraint functionals were not independently chosen. Here we study a problem of minimizing the energy functional subject to three independent $L^2$ mass constraints and establish existence and stability results for a true three-parameter family of solitary waves.
Citation: Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789
References:
[1]

J. Albert and J. Angulo, Existence and stability of ground-state solutions of a Schrödinger-KdV system,, Proc. Royal Soc. of Edinburgh A, 133 (2003), 987. doi: 10.1017/S030821050000278X. Google Scholar

[2]

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

[3]

J. Albert, J. Bona and J.-C. Saut, Model equations for waves in stratified fluids,, Proc. Royal. Soc. of Edinburgh, 453 (1997), 1233. doi: 10.1098/rspa.1997.0068. Google Scholar

[4]

T. B. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. London Ser. A, 328 (1972), 153. doi: 10.1098/rspa.1972.0074. Google Scholar

[5]

S. Bhattarai, Solitary waves and a stability analysis for an equation of short and long dispersive waves,, Nonlinear Anal., 75 (2012), 6506. doi: 10.1016/j.na.2012.07.026. Google Scholar

[6]

S. Bhattarai, Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities,, Adv. Nonlinear Anal., 4 (2015), 73. doi: 10.1515/anona-2014-0058. Google Scholar

[7]

J. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. London Ser. A, 344 (1975), 363. doi: 10.1098/rspa.1975.0106. Google Scholar

[8]

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

[9]

T. Cazenave, Semilinear Schrödinger Equations,, 10, 10 (2003). Google Scholar

[10]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549. doi: 10.1007/BF01403504. Google Scholar

[11]

S. Chakravarty, M. J. Ablowitz, J. R. Sauer and R. B. Jenkins, Multisoliton interactions and wavelength-division multiplexing,, Opt. Lett., 20 (1995), 136. doi: 10.1364/OL.20.000136. Google Scholar

[12]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases,, Rev. Mod. Phys., 71 (1999), 463. doi: 10.1103/RevModPhys.71.463. Google Scholar

[13]

T.-L. Ho, Spinor Bose condensates in optical traps,, Phys. Rev. Lett., 81 (1998). doi: 10.1103/PhysRevLett.81.742. Google Scholar

[14]

Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates,, Phys. Reports, 520 (2012), 253. doi: 10.1016/j.physrep.2012.07.005. Google Scholar

[15]

E. H. Lieb and M. Loss, Analysis, 2nd ed., 14,, AMS-Grad. Stud. Math., (2001). doi: 10.1090/gsm/014. Google Scholar

[16]

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

[17]

L. F. Mollenauer, S. G. Evangelides and J. P. Gordon, Wavelength division multiplexing with solitons in ultra-long transmission using lumped amplifiers,, J. Lightwave Technol., 9 (1991), 362. doi: 10.1109/50.70013. Google Scholar

[18]

N. V. Nguyen, R.-S. Tian, B. Deconinck and N. Sheils, Global existence for a system of Schrödinger equations with power-type nonlinearities,, Jour. Math. Phys., 54 (2013). doi: 10.1063/1.4774149. Google Scholar

[19]

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

[20]

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

[21]

N. V. Nguyen and Z-Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system,, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 36 (2016), 1005. doi: 10.3934/dcds.2016.36.1005. Google Scholar

[22]

N. V. Nguyen, R. Tian and Z.-Q. Wang, Stability of traveling-wave solutions for a Schrödinger system with power-type nonlinearities,, preprint., (). Google Scholar

[23]

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

[24]

A. C. Scott, Launching a davydov soliton: I. soliton analysis,, Phys. Scr., 29 (1984). doi: 10.1088/0031-8949/29/3/016. Google Scholar

[25]

B. K. Som, M. R. Gupta and B. Dasgupta, Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,, Phys. Lett. A, 72 (1979), 111. doi: 10.1016/0375-9601(79)90663-7. Google Scholar

[26]

J. Q. Sun, Z. Q. Ma and M. Z. Qin, Simulation of envelope Rossby solitons in a pair of cubic Schrödinger equations,, Appl. Math. Comput., 183 (2006), 946. doi: 10.1016/j.amc.2006.06.041. Google Scholar

[27]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, 106, AMS-CBMS, (2006). Google Scholar

[28]

C. Yeh and L. Bergman, Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,, Phys. Rev. E, 57 (1998). doi: 10.1103/PhysRevE.57.2398. Google Scholar

show all references

References:
[1]

J. Albert and J. Angulo, Existence and stability of ground-state solutions of a Schrödinger-KdV system,, Proc. Royal Soc. of Edinburgh A, 133 (2003), 987. doi: 10.1017/S030821050000278X. Google Scholar

[2]

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

[3]

J. Albert, J. Bona and J.-C. Saut, Model equations for waves in stratified fluids,, Proc. Royal. Soc. of Edinburgh, 453 (1997), 1233. doi: 10.1098/rspa.1997.0068. Google Scholar

[4]

T. B. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. London Ser. A, 328 (1972), 153. doi: 10.1098/rspa.1972.0074. Google Scholar

[5]

S. Bhattarai, Solitary waves and a stability analysis for an equation of short and long dispersive waves,, Nonlinear Anal., 75 (2012), 6506. doi: 10.1016/j.na.2012.07.026. Google Scholar

[6]

S. Bhattarai, Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities,, Adv. Nonlinear Anal., 4 (2015), 73. doi: 10.1515/anona-2014-0058. Google Scholar

[7]

J. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. London Ser. A, 344 (1975), 363. doi: 10.1098/rspa.1975.0106. Google Scholar

[8]

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

[9]

T. Cazenave, Semilinear Schrödinger Equations,, 10, 10 (2003). Google Scholar

[10]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549. doi: 10.1007/BF01403504. Google Scholar

[11]

S. Chakravarty, M. J. Ablowitz, J. R. Sauer and R. B. Jenkins, Multisoliton interactions and wavelength-division multiplexing,, Opt. Lett., 20 (1995), 136. doi: 10.1364/OL.20.000136. Google Scholar

[12]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases,, Rev. Mod. Phys., 71 (1999), 463. doi: 10.1103/RevModPhys.71.463. Google Scholar

[13]

T.-L. Ho, Spinor Bose condensates in optical traps,, Phys. Rev. Lett., 81 (1998). doi: 10.1103/PhysRevLett.81.742. Google Scholar

[14]

Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates,, Phys. Reports, 520 (2012), 253. doi: 10.1016/j.physrep.2012.07.005. Google Scholar

[15]

E. H. Lieb and M. Loss, Analysis, 2nd ed., 14,, AMS-Grad. Stud. Math., (2001). doi: 10.1090/gsm/014. Google Scholar

[16]

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

[17]

L. F. Mollenauer, S. G. Evangelides and J. P. Gordon, Wavelength division multiplexing with solitons in ultra-long transmission using lumped amplifiers,, J. Lightwave Technol., 9 (1991), 362. doi: 10.1109/50.70013. Google Scholar

[18]

N. V. Nguyen, R.-S. Tian, B. Deconinck and N. Sheils, Global existence for a system of Schrödinger equations with power-type nonlinearities,, Jour. Math. Phys., 54 (2013). doi: 10.1063/1.4774149. Google Scholar

[19]

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

[20]

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

[21]

N. V. Nguyen and Z-Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system,, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 36 (2016), 1005. doi: 10.3934/dcds.2016.36.1005. Google Scholar

[22]

N. V. Nguyen, R. Tian and Z.-Q. Wang, Stability of traveling-wave solutions for a Schrödinger system with power-type nonlinearities,, preprint., (). Google Scholar

[23]

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

[24]

A. C. Scott, Launching a davydov soliton: I. soliton analysis,, Phys. Scr., 29 (1984). doi: 10.1088/0031-8949/29/3/016. Google Scholar

[25]

B. K. Som, M. R. Gupta and B. Dasgupta, Coupled nonlinear Schrödinger equation for Langmuir and dispersive ion acoustic waves,, Phys. Lett. A, 72 (1979), 111. doi: 10.1016/0375-9601(79)90663-7. Google Scholar

[26]

J. Q. Sun, Z. Q. Ma and M. Z. Qin, Simulation of envelope Rossby solitons in a pair of cubic Schrödinger equations,, Appl. Math. Comput., 183 (2006), 946. doi: 10.1016/j.amc.2006.06.041. Google Scholar

[27]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, 106, AMS-CBMS, (2006). Google Scholar

[28]

C. Yeh and L. Bergman, Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,, Phys. Rev. E, 57 (1998). doi: 10.1103/PhysRevE.57.2398. Google Scholar

[1]

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

[2]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048

[3]

Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6085-6112. doi: 10.3934/dcds.2015.35.6085

[4]

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

[5]

Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431

[6]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241

[7]

Chunhua Li. Decay of solutions for a system of nonlinear Schrödinger equations in 2D. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4265-4285. doi: 10.3934/dcds.2012.32.4265

[8]

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

[9]

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

[10]

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

[11]

Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043

[12]

Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018

[13]

Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007

[14]

David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327

[15]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[16]

Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003

[17]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911

[18]

Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138

[19]

Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413

[20]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]