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

Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

Santosh Bhattarai 1,

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.
Keywords: Nonlinear Schrödinger system, solitary waves, $L^2$ normalized solutions, ground states, stability., positive solutions.
Mathematics Subject Classification: 35Q55, 35B35, 35A1.
Citation: Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 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), 987-1029. doi: 10.1017/S030821050000278X.

[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), 1129-1164.

[3]

J. Albert, J. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Royal. Soc. of Edinburgh, Sect. A 453 (1997), 1233-1260. doi: 10.1098/rspa.1997.0068.

[4]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.

[5]

S. Bhattarai, Solitary waves and a stability analysis for an equation of short and long dispersive waves, Nonlinear Anal., 75 (2012), 6506-6519. doi: 10.1016/j.na.2012.07.026.

[6]

S. Bhattarai, Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities, Adv. Nonlinear Anal., 4 (2015), 73-90. doi: 10.1515/anona-2014-0058.

[7]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), 363-374. doi: 10.1098/rspa.1975.0106.

[8]

J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. Differential Eqns., 163 (2000), 429-474. doi: 10.1006/jdeq.1999.3737.

[9]

T. Cazenave, Semilinear Schrödinger Equations, 10, AMS-Courant Lect. Notes in Math., 2003.

[10]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.

[11]

S. Chakravarty, M. J. Ablowitz, J. R. Sauer and R. B. Jenkins, Multisoliton interactions and wavelength-division multiplexing, Opt. Lett., 20 (1995), 136-138. doi: 10.1364/OL.20.000136.

[12]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. doi: 10.1103/RevModPhys.71.463.

[13]

T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), p742. doi: 10.1103/PhysRevLett.81.742.

[14]

Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Reports, 520 (2012), 253-381. doi: 10.1016/j.physrep.2012.07.005.

[15]

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

[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), 109-145.

[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), 362-367. doi: 10.1109/50.70013.

[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), 011503. doi: 10.1063/1.4774149.

[19]

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

[20]

N. V. Nguyen and Z-Q. Wang, Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90 (2013), 1-26. doi: 10.1016/j.na.2013.05.027.

[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), 1005-1021. doi: 10.3934/dcds.2016.36.1005.

[22]

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

[23]

M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26 (1996), 933-939. doi: 10.1016/0362-546X(94)00340-8.

[24]

A. C. Scott, Launching a davydov soliton: I. soliton analysis, Phys. Scr., 29 (1984), p279. doi: 10.1088/0031-8949/29/3/016.

[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), 111-114. doi: 10.1016/0375-9601(79)90663-7.

[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), 946-952. doi: 10.1016/j.amc.2006.06.041.

[27]

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

[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), p2398. doi: 10.1103/PhysRevE.57.2398.

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), 987-1029. doi: 10.1017/S030821050000278X.

[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), 1129-1164.

[3]

J. Albert, J. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Royal. Soc. of Edinburgh, Sect. A 453 (1997), 1233-1260. doi: 10.1098/rspa.1997.0068.

[4]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.

[5]

S. Bhattarai, Solitary waves and a stability analysis for an equation of short and long dispersive waves, Nonlinear Anal., 75 (2012), 6506-6519. doi: 10.1016/j.na.2012.07.026.

[6]

S. Bhattarai, Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities, Adv. Nonlinear Anal., 4 (2015), 73-90. doi: 10.1515/anona-2014-0058.

[7]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A, 344 (1975), 363-374. doi: 10.1098/rspa.1975.0106.

[8]

J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. Differential Eqns., 163 (2000), 429-474. doi: 10.1006/jdeq.1999.3737.

[9]

T. Cazenave, Semilinear Schrödinger Equations, 10, AMS-Courant Lect. Notes in Math., 2003.

[10]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.

[11]

S. Chakravarty, M. J. Ablowitz, J. R. Sauer and R. B. Jenkins, Multisoliton interactions and wavelength-division multiplexing, Opt. Lett., 20 (1995), 136-138. doi: 10.1364/OL.20.000136.

[12]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. doi: 10.1103/RevModPhys.71.463.

[13]

T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), p742. doi: 10.1103/PhysRevLett.81.742.

[14]

Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Reports, 520 (2012), 253-381. doi: 10.1016/j.physrep.2012.07.005.

[15]

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

[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), 109-145.

[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), 362-367. doi: 10.1109/50.70013.

[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), 011503. doi: 10.1063/1.4774149.

[19]

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

[20]

N. V. Nguyen and Z-Q. Wang, Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90 (2013), 1-26. doi: 10.1016/j.na.2013.05.027.

[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), 1005-1021. doi: 10.3934/dcds.2016.36.1005.

[22]

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

[23]

M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26 (1996), 933-939. doi: 10.1016/0362-546X(94)00340-8.

[24]

A. C. Scott, Launching a davydov soliton: I. soliton analysis, Phys. Scr., 29 (1984), p279. doi: 10.1088/0031-8949/29/3/016.

[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), 111-114. doi: 10.1016/0375-9601(79)90663-7.

[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), 946-952. doi: 10.1016/j.amc.2006.06.041.

[27]

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

[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), p2398. doi: 10.1103/PhysRevE.57.2398.

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