# American Institute of Mathematical Sciences

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, 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.  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), 1129-1164.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. Bhattarai, Solitary waves and a stability analysis for an equation of short and long dispersive waves, Nonlinear Anal., 75 (2012), 6506-6519. 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), 73-90. 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), 363-374. 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), 429-474. doi: 10.1006/jdeq.1999.3737.  Google Scholar [9] T. Cazenave, Semilinear Schrödinger Equations, 10, AMS-Courant Lect. Notes in Math., 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), 549-561. 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), 136-138. 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), 463-512. doi: 10.1103/RevModPhys.71.463.  Google Scholar [13] T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), p742. doi: 10.1103/PhysRevLett.81.742.  Google Scholar [14] Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Reports, 520 (2012), 253-381. 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), 109-145.  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), 362-367. 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), 011503. 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), 977-1000.  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), 1-26. 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), 1005-1021. 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), 933-939. 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), p279. 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), 111-114. 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), 946-952. 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), p2398. doi: 10.1103/PhysRevE.57.2398.  Google Scholar

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##### 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.  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), 1129-1164.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. Bhattarai, Solitary waves and a stability analysis for an equation of short and long dispersive waves, Nonlinear Anal., 75 (2012), 6506-6519. 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), 73-90. 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), 363-374. 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), 429-474. doi: 10.1006/jdeq.1999.3737.  Google Scholar [9] T. Cazenave, Semilinear Schrödinger Equations, 10, AMS-Courant Lect. Notes in Math., 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), 549-561. 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), 136-138. 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), 463-512. doi: 10.1103/RevModPhys.71.463.  Google Scholar [13] T.-L. Ho, Spinor Bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), p742. doi: 10.1103/PhysRevLett.81.742.  Google Scholar [14] Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein condensates, Phys. Reports, 520 (2012), 253-381. 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), 109-145.  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), 362-367. 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), 011503. 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), 977-1000.  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), 1-26. 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), 1005-1021. 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), 933-939. 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), p279. 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), 111-114. 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), 946-952. 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), p2398. doi: 10.1103/PhysRevE.57.2398.  Google Scholar
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