EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHR¨|ODINGER SYSTEM

doi:10.1016/S0252-9602(14)60138-7

Abstract

Abstract:

We study the existence and stability of the standing waves of two coupled Schr¨odinger equations with potentials |x|bi (bi 2 R, i = 1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the Schr¨odinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

Key words: nonlinear Schr¨odinger system, constrained variational problem, standing waves, orbital stubility

CLC Number:

32J20

ZENG XiaoYu,ZHANG Yi Ming, ZHOU Huang Song. EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHR¨|ODINGER SYSTEM[J].Acta mathematica scientia,Series B, 2015, 35(1): 45-70.

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References

[1] Ambrosetti A, Colorado E. Standing waves of some coupled nonlinear Schr¨odinger equations. J London Math Soc, 2007, 75: 67–82

[2] Bartsch T, Wang Z Q. Note on ground state of nonlinear Schr¨odinger systems. J Partial Diff Eqs, 2007,75: 67–82

[3] Berestycki H, Lions P L. Nonlinear scalar fields equation I. Existence of a ground state. Arch Ration MechAnal, 1983, 82: 313–346

[4] Cazenave T. Semilinear Schr¨odinger Equations. Courant Lecture Notes in Mathematics 10. New York: Courant Institute of Mathematical Science/AMS, 2003

[5] Cazenave T, Lions P L. Orbital stability of standing waves for some nonlinear Schr¨odinger equations. Comm
Math Phys, 1982, 85: 549–561

[6] Chen Z J, Zou W M. Standing waves for coupled Schr¨odinger equations with decaying potentials. J Math
Phys, 2013, 54: 111505

[7] Cipolatti R, Zumpichiatti W. Orbitally stable standing waves for a system of coupled nonlinear Schr¨odinger
equations. Nonlinear Anal, 2000, 42: 445–461

[8] Del Pino M, Felmer P. Local mountain passes for semi-linear elliptic problems in unbounded domains. Calc
Var Partial Differ Equ, 1996, 4: 121–137

[9] Ding Y H, Li S J. Homoclinic orbits for first order Hamiltonian systems. J Math Anal Appl, 1995, 189:
585–601

[10] Esry B D, Greene C H, Burke Jr J P, Bohn J L. Hartree-Fock theory for double condensates. Phys Rev
Lett, 1997, 78: 3594–3597

[11] Fukuizumi R. Stability and instability of standing waves for the nonlinear Schr¨odinger equation with harmonic
potential. Discrete Contin Dyn Syst, 2001, 7: 525–544

[12] Grillakis M, Shatah J, Strauss W. Stability theory of solitary waves in the presence of symmetry, I. J Funct
Anal, 1982, 74: 160–197

[13] Lin T C, Wei J C. Grond state of N coupled nonlinear Schr¨odinger equations in Rn, n 3. Comm Math
Phys, 2005, 255: 629–653

[14] Lin T C, Wei J C. Spikes in two-component systems of nonlinear Schr¨odinger equations with trapping
potentials. J Differ Equ, 2006, 229: 538–569

[15] Lions P L. The concentration-compactness principle in the caclulus of variations. The locally compact case
I. Ann Inst H Poincar´e Anal Non Lin´eaire, 1984, 1: 109–145

[16] Lions P L. The concentration-compactness principle in the caclulus of variations. The locally compact case
II. Ann Inst H Poincar´e Anal Non Lin´eaire, 1984, 1: 223–283

[17] Liu Y, Wang X P, Wang K. Instability of standing waves of the Schr¨odinger equation with inhomogeneous
nonlinearity. Trans Amer Math Soc, 2005, 358: 2105–2122

[18] Maia L A, Montefusco E, Pellacci B. Positive solutions for a weakly coupled nonlinear Schr¨odinger system.
J Differ Equ, 2006, 229: 743–767

[19] Maia L A, Montefusco E, Pellacci B. Orbital stability property for coupled nonlinear Schr¨odinger equations.
Adv Nonlinear Stud, 2010, 10: 681–705

[20] Menyuk C R. Nonlinear pulse propagation in birefringent optical fibers. IEEE J Quantum Electron, 1987,
23: 174–176

[21] Montefusco E, Pellacci B, Squassina M. Semiclassical states for weakly coupled nonlinear Schr¨odinger
systems. J Eur Math Soc, 2008, 10: 47–71

[22] Nguyen N V, Tian R T, Wang Z Q. Stability of traveling-wave solutions for a Schr¨odinger system with
power-type nonlinearities. Preprint

[23] Nguyen N V, Wang Z Q. Orbital stability of solitary waves for a nonlinear Schr¨odinger system. Adv Differ
Equ, 2011, 16: 977–1000

[24] Nguyen N V, Wang Z Q. Existence and stability of a two-parameter family of solitary waves for a 2-coupled
nonlinear Schrodinger system. Preprint

[25] Ohta M. Stability of solitary waves for coupled nonlinear Schr¨odinger equations. Nonlinear Anal, 1996,
26(5): 933–939

[26] Ozawa T. Remarks on proofs of conservation laws for nonlinear Schr¨oinger equations. Calc Var Partial
Differ Equ, 2006, 25: 403–408

[27] Reed M, Simon B. Methods of ModernMathematical Physics. IV. Analysis of Operators. New York, London:
Academic Press, 1978

[28] Shatah J, Strauss W. Instability of bound states. Comm Math Phys, 1985, 100: 173–190
[29] Sirakov B. Least energy solitary waves for a system of nonlinear Schr¨odinger equations in Rn. Comm Math
Phys, 2007, 271: 199–221

[30] Song X F. Stability and instability of standing waves to a system of Schr¨odinger equations with combined
power-type nonlinearities. J Math Anal Appl, 2010, 366: 345–359

[31] Strauss W A. Existence of solitary waves in higher dimensions. Comm Math Phys, 1977, 55: 149–162

[32] Weinstein M I. Nonlinear Schr¨odinger equations and sharp interpolations estimates. Comm Math Phys,
1983, 87: 567–576