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Generalized homoclinic solutions of a coupled Schrödinger system under a small perturbation. (English) Zbl 1257.35155

Summary: This paper is devoted to study a coupled Schrödinger system with a small perturbation \[ \begin{aligned} u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad \mathrm {in} \, \mathbf {R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad \mathrm {in} \, \mathbf {R} \end{aligned} \] where \(\beta \) is a constant and \(\epsilon \) is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schrödinger-KdV system.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Ambrosetti A., Colorado E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 342, 53–458 (2006) · Zbl 1094.35112
[2] Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on R n . Progress in Math. 240, Birkhäuser (2005) · Zbl 1115.35004
[3] Ambrosetti A., Malchiodi A., Ni W.M.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I. Commun. Math. Phys. 235, 427–466 (2003) · Zbl 1072.35019 · doi:10.1007/s00220-003-0811-y
[4] Ambrosetti A., Malchiodi A., Secchi S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Rat. Mech. Anal. 159, 253–271 (2001) · Zbl 1040.35107 · doi:10.1007/s002050100152
[5] Belmonte-Beitia J.: A note on radial nonlinear Schrödinger systems with nonlinearity spatially modulated. Electron. J. Diff. Equ. 148, 1–6 (2008) · Zbl 1172.34309
[6] Belmonte-Beitia J., Perez-Garcia V.M., Torres P.J.: Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients. J. Nonlinear Sci. 19, 437–451 (2009) · Zbl 1361.34043 · doi:10.1007/s00332-008-9037-7
[7] Bernard P.: Homoclinic orbit to a center manifold. Calc. Var. Partial Differ. Equ. 17, 121–157 (2003) · Zbl 1187.70039 · doi:10.1007/s00526-002-0162-0
[8] Brezzi F., Markowich P.A.: The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation. Math. Mod. Meth. Appl. Sci. 14, 35–61 (1991) · Zbl 0739.35080 · doi:10.1002/mma.1670140103
[9] Champneys A.R.: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys. D 112, 158–186 (1998) · Zbl 1194.37154 · doi:10.1016/S0167-2789(97)00209-1
[10] Champneys A.R.: Homoclinic orbits in reversible systems II: Multi-bumps and saddle-centres. CWI Quart 12, 185–212 (1999) · Zbl 0998.37011
[11] Champneys A.R., Harterich J.: Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems. Dyn. Stab. Syst. 15, 231–252 (2000) · Zbl 1003.37033 · doi:10.1080/026811100418701
[12] Champneys A.R., Malomed B.A., Yang J., Kaup D.J.: Embedded solitons: solitary waves in resonance with the linear spectrum. Proc. Royal Soc. Edinburgh Sect. D 152–153(153), 340–354 (2001) · Zbl 0976.35087
[13] Cingolani S., Nolasco M.: Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations. Rev. Mod. Phys. A 128, 1249–1260 (1998) · Zbl 0922.35158
[14] Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1995) · Zbl 0064.33002
[15] Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999) · doi:10.1103/RevModPhys.71.463
[16] Davydov A.S.: Solitons in Molecular Systems. Reidel, Dordrecht (1985) · Zbl 0597.35001
[17] Deng S., Sun S.: Existence of three-dimensional generalized solitary waves with gravity and small surface tension. Phys. D 238, 1735–1751 (2009) · Zbl 1181.37022 · doi:10.1016/j.physd.2009.05.012
[18] Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[19] Groves M.D., Mielke A.: A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc. R. Soc. Edinburgh Sect. 131, 83–136 (2001) · Zbl 0976.76012 · doi:10.1017/S0308210500000809
[20] Kielhöfer H.: Bifurcation Theory: an Introduction with Applications to PDEs. Springer, (2003) · Zbl 1032.35001
[21] Kivshar Y., Agrawal G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003)
[22] Lerman L.M.: Hamiltonian systems with a separatrix loop of a saddle-center. Selecta. Math. Sov. 10, 297–306 (1991) · Zbl 0743.58017
[23] Lin T.C., Wei J.: Solitary and self-similar soltuions of two-component system of nonlinear Schrödinger equations. Phys. D 220, 99–115 (2006) · Zbl 1105.35116 · doi:10.1016/j.physd.2006.07.009
[24] Lin T.C., Wei J.: Half-skyrmions and spike-vortex solutions of two-component nonlinear Schrödinger systems. J. Math. Phys. 48, 053518 (2007) · Zbl 1144.81377 · doi:10.1063/1.2722559
[25] Maia L.A., Montefusco E., Pellacci B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Diff. Equ. 229, 743–767 (2006) · Zbl 1104.35053 · doi:10.1016/j.jde.2006.07.002
[26] Mielke A., Holmes P., O’Reilly O.: Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center. J. Dynam. Differ. Equ. 4, 95–126 (1992) · Zbl 0749.58022 · doi:10.1007/BF01048157
[27] Noris B., Ramos M.: Existence and bounds of positive solutions for a nonlinear Schrödinger system. Proc. Am. Math. Soc. 138, 1681–1692 (2010) · Zbl 1189.35086 · doi:10.1090/S0002-9939-10-10231-7
[28] Noris B., Trvares H., Terracini S., Verzini G.: Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 6, 267–302 (2010) · Zbl 1189.35314
[29] Oh Y-G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials. Commun. Math. Phys. 131, 223–253 (1990) · Zbl 0753.35097 · doi:10.1007/BF02161413
[30] Peletier L.A., Rodrígues J.A.: Homoclinic orbits to a saddle-center in a fourth-order differential equation. J. Differ. Equ. 203, 185–215 (2004) · Zbl 1080.34030 · doi:10.1016/j.jde.2004.03.036
[31] Pitaevskii L., Stringari S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)
[32] Ragazzo C.G.: Irregular dynamics and homoclinic orbits to Hamiltonian saddle-centers. Commun. Pure. Appl. Math. 50, 105–147 (1997) · Zbl 0884.58043 · doi:10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G
[33] Shatah J., Zeng C.: Orbits homoclinic to center manifolds of conservative PDEs. Nonlinearity 16, 591–614 (2003) · Zbl 1026.35092 · doi:10.1088/0951-7715/16/2/314
[34] Sirakov B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in R n . Commun. Math. Phys. 271, 199–221 (2007) · Zbl 1147.35098 · doi:10.1007/s00220-006-0179-x
[35] Stwalley W.C.: Stability of spin-eqnarrayed hydrogen at low temperatures and high magnetic fields: new field-dependent scattering resonances and predissociations. Phys. Rev. Lett. 37, 1628–1631 (1976) · doi:10.1103/PhysRevLett.37.1628
[36] Sulem C., Sulem P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, Berlin (2000) · Zbl 0928.35157
[37] Vázquez L., Streit L., Pérez-García V.M. (eds.): Nonlinear Klein-Gordon and Schrödinger Systems: Theory and Applications. World Scientific, Singapur (1997)
[38] Wagenknecht T., Champneys A.R.: When gap solitons become embedded solitons: a generic unfolding. Phys. D 177, 50–70 (2003) · Zbl 1011.37037 · doi:10.1016/S0167-2789(02)00773-X
[39] Walter W.: Gewöhnliche Differentialgleichungen. Springer-Verlag, New York/Berlin (1972) · Zbl 0247.34001
[40] Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications. 24, Birkäuser Boston, MA (1996)
[41] Yagasaki K.: Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom. Hamiltonian systems with saddle-centres. Nonlinearity 18, 1331–1350 (2005) · Zbl 1125.37043 · doi:10.1088/0951-7715/18/3/020
[42] Yang J.: Dynamics of embedded solitons in the extended KdV equations. Stud. Appl. Math. 106, 337–366 (2001) · Zbl 1152.76344 · doi:10.1111/1467-9590.00169
[43] Coti Zelati V., Macrì M.: Homoclinic solutions to invariant tori in a center manifold. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19, 103–134 (2008) · Zbl 1223.35062 · doi:10.4171/RLM/511
[44] Coti Zelati V., Macrì M.: Multibump solutions homoclinic to periodic orbits of large energy in a centre manifold. Nonlinearity 18, 2409–2445 (2005) · Zbl 1125.37042 · doi:10.1088/0951-7715/18/6/001
[45] Coti Zelati V., Macrì M.: Existence of homoclinic solutions to periodic orbits in a center manifold. J. Differ. Equ. 202, 158–182 (2004) · Zbl 1059.37046 · doi:10.1016/j.jde.2004.03.030
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