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Generalized Homoclinic Solutions of a Coupled Schrödinger System Under a Small Perturbation

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Abstract

This paper is devoted to study a coupled Schrödinger system with a small perturbation

$$\begin{array}{ll}u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R} \end{array}$$

where β is a constant and ε 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.

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References

  1. Ambrosetti A., Colorado E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 342, 53–458 (2006)

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on R n. Progress in Math. 240, Birkhäuser (2005)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosetti A., Malchiodi A., Secchi S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Rat. Mech. Anal. 159, 253–271 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Belmonte-Beitia J.: A note on radial nonlinear Schrödinger systems with nonlinearity spatially modulated. Electron. J. Diff. Equ. 148, 1–6 (2008)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bernard P.: Homoclinic orbit to a center manifold. Calc. Var. Partial Differ. Equ. 17, 121–157 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  9. Champneys A.R.: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys. D 112, 158–186 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Champneys A.R.: Homoclinic orbits in reversible systems II: Multi-bumps and saddle-centres. CWI Quart 12, 185–212 (1999)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  14. Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1995)

    Google Scholar 

  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)

    Article  Google Scholar 

  16. Davydov A.S.: Solitons in Molecular Systems. Reidel, Dordrecht (1985)

    MATH  Google Scholar 

  17. Deng S., Sun S.: Existence of three-dimensional generalized solitary waves with gravity and small surface tension. Phys. D 238, 1735–1751 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kielhöfer H.: Bifurcation Theory: an Introduction with Applications to PDEs. Springer, (2003)

  21. Kivshar Y., Agrawal G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003)

    Google Scholar 

  22. Lerman L.M.: Hamiltonian systems with a separatrix loop of a saddle-center. Selecta. Math. Sov. 10, 297–306 (1991)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin T.C., Wei J.: Half-skyrmions and spike-vortex solutions of two-component nonlinear Schrödinger systems. J. Math. Phys. 48, 053518 (2007)

    Article  MathSciNet  Google Scholar 

  25. Maia L.A., Montefusco E., Pellacci B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Diff. Equ. 229, 743–767 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Noris B., Ramos M.: Existence and bounds of positive solutions for a nonlinear Schrödinger system. Proc. Am. Math. Soc. 138, 1681–1692 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  31. Pitaevskii L., Stringari S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  32. Ragazzo C.G.: Irregular dynamics and homoclinic orbits to Hamiltonian saddle-centers. Commun. Pure. Appl. Math. 50, 105–147 (1997)

    Article  MATH  Google Scholar 

  33. Shatah J., Zeng C.: Orbits homoclinic to center manifolds of conservative PDEs. Nonlinearity 16, 591–614 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sirakov B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in R n. Commun. Math. Phys. 271, 199–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  36. Sulem C., Sulem P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, Berlin (2000)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Walter W.: Gewöhnliche Differentialgleichungen. Springer-Verlag, New York/Berlin (1972)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang J.: Dynamics of embedded solitons in the extended KdV equations. Stud. Appl. Math. 106, 337–366 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Coti Zelati V., Macrì M.: Multibump solutions homoclinic to periodic orbits of large energy in a centre manifold. Nonlinearity 18, 2409–2445 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. Coti Zelati V., Macrì M.: Existence of homoclinic solutions to periodic orbits in a center manifold. J. Differ. Equ. 202, 158–182 (2004)

    Article  MATH  Google Scholar 

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Correspondence to Shengfu Deng.

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Deng, S., Guo, B. Generalized Homoclinic Solutions of a Coupled Schrödinger System Under a Small Perturbation. J Dyn Diff Equat 24, 761–776 (2012). https://doi.org/10.1007/s10884-012-9274-1

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  • DOI: https://doi.org/10.1007/s10884-012-9274-1

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