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Strong Instability of Standing Waves for a System NLS with Quadratic Interaction

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Abstract

We study the strong instability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction under the mass resonance condition in dimension d = 5.

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References

  1. Cazenave T. Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. Courant Institute of Mathematical Sciences: American Mathematical Society, 2003

    Google Scholar 

  2. Colin M, Colin T, Ohta M. Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction. Funkcial Ekvac, 2009, 52: 371–380

    Article  MathSciNet  Google Scholar 

  3. Colin M, Di Menza L, Saut J C. Solitons in quadratic media. Nonlinearity, 2016, 29: 1000–1035

    Article  MathSciNet  Google Scholar 

  4. Dinh V D. Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction. Nonlinear Anal, 2020, 190: 111589

    Article  MathSciNet  Google Scholar 

  5. Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J Math Phys, 1977, 18: 1794–1797

    Article  MathSciNet  Google Scholar 

  6. Hamano M. Global dynamics below the ground state for the quadratic Schrödinger system in 5D. Preprint arXiv:1805.12245, 2018

    Google Scholar 

  7. Hoshino G, Ozawa T. Analytic smoothing effect for a system of nonlinear Schrödinger equations. Differ Equ Appl, 2013, 5: 395–408

    MathSciNet  MATH  Google Scholar 

  8. Hoshino G, Ozawa T. Analytic smoothing effect for a system of Schrödinger equations with two wave interaction. Adv Differential Equations, 2015, 20: 697–716

    MathSciNet  MATH  Google Scholar 

  9. Hoshino G. Space-time analytic smoothing effect for global solutions to a system of nonlinear Schrödinger equations with large data. Ann Henri Poincaré, 2018, 19: 2101–2114

    Article  MathSciNet  Google Scholar 

  10. Hayashi N, Li C, Naumkin P I. On a system of nonlinear Schrödinger equation in 2D. Differential Integral Equations, 2011, 24: 417–434

    MathSciNet  MATH  Google Scholar 

  11. Hayashi N, Li C, Naumkin P I. Modified wave operator for a system of nonlinear Schrödinger equations in 2D. Commun Partial Differential Equations, 2012, 37: 947–968

    Article  Google Scholar 

  12. Hayashi N, Li C, Ozawa T. Small data scattering for a system of nonlinear Schrödinger equations. Differ Equ Appl, 2011, 3: 415–426

    MathSciNet  MATH  Google Scholar 

  13. Hayashi N, Ozawa T, Tanaka K. On a system of nonlinear Schrödinger equations with quadratic interaction. Ann Inst H Poincaré Anal Non Linéaire, 2013, 30: 661–690

    Article  MathSciNet  Google Scholar 

  14. Ogawa T, Uriya K. Final state problem for a quadratic nonlinear Schrödinger system in two space dimensions with mass resonance. J Differential Equations, 2015, 258: 483–583

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The author would like to express his deep gratitude to his wife - Uyen Cong for her encouragement and support.

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Authors and Affiliations

  1. Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655, Villeneuve d’Ascq, Cedex, France

    Van Duong Dinh

  2. Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

    Van Duong Dinh

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  1. Van Duong Dinh
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Correspondence to Van Duong Dinh.

Additional information

The author is supported by the Labex CEMPI (ANR-11-LABX-0007-01).

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Dinh, V.D. Strong Instability of Standing Waves for a System NLS with Quadratic Interaction. Acta Math Sci 40, 515–528 (2020). https://doi.org/10.1007/s10473-020-0214-6

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  • DOI: https://doi.org/10.1007/s10473-020-0214-6

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