doi: 10.3934/dcdsb.2022101
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Periodic waves for the cubic-quintic nonlinear Schrodinger equation: Existence and orbital stability

1. 

Centro de Ciências Exatas Naturais e Tecnológicas, Universidade Estadual da Região Tocantina do Maranhão, Imperatriz, Maranhão, 65900-000, Brazil

2. 

Departamento de Matemática, Universidade Estadual de Maringá, Maringá, Paraná, 87020-900, Brazil

*Corresponding author: fmanatali@uem.br (F. Natali)

Received  December 2022 Early access June 2022

Fund Project: The second author is partially supported by CNPq grant 303907/2021-5, Fundação Araucária grant 002/2017 and CAPES MathAmSud grant 88881.520205/2020-01

In this paper, we prove existence and orbital stability results of periodic standing waves for the cubic-quintic nonlinear Schrödinger equation. We use the implicit function theorem to construct a smooth curve of explicit periodic waves with dnoidal profile and such construction can be used to prove that the associated period map is strictly increasing in terms of the energy levels. The monotonicity is also useful to obtain the behaviour of the non-positive spectrum for the associated linearized operator around the wave. Concerning the stability, we prove that the dnoidal waves are orbitally stable in the energy space restricted to the even functions.

Citation: Giovana Alves, Fábio Natali. Periodic waves for the cubic-quintic nonlinear Schrodinger equation: Existence and orbital stability. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022101
References:
[1]

J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Diff. Equat., 235 (2007), 1-30.  doi: 10.1016/j.jde.2007.01.003.

[2]

J. Angulo Pava and F. Natali, Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations, Phys. D, 238 (2009), 603-621.  doi: 10.1016/j.physd.2008.12.011.

[3]

J. Angulo Pava and F. Natali, Positivity properties of the Fourier transform and the stability of periodic travelling-wave solutions, SIAM J. Math. Anal., 40 (2008), 1123-1151.  doi: 10.1137/080718450.

[4]

I. V. Barashenkov, A. D. Gocheva, V. G. Makhankov and I. V. Puzynin, Stability of the soliton-like "bubbles", Physica D, 34 (1989), 240–254. doi: 10.1016/0167-2789(89)90237-6.

[5]

J. Bona, On the stability theory of solitary waves, Proc. R. Soc. Lond. Ser. A, 344 (1975), 363-374.  doi: 10.1098/rspa.1975.0106.

[6]

P. F. Byrd and M. D. Friedmann, Handbok of Elliptical Integrals for Enginners and Scientist, Springer, New York, 1971.

[7]

M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.

[8]

T. Gallay and M. Hǎrǎguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Diff. Equat., 234 (2007), 544-581.  doi: 10.1016/j.jde.2006.12.007.

[9]

T. Gallay and M. Hǎrǎguş, Orbital stability of periodic waves for the nonlinear Schrödinger equation, J. Dyn. Dif. Equat., 19 (2007), 825-865.  doi: 10.1007/s10884-007-9071-4.

[10]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.

[11]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[12]

S. Gustafson, S. Le Coz and T.-P. Tsai, Stability of periodic waves of 1D cubic nonlinear Schrödinger equations, Appl. Math. Res. Express, 2017 (2017), 431–487. doi: 10.1093/amrx/abx004.

[13]

S. HakkaevM. Stanislavova and A. Stefanov, On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS, J. Nonl. Sci., 31 (2021), 54.  doi: 10.1007/s00332-021-09712-6.

[14]

C. A. Hernández Melo, Estabilidade de Ondas Viajantes Para Equações de Schrödinger Do Tipo Cúbica-quíntica, Ph.D Thesis, State University of São Paulo, 2012.

[15]

R. Iório, Jr and V. M. Iório, Fourier Analysis and Partial Differential Equations, Cambridge, UK, 2001. doi: 10.1017/CBO9780511623745.

[16]

G. Loreno, G. E. B. Moraes, F. Natali and A. Pastor, Cnoidal waves for the cubic nonlinear Klein-Gordon and Schrödinger equations, preprint, 2021, arXiv: 2105.02299.

[17]

W. Magnus and S. Winkler, Hill's Equation, Wiley, New York, 1966.

[18]

F. Natali and A. Pastor, The fourth-order dispersive nonlinear Schrödinger equation: Orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1346.  doi: 10.1137/151004884.

[19]

F. Natali and A. Neves, Orbital stability of solitary waves, IMA J. Appl. Math., 79 (2014), 1161-1179.  doi: 10.1093/imamat/hxt018.

[20]

A. Neves, Floquet's theorem and stability of periodic solitary waves, J. Dyn. Diff. Equat., 21 (2009), 555-565.  doi: 10.1007/s10884-009-9143-8.

[21]

M. Ohta, Stability and Instability of standing waves for one dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74.  doi: 10.2996/kmj/1138043354.

[22]

M. I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, SIAM J. Math, 16 (1985), 472-490.  doi: 10.1137/0516034.

show all references

References:
[1]

J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Diff. Equat., 235 (2007), 1-30.  doi: 10.1016/j.jde.2007.01.003.

[2]

J. Angulo Pava and F. Natali, Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations, Phys. D, 238 (2009), 603-621.  doi: 10.1016/j.physd.2008.12.011.

[3]

J. Angulo Pava and F. Natali, Positivity properties of the Fourier transform and the stability of periodic travelling-wave solutions, SIAM J. Math. Anal., 40 (2008), 1123-1151.  doi: 10.1137/080718450.

[4]

I. V. Barashenkov, A. D. Gocheva, V. G. Makhankov and I. V. Puzynin, Stability of the soliton-like "bubbles", Physica D, 34 (1989), 240–254. doi: 10.1016/0167-2789(89)90237-6.

[5]

J. Bona, On the stability theory of solitary waves, Proc. R. Soc. Lond. Ser. A, 344 (1975), 363-374.  doi: 10.1098/rspa.1975.0106.

[6]

P. F. Byrd and M. D. Friedmann, Handbok of Elliptical Integrals for Enginners and Scientist, Springer, New York, 1971.

[7]

M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.

[8]

T. Gallay and M. Hǎrǎguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Diff. Equat., 234 (2007), 544-581.  doi: 10.1016/j.jde.2006.12.007.

[9]

T. Gallay and M. Hǎrǎguş, Orbital stability of periodic waves for the nonlinear Schrödinger equation, J. Dyn. Dif. Equat., 19 (2007), 825-865.  doi: 10.1007/s10884-007-9071-4.

[10]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.

[11]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[12]

S. Gustafson, S. Le Coz and T.-P. Tsai, Stability of periodic waves of 1D cubic nonlinear Schrödinger equations, Appl. Math. Res. Express, 2017 (2017), 431–487. doi: 10.1093/amrx/abx004.

[13]

S. HakkaevM. Stanislavova and A. Stefanov, On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS, J. Nonl. Sci., 31 (2021), 54.  doi: 10.1007/s00332-021-09712-6.

[14]

C. A. Hernández Melo, Estabilidade de Ondas Viajantes Para Equações de Schrödinger Do Tipo Cúbica-quíntica, Ph.D Thesis, State University of São Paulo, 2012.

[15]

R. Iório, Jr and V. M. Iório, Fourier Analysis and Partial Differential Equations, Cambridge, UK, 2001. doi: 10.1017/CBO9780511623745.

[16]

G. Loreno, G. E. B. Moraes, F. Natali and A. Pastor, Cnoidal waves for the cubic nonlinear Klein-Gordon and Schrödinger equations, preprint, 2021, arXiv: 2105.02299.

[17]

W. Magnus and S. Winkler, Hill's Equation, Wiley, New York, 1966.

[18]

F. Natali and A. Pastor, The fourth-order dispersive nonlinear Schrödinger equation: Orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1346.  doi: 10.1137/151004884.

[19]

F. Natali and A. Neves, Orbital stability of solitary waves, IMA J. Appl. Math., 79 (2014), 1161-1179.  doi: 10.1093/imamat/hxt018.

[20]

A. Neves, Floquet's theorem and stability of periodic solitary waves, J. Dyn. Diff. Equat., 21 (2009), 555-565.  doi: 10.1007/s10884-009-9143-8.

[21]

M. Ohta, Stability and Instability of standing waves for one dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74.  doi: 10.2996/kmj/1138043354.

[22]

M. I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, SIAM J. Math, 16 (1985), 472-490.  doi: 10.1137/0516034.

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