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July  2017, 37(7): 4091-4108. doi: 10.3934/dcds.2017174

Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation

Zhong Wang ,

School of Mathematics and Big Data, Foshan University, Foshan, Guangdong 528000, China

* Corresponding author: Zhong Wang.

Received  December 2015 Revised  April 2016 Published  April 2017

Fund Project: This work is supported by the China National Natural Science Foundation under grant number 11571381.

In this article we investigate the nonlinear stability of Hasimoto solitons, in energy space, for a fourth order Schrödinger equation (4NLS) which arises in the context of the vortex filament. The proof relies on a suitable Lyapunov functional, at the $H^2$ level, which allows us to describe the dynamics of small perturbations. This stability result is also extended to Sobolev spaces $H^m$ for all $m∈\mathbb{Z}_+$ by employing the infinite conservation laws of 4NLS.

Keywords: Orbital stability, fourth order, Schrödinger equation, vortex filament.
Mathematics Subject Classification: Primary: 35Q35, 76W05; Secondary: 35B65.
Citation: Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
References:
[1]

J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Commun. Partial Differential Equations, 17 (1992), 1-22.  doi: 10.1080/03605309208820831.  Google Scholar

[2]

J. P. Albert and J. L. Bona, Total positivity and the stability of internal waves in stratified fluids of finite depth, IMA J. Appl. Math., 46 (1991), 1-19.  doi: 10.1093/imamat/46.1-2.1.  Google Scholar

[3]

J. P. Albert, J. L. Bona and D. Henry, Sufficient conditions for instability of solitary-wave solutions of model equation for long waves, Physica D, 24 (1987), 343-366.  doi: 10.1016/0167-2789(87)90084-4.  Google Scholar

[4]

T. B. Benjamin, The stability of solitary waves, Proc. R. Soc. London, Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.  Google Scholar

[5]

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

[6]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[7]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo., 22 (1906), 117-135 (in Italian).   Google Scholar

[8]

Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part Ⅰ. A higher-order asymptotic formula for the velocity, J. Fluid Mech., 417 (2000), 1-45.  doi: 10.1017/S0022112000008995.  Google Scholar

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L. Greenberg, An oscillation method for fourth order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues, SIAM J. Math. Anal., 22 (1991), 1021-1042.  doi: 10.1137/0522067.  Google Scholar

[10]

M. Grillakis, J. 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.  Google Scholar

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M. Grillakis, J. 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.  Google Scholar

[12]

H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.  doi: 10.1017/S0022112072002307.  Google Scholar

[13]

S. M. Hoseini and T. R. Marchant, Solitary wave interaction for a higher-order nonlinear Schrödinger equation, IMA J. Appl. Math., 72 (2007), 206-222.  doi: 10.1093/imamat/hxl034.  Google Scholar

[14]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differential Equations, 214 (2005), 1-35.  doi: 10.1016/j.jde.2004.09.005.  Google Scholar

[15]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510.  doi: 10.1080/03605300701629385.  Google Scholar

[16]

J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93.  doi: 10.1007/BF01209148.  Google Scholar

[17]

M. Maeda and J. Segata, Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament, Funkcial. Ekvac., 54 (2011), 1-14.  doi: 10.1619/fesi.54.1.  Google Scholar

[18]

J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math., 46 (1993), 867-901.  doi: 10.1002/cpa.3160460604.  Google Scholar

[19]

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), 1326-1347.  doi: 10.1137/151004884.  Google Scholar

[20]

A. Neves and O. Lopes, Orbital stability of double solitons for the Benjamin-Ono equation, Comm. Math. Phys., 262 (2006), 757-791.  doi: 10.1007/s00220-005-1484-5.  Google Scholar

[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978.  Google Scholar

[22]

R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dyn. Res., 18 (1996), 245-268.  doi: 10.1016/0169-5983(96)82495-6.  Google Scholar

[23]

J. Segata, Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 27 (2010), 1093-1105.  doi: 10.3934/dcds.2010.27.1093.  Google Scholar

[24]

J. Segata, Orbital stability of a two parameter family of solitary waves for a fourth order nonlinear Schrödinger type equation, J. Math. Phys., 54 (2013), 061503, 6 pp. doi: 10. 1063/1. 4811522.  Google Scholar

[25]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.  Google Scholar

[26]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

[27]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.   Google Scholar

show all references

References:
[1]

J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Commun. Partial Differential Equations, 17 (1992), 1-22.  doi: 10.1080/03605309208820831.  Google Scholar

[2]

J. P. Albert and J. L. Bona, Total positivity and the stability of internal waves in stratified fluids of finite depth, IMA J. Appl. Math., 46 (1991), 1-19.  doi: 10.1093/imamat/46.1-2.1.  Google Scholar

[3]

J. P. Albert, J. L. Bona and D. Henry, Sufficient conditions for instability of solitary-wave solutions of model equation for long waves, Physica D, 24 (1987), 343-366.  doi: 10.1016/0167-2789(87)90084-4.  Google Scholar

[4]

T. B. Benjamin, The stability of solitary waves, Proc. R. Soc. London, Ser. A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074.  Google Scholar

[5]

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

[6]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[7]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo., 22 (1906), 117-135 (in Italian).   Google Scholar

[8]

Y. Fukumoto and H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part Ⅰ. A higher-order asymptotic formula for the velocity, J. Fluid Mech., 417 (2000), 1-45.  doi: 10.1017/S0022112000008995.  Google Scholar

[9]

L. Greenberg, An oscillation method for fourth order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues, SIAM J. Math. Anal., 22 (1991), 1021-1042.  doi: 10.1137/0522067.  Google Scholar

[10]

M. Grillakis, J. 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.  Google Scholar

[11]

M. Grillakis, J. 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.  Google Scholar

[12]

H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.  doi: 10.1017/S0022112072002307.  Google Scholar

[13]

S. M. Hoseini and T. R. Marchant, Solitary wave interaction for a higher-order nonlinear Schrödinger equation, IMA J. Appl. Math., 72 (2007), 206-222.  doi: 10.1093/imamat/hxl034.  Google Scholar

[14]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differential Equations, 214 (2005), 1-35.  doi: 10.1016/j.jde.2004.09.005.  Google Scholar

[15]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510.  doi: 10.1080/03605300701629385.  Google Scholar

[16]

J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93.  doi: 10.1007/BF01209148.  Google Scholar

[17]

M. Maeda and J. Segata, Existence and stability of standing waves of fourth order nonlinear Schrödinger type equation related to vortex filament, Funkcial. Ekvac., 54 (2011), 1-14.  doi: 10.1619/fesi.54.1.  Google Scholar

[18]

J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math., 46 (1993), 867-901.  doi: 10.1002/cpa.3160460604.  Google Scholar

[19]

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), 1326-1347.  doi: 10.1137/151004884.  Google Scholar

[20]

A. Neves and O. Lopes, Orbital stability of double solitons for the Benjamin-Ono equation, Comm. Math. Phys., 262 (2006), 757-791.  doi: 10.1007/s00220-005-1484-5.  Google Scholar

[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978.  Google Scholar

[22]

R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dyn. Res., 18 (1996), 245-268.  doi: 10.1016/0169-5983(96)82495-6.  Google Scholar

[23]

J. Segata, Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 27 (2010), 1093-1105.  doi: 10.3934/dcds.2010.27.1093.  Google Scholar

[24]

J. Segata, Orbital stability of a two parameter family of solitary waves for a fourth order nonlinear Schrödinger type equation, J. Math. Phys., 54 (2013), 061503, 6 pp. doi: 10. 1063/1. 4811522.  Google Scholar

[25]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.  Google Scholar

[26]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

[27]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.   Google Scholar

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