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Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation
School of Mathematics and Big Data, Foshan University, Foshan, Guangdong 528000, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[12] |
H. Hasimoto,
A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.
doi: 10.1017/S0022112072002307. |
[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. |
[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. |
[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. |
[16] |
J. Langer and R. Perline,
Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93.
doi: 10.1007/BF01209148. |
[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. |
[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. |
[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. |
[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. |
[21] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[12] |
H. Hasimoto,
A soliton on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.
doi: 10.1017/S0022112072002307. |
[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. |
[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. |
[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. |
[16] |
J. Langer and R. Perline,
Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93.
doi: 10.1007/BF01209148. |
[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. |
[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. |
[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. |
[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. |
[21] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York, 1978. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
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