-
Previous Article
Center conditions for generalized polynomial kukles systems
- CPAA Home
- This Issue
-
Next Article
Singular periodic solutions for the p-laplacian ina punctured domain
Long-term stability for kdv solitons in weighted Hs spaces
| 1. | Department of Mathematics Wofford College, 429 North Church Street, Spartanburg, SC 29303 |
| 2. | Department of Mathematics and Statistics, Wake Forest University, P.O. Box 7388, Winston Salem, NC 27109 |
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.
References:
| [1] |
T. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183.
doi: 10.1098/rspa.1972.0074. |
| [2] |
J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363-374.
doi: 10.1098/rspa.1975.0106. |
| [3] |
J. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. (London) Ser. A, 411 (1987), 395-412.
doi: 10.1098/rspa.1987.0073. |
| [4] |
T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 |
| [5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Fund. Anal., 211 (2004), 173-218. |
| [6] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrodinger equation below the energy norm, Commun. Pure. Appl. Anal., 2 (2003), 33-50. |
| [7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54. |
| [8] |
Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901. |
| [9] |
N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. |
| [10] |
Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219-254.
doi: 10.1007/s002050100138. |
| [11] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited, Nonlinearity, 18 (2005), 391-427.
doi: 10.1088/0951-7715/18/1/004. |
| [12] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391-427.
doi: 10.1007/s00208-007-0194-z. |
| [13] |
F. Merle and L. Vega, L2 stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753.
doi: 10.1155/S1073792803208060. |
| [14] |
T. Mizumachi, Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., 32 (2001), 1050-1080. |
| [15] |
T. Mizumachi and N. Tzvetkov, L2-stability of solitary waves for the KdV equation via Pego and Weinstein's method, preprint, arXiv: 1403.5321. |
| [16] |
L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776. |
| [17] |
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005. |
| [18] |
R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349. |
| [19] |
B. Pigott, Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations, Commun. Pure. Appl. Anal., 13 (2014), 389-418.
doi: 10.3934/cpaa.2014.13.389. |
| [20] |
B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). |
| [21] |
S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation, Commun. Pure. Appl. Anal., 2 (2003), 277-296.
doi: 10.3934/cpaa.2003.2.277. |
| [22] |
M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math, 39 (1986), 51-67.
doi: 10.1002/cpa.3160390103. |
show all references
References:
| [1] |
T. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183.
doi: 10.1098/rspa.1972.0074. |
| [2] |
J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363-374.
doi: 10.1098/rspa.1975.0106. |
| [3] |
J. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. (London) Ser. A, 411 (1987), 395-412.
doi: 10.1098/rspa.1987.0073. |
| [4] |
T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 |
| [5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Fund. Anal., 211 (2004), 173-218. |
| [6] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrodinger equation below the energy norm, Commun. Pure. Appl. Anal., 2 (2003), 33-50. |
| [7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54. |
| [8] |
Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901. |
| [9] |
N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. |
| [10] |
Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219-254.
doi: 10.1007/s002050100138. |
| [11] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited, Nonlinearity, 18 (2005), 391-427.
doi: 10.1088/0951-7715/18/1/004. |
| [12] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391-427.
doi: 10.1007/s00208-007-0194-z. |
| [13] |
F. Merle and L. Vega, L2 stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753.
doi: 10.1155/S1073792803208060. |
| [14] |
T. Mizumachi, Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., 32 (2001), 1050-1080. |
| [15] |
T. Mizumachi and N. Tzvetkov, L2-stability of solitary waves for the KdV equation via Pego and Weinstein's method, preprint, arXiv: 1403.5321. |
| [16] |
L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776. |
| [17] |
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005. |
| [18] |
R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349. |
| [19] |
B. Pigott, Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations, Commun. Pure. Appl. Anal., 13 (2014), 389-418.
doi: 10.3934/cpaa.2014.13.389. |
| [20] |
B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). |
| [21] |
S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation, Commun. Pure. Appl. Anal., 2 (2003), 277-296.
doi: 10.3934/cpaa.2003.2.277. |
| [22] |
M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math, 39 (1986), 51-67.
doi: 10.1002/cpa.3160390103. |
| [1] |
Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509 |
| [2] |
Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021039 |
| [3] |
John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149 |
| [4] |
Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control and Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45 |
| [5] |
M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22 |
| [6] |
Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 |
| [7] |
Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655 |
| [8] |
Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069 |
| [9] |
Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021052 |
| [10] |
Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61 |
| [11] |
Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625 |
| [12] |
Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations and Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028 |
| [13] |
Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061 |
| [14] |
Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761 |
| [15] |
Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control and Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024 |
| [16] |
Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097 |
| [17] |
Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure and Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046 |
| [18] |
Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857 |
| [19] |
Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312 |
| [20] |
Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]