American Institute of Mathematical Sciences

Advanced
March  2017, 16(2): 393-416. doi: 10.3934/cpaa.2017020

Long-term stability for kdv solitons in weighted Hs spaces

Brian Pigott 1, and  Sarah Raynor 2,,

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

* Corresponding author: Sarah Raynor.

Received  December 2015 Revised  October 2016 Published  January 2017

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.

Keywords: Korteweg-de Vries equation, soliton, asymptotic stability, traveling waves, stability.
Mathematics Subject Classification: Primary: 35Q53, 35B35; Secondary: 37K40, 35B40, 37K45, 35Q51.
Citation: Brian Pigott, Sarah Raynor. Long-term stability for kdv solitons in weighted Hs spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 393-416. doi: 10.3934/cpaa.2017020
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[18]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349.   Google Scholar

[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.  Google Scholar

[20]

B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[18]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349.   Google Scholar

[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.  Google Scholar

[20]

B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[2]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[3]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[4]

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
[5]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[6]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[7]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[8]

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 & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[9]

Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625
[10]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[11]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[12]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[13]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[14]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[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 & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[16]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020028
[17]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[18]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442
[19]

Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389
[20]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (14)
  • HTML views (4)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences